s

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www.seasolve.com

PeakFit
®

PEAK SEPARATION AND ANALYSIS SOFTWARE

User’s Manual
For more information about SeaSolve Software Inc. products, please visit our web site at
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SeaSolve Software Inc.

235 Walnut St., Suite 7
Framingham, MA 01702
Telephone: 508-872-5100
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PeakFitä v4 Users Guide

Copyright © 2003 by SeaSolve Software Inc. All rights reserved.

Printed in the United States of America.

No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any
form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the
prior written permission of the publisher.

1 2 3 4 5 6 7 8 9 0 07 06 05 04 03

ISBN 81-88341-07-X
 Contents

Contents

PeakFit Concepts . . . . . . . . . . . . . . . . . . . . . . . . . 1
Hidden Peaks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1
Residuals Method For Finding Hidden Peaks . . . . . . . . . . . . . 1-2
Second Derivative Method For Finding Hidden Peaks . . . . . . . . . 1-3
Deconvolution Method For Finding Hidden Peaks . . . . . . . . . . . 1-4
Convolution, Deconvolution, and Convolution Models . . . . . . . . 1-5
Deconvolving a Spectral Instrument Response Function . . . . . . . 1-6
Deconvolving an Exponential Detector Response Function . . . . . . 1-7
Deconvolving Intrinsic Peak Skew . . . . . . . . . . . . . . . . . . . 1-8
Non-Parametric Digital Filtering and Enhancement. . . . . . . . . . 1-10
Data Smoothing and Filtering . . . . . . . . . . . . . . . . . . . . 1-11
Sectioning and Uniformly Spaced X-Values . . . . . . . . . . . . . 1-12
Baseline Processing . . . . . . . . . . . . . . . . . . . . . . . . . 1-13
AutoScan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-14
Levels of Peak Placement . . . . . . . . . . . . . . . . . . . . . . 1-14
Non-Linear Peak Fitting . . . . . . . . . . . . . . . . . . . . . . . 1-15

Quick Start Tour . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Loading PeakFit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1
Hidden Peak Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1
Importing SAMPLE.XLS . . . . . . . . . . . . . . . . . . . . . . . . 2-1
Automated Fitting Using Residuals . . . . . . . . . . . . . . . . . . 2-2
Automated Fitting Using the Second Derivative . . . . . . . . . . . . 2-6
Automated Fitting Using Deconvolution . . . . . . . . . . . . . . . . 2-9

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Contents

PeakFit Common Elements . . . . . . . . . . . . . . . . . . . . 3

PeakFit Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1
PeakFit Text List Windows . . . . . . . . . . . . . . . . . . . . . . 3-20
ASCII Text Editor . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-22
Function Insert Feature . . . . . . . . . . . . . . . . . . . . . . . . 3-24
Evaluation Procedure. . . . . . . . . . . . . . . . . . . . . . . . . 3-25

Data Import and Entry . . . . . . . . . . . . . . . . . . . . . . . 4

File Menu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-1
Edit Menu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-1
PeakFit Import Options . . . . . . . . . . . . . . . . . . . . . . . . 4-2
Supported File Formats . . . . . . . . . . . . . . . . . . . . . . . . 4-2
Importing File Data . . . . . . . . . . . . . . . . . . . . . . . . . . 4-3
Importing Worksheet Data Into PeakFit . . . . . . . . . . . . . . . . 4-4
Importing Data From SigmaPlot Files . . . . . . . . . . . . . . . . . 4-7
Importing AIA Chromatography Files . . . . . . . . . . . . . . . . . 4-7
Importing Data From dBase Files . . . . . . . . . . . . . . . . . . . 4-7
Importing Data From ASCII Files . . . . . . . . . . . . . . . . . . . 4-8
Single And Multi-Column ASCII Files. . . . . . . . . . . . . . . . . 4-10
Data Interchange Format (DIF) Files . . . . . . . . . . . . . . . . . 4-11
Importing Data From The Windows Clipboard . . . . . . . . . . . . 4-12
Saving X-Y Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-13
ASCII List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-14
Reset All Defaults . . . . . . . . . . . . . . . . . . . . . . . . . . 4-14
Previous Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-15
Drag And Drop Files . . . . . . . . . . . . . . . . . . . . . . . . . 4-15
The PeakFit Editor . . . . . . . . . . . . . . . . . . . . . . . . . . 4-16
Weighting Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-20
ASCII Editor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-21
Copying Data Table To Clipboard. . . . . . . . . . . . . . . . . . . 4-22
Tool Bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-23
Status Bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-23

iv
 Contents

Preparing Data . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Data Menu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-1
Prepare Menu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-2
Compare with Reference. . . . . . . . . . . . . . . . . . . . . . . . 5-3
New X-Y Titles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-4
Enter Calculation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-5
Apply to X-Y Table . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-7
Cancel Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-7
Zero Negative Data . . . . . . . . . . . . . . . . . . . . . . . . . . 5-8
Area Normalize . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-9
Cumulative Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-9
Non-Parametric Digital Filter . . . . . . . . . . . . . . . . . . . . . 5-10
Clear Inactive Points . . . . . . . . . . . . . . . . . . . . . . . . . 5-12
Clear X-Y Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-12
Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-13
Smooth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-16
Fourier Domain Editing . . . . . . . . . . . . . . . . . . . . . . . . 5-21
Deconvolve Gaussian IRF . . . . . . . . . . . . . . . . . . . . . . 5-24
Deconvolve Exponential IRF . . . . . . . . . . . . . . . . . . . . . 5-28
Import and Subtract Baseline . . . . . . . . . . . . . . . . . . . . 5-32
Inspect 2nd Derivative . . . . . . . . . . . . . . . . . . . . . . . . 5-33
Inspect 4th Derivative . . . . . . . . . . . . . . . . . . . . . . . . 5-35
Inspect Function(X) . . . . . . . . . . . . . . . . . . . . . . . . . 5-37

Automated Peak Fitting . . . . . . . . . . . . . . . . . . . . . . 6

AutoFit Baseline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-1
AutoFit Peaks Overview . . . . . . . . . . . . . . . . . . . . . . . . 6-7
AutoFit Peaks I Residuals . . . . . . . . . . . . . . . . . . . . . . 6-10
AutoFit Peaks II Second Derivative . . . . . . . . . . . . . . . . . . 6-20
AutoFit Peaks III Deconvolution . . . . . . . . . . . . . . . . . . . 6-31
Peak Fit Preferences . . . . . . . . . . . . . . . . . . . . . . . . . 6-42
Peak Adjust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-47
Common Estimates . . . . . . . . . . . . . . . . . . . . . . . . . 6-48

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Contents

Fast Peak Fit with Numerical Update . . . . . . . . . . . . . . . . . 6-49

Full Peak Fit with Graphical Update. . . . . . . . . . . . . . . . . . 6-52
Peak Fit Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-56
User-Defined Functions . . . . . . . . . . . . . . . . . . . . . . . 6-69
Last Session User Functions . . . . . . . . . . . . . . . . . . . . . 6-78
Robust Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-79

PeakFit Functions . . . . . . . . . . . . . . . . . . . . . . . . . 7
Gaussian (Amplitude) . . . . . . . . . . . . . . . . . . . . . . . . . 7-1
Gaussian (Area) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-1
Lorentzian (Amplitude) . . . . . . . . . . . . . . . . . . . . . . . . 7-4
Lorentzian (Area) . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-4
Voigt (Amplitude) . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-7
Voigt (Area) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-7
Voigt (Amplitude, Gaussian/Lorentzian Widths) . . . . . . . . . . . . 7-8
Voigt (Area, Gaussian/Lorentzian Widths) . . . . . . . . . . . . . . . 7-8
Pearson VII (Amplitude) . . . . . . . . . . . . . . . . . . . . . . . 7-10
Pearson VII (Area) . . . . . . . . . . . . . . . . . . . . . . . . . . 7-10
Gaussian-Lorentzian Sum (Amplitude) . . . . . . . . . . . . . . . . 7-11
Gaussian-Lorentzian Sum (Area). . . . . . . . . . . . . . . . . . . 7-11
Gaussian-Lorentzian Cross Product (Amplitude). . . . . . . . . . . 7-12
Constrained Gaussian (Amplitude) . . . . . . . . . . . . . . . . . . 7-12
Constrained Gaussian (Area) . . . . . . . . . . . . . . . . . . . . . 7-13
Gamma Ray Peak (Gaussian + Compton Edge) . . . . . . . . . . . 7-14
Compton Edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-16
Notes on Chromatography Functions . . . . . . . . . . . . . . . . 7-17
HVL (Haarhoff-Van der Linde) . . . . . . . . . . . . . . . . . . . . 7-21
Giddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-23
NLC (Non-Linear Chromatography) . . . . . . . . . . . . . . . . . 7-25
EMG (Exponentially Modified Gaussian) . . . . . . . . . . . . . . . 7-28
GMG (Half-Gaussian Modified Gaussian) . . . . . . . . . . . . . . 7-31
GEMG4 (4 Parameter EMG-GMG Hybrid) . . . . . . . . . . . . . . 7-34
GEMG5 (5 Parameter EMG-GMG Hybrid) . . . . . . . . . . . . . . 7-34
EMG+GMG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-35

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 Contents

Log Normal (Amplitude) . . . . . . . . . . . . . . . . . . . . . . . 7-37

Log Normal (Area) . . . . . . . . . . . . . . . . . . . . . . . . . . 7-37
Log Normal 4-Parameter (Amplitude) . . . . . . . . . . . . . . . . 7-38
Log Normal 4-Parameter (Area) . . . . . . . . . . . . . . . . . . . 7-38
Extreme Value (Amplitude) . . . . . . . . . . . . . . . . . . . . . . 7-39
Extreme Value (Area). . . . . . . . . . . . . . . . . . . . . . . . . 7-39
Extreme Value 4 Parameter Tailed (Amplitude) . . . . . . . . . . . 7-40
Extreme Value 4 Parameter Tailed (Area) . . . . . . . . . . . . . . 7-40
Extreme Value 4 Parameter Fronted (Amplitude). . . . . . . . . . . 7-41
Extreme Value 4 Parameter Fronted (Area). . . . . . . . . . . . . . 7-41
Logistic (Amplitude) . . . . . . . . . . . . . . . . . . . . . . . . . 7-42
Logistic (Area) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-42
Laplace or Double Exponential (Amplitude) . . . . . . . . . . . . . 7-43
Laplace or Double Exponential (Area) . . . . . . . . . . . . . . . . 7-43
Error (Amplitude). . . . . . . . . . . . . . . . . . . . . . . . . . . 7-44
Error (Area) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-44
Student t (Amplitude) . . . . . . . . . . . . . . . . . . . . . . . . 7-45
Student t (Area) . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-45
Gamma (Amplitude) . . . . . . . . . . . . . . . . . . . . . . . . . 7-46
Gamma (Area) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-46
Weibull (Amplitude) . . . . . . . . . . . . . . . . . . . . . . . . . 7-47
Weibull (Area) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-47
Beta (Amplitude) . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-48
Beta (Area) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-48
Inverted Gamma (Amplitude) . . . . . . . . . . . . . . . . . . . . 7-49
Inverted Gamma (Area) . . . . . . . . . . . . . . . . . . . . . . . 7-49
F-Variance (Amplitude). . . . . . . . . . . . . . . . . . . . . . . . 7-50
F-Variance (Area). . . . . . . . . . . . . . . . . . . . . . . . . . . 7-50
Chi-Squared (Amplitude) . . . . . . . . . . . . . . . . . . . . . . . 7-51
Chi-Squared (Area). . . . . . . . . . . . . . . . . . . . . . . . . . 7-51
Pearson IV (Amplitude) . . . . . . . . . . . . . . . . . . . . . . . 7-52
a3=1 Pearson IV (Area). . . . . . . . . . . . . . . . . . . . . . . . 7-53
a3=2 Pearson IV (Area). . . . . . . . . . . . . . . . . . . . . . . . 7-54

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Erfc Peak (Amplitude) . . . . . . . . . . . . . . . . . . . . . . . . 7-55

Pulse Peak (Amplitude) . . . . . . . . . . . . . . . . . . . . . . . 7-55
Logistic Dose Response Peak (Amplitude) . . . . . . . . . . . . . . 7-56
Asymmetric Logistic (Amplitude) . . . . . . . . . . . . . . . . . . 7-56
Logistic Power Peak (Amplitude). . . . . . . . . . . . . . . . . . . 7-57
Pulse Peak Modified with Power Term (Amplitude) . . . . . . . . . 7-57
Pulse Peak with Second Width Term (Amplitude) . . . . . . . . . . 7-58
Intermediate Peak . . . . . . . . . . . . . . . . . . . . . . . . . . 7-58
Symmetric Double Sigmoidal . . . . . . . . . . . . . . . . . . . . 7-59
Asymmetric Double Sigmoidal . . . . . . . . . . . . . . . . . . . . 7-59
Symmetric Double Gaussian Cumulative . . . . . . . . . . . . . . . 7-60
Asymmetric Double Gaussian Cumulative . . . . . . . . . . . . . . 7-60
Sigmoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-61
Gaussian Cumulative . . . . . . . . . . . . . . . . . . . . . . . . . 7-61
Lorentzian Cumulative . . . . . . . . . . . . . . . . . . . . . . . . 7-61
Logistic Dose Response . . . . . . . . . . . . . . . . . . . . . . . 7-61
Log-Normal Cumulative . . . . . . . . . . . . . . . . . . . . . . . 7-62
Extreme Value Cumulative . . . . . . . . . . . . . . . . . . . . . . 7-62
Pulse Cumulative . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-62

Appendix A: Functions . . . . . . . . . . . . . . . . . . . . . A-1

General Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . A-1
Conditional Expressions . . . . . . . . . . . . . . . . . . . . . . . . A-2
Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . A-2
Statistical Functions . . . . . . . . . . . . . . . . . . . . . . . . . . A-4
Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . A-5
Integral, Derivative, And Summation Functions . . . . . . . . . . . . A-7
XY Data Table Constants and Functions . . . . . . . . . . . . . . . . A-8
Built-in Peak Functions. . . . . . . . . . . . . . . . . . . . . . . . A-10
Built-in Transition and Supplementary Functions . . . . . . . . . . A-13

Appendix B: Statistics of Fit . . . . . . . . . . . . . . . . . . . B-1

viii
 PeakFit Concepts

1 PeakFit Concepts

PeakFit for Windows introduces automated peak fitting science to

spectroscopy and chromatography. To use PeakFit effectively from the start,
it is important to understand some key concepts.

Hidden Peaks
PeakFit defines a hidden peak as one which is not responsible for a local
maximum in the data stream. This does not mean that a hidden peak is not
discernible to your perception. The following example illustrates data
containing one local maximum peak and two hidden peaks. The hidden peak
on the left just barely misses producing this local maximum in the data
stream. The hidden peak on the right is far less apparent in the data stream,
positioned within the shoulder of the principal peak. A hidden peak does
not result in a sign change in the data’s first derivative.

Hidden Peaks 1-1

PeakFit Concepts

Residuals Method For Finding Hidden Peaks

A residual is simply the difference in y-value between a data point and the
sum of component peaks evaluated at the data point’s x-value. By placing
peaks in such a way that their total area equals the area of the data, hidden
peaks are revealed by residuals.

The upper graph’s data reflects five local maxima peaks and two hidden
peaks. When the five local maxima peaks are placed so as to conserve the
data area, the residuals in the lower graph clearly reveal the hidden peaks.

Finding hidden peaks through the information content of residuals is the

premise behind AutoFit Peaks I Residuals option in the AutoFit menu.

1-2 Residuals Method For Finding Hidden Peaks

 PeakFit Concepts

Second Derivative Method For Finding Hidden Peaks

A smoothed second derivative of the data will contain local minima at peak
positions. In many cases, the hidden peaks which evidence no local maxima
in the original data stream do appear as local minima in a smoothed second
derivative.

The same data with the five local maxima and two hidden peaks is shown in
the upper plot. A smoothed second derivative is shown in the lower plot.
Note that the two hidden peaks are easily detected within the second
derivative data stream.

Finding hidden peaks through minima in a smoothed second derivative is

the premise behind the AutoFit Peaks II Second Derivative option in the
AutoFit menu.

Second Derivative Method For Finding Hidden Peaks 1-3

PeakFit Concepts

Deconvolution Method For Finding Hidden Peaks

Deconvolution is a mathematical procedure which is often used to remove
the smearing or broadening of peaks arising because of the imperfections in
an instrument’s measuring system. When an instrument’s response function
is deconvolved from data, peaks are “sharpened”. Hidden peaks which
evidence no local maxima in the original data stream may do so once the
data has been deconvolved and filtered.

Again, this same data set with the five local maxima peaks and the two
hidden peaks is shown in the upper plot. The lower plot consists of a good
Gaussian deconvolution and Fourier-domain filtration of this data. Note
that the peaks are indeed sharpened, and that the two previously hidden
peaks now clearly evidence a local maximum.

Finding hidden peaks through maxima in a deconvolved and filtered data

stream is the premise behind the AutoFit Peaks III Deconvolution option
in the AutoFit menu.

1-4 Deconvolution Method For Finding Hidden Peaks

 PeakFit Concepts

Convolution, Deconvolution, and Convolution Models

Convolution When different processes broaden a peak according to two different and
independent spread functions, the resultant peak is said to consist of the
convolution of the two different functional forms. The mathematical form
of a convolution consists of an integral:
∞ ∞
f (x) =∫
−∞
f
 ( x − t ) f ( t )dt = ∫−∞ f  ( t ) f ( x − t )dt
For example, many spectral peaks are formed by a “natural” Lorentzian
broadening and an independent Gaussian instrumental broadening. The
convolution of a Gaussian and Lorentzian is the Voigt function, a model
considerably more complex than the individual components comprising the
convolution.

Convolution Models Most convolutions of two different peak functions lack a closed form
solution for the convolution integral. PeakFit’s set of built-in peak
functions contain three different convolution models having analytical
forms, the Voigt within the spectroscopy function set, and the EMG and
GMG within the chromatography functions.

The convolution models are unique in that the parameters of the fitted
functions directly describe the components within the convolution. This is
the only form of deconvolution that actually occurs within peak fitting. It is
a most attractive approach, since it accurately resolves both components in
the convolution product without introducing noise and can do so for
overlapping and hidden peaks.

Deconvolution Deconvolution is essentially the undoing or reversing of the convolution. It

is normally done with discrete data in the Fourier domain, and usually to
remove an instrument response function. The FFT form of deconvolution
involves a major introduction of noise that must be dealt with using some
effective form of frequency domain filtration.

Instrument Response When an imperfect measuring instrument contributes to peak broadening,

Functions the function describing this effect is known as an instrument response
function. The goal of most deconvolution is to remove the instrumental
smearing so as to produce the “true” signal, free of instrumental distortion.

Convolution,Deconvolution, and Convolution Models 1-5

PeakFit Concepts

Deconvolving a Spectral Instrument Response Function

Gaussian Instrument Most spectral instrument response functions tend to be Gaussian in form.
Response As such, most spectral peaks will consist of a convolution of their “natural”
Lorentzian forms and this Gaussian smearing. The independent Gaussian
and Lorentzian broadening can be resolved by fitting the Voigt model. You
can also remove the Gaussian smearing by first deconvolving the data in
the Fourier domain using a Gaussian response function. The Deconvolve
Gaussian IRF option, found in the Prepare menu, offers an automated
FFT Gaussian deconvolution procedure that can be used to remove
instrumental effects prior to fitting.

Convolution is Area When a response function has unit area, as is true of instrument response
Invariant functions, a convolution is area invariant. The observed peak will have the
same area as the “true” peak. It is simply greater in width and smaller in
amplitude. If your only aim is to resolve areas and emission frequencies, it is
not necessary to concern yourself with instrumental effects.

The above graph illustrates data consisting of three overlapping Voigt

functions. The deconvolution shown effectively removes most of the
Gaussian instrumental smearing. Note that the result of deconvolution is
not three separate Lorentzian component peaks. It is rather a single data
stream that has been sharpened in the sense that the component peaks

1-6 Deconvolving a Spectral Instrument Response Function

 PeakFit Concepts

comprising it are more evident and that greater amplitudes and narrower
widths are suggested for each of the peaks comprising the data.

Deconvolution and By removing the instrumental smearing, however, deconvolution may

Peak Separation reduce or eliminate the overlap of adjacent peaks. It is for this reason that
deconvolution is thought of as a peak separation procedure. Peaks are
separated only if the overlap is modest enough that the deconvolution
resolves the original data into a new data stream where no peaks overlap.
This is very different from resolving component peaks via non-linear fitting
procedures, the essential function of PeakFit.

Deconvolving an Exponential Detector Response Function

Exponential Material arriving at a chromatographic detector is not instantly sensed. In
Instrument Response many cases, detector responses have been well described with a first order
or exponential decay model. Unlike spectroscopy, where the instrument
response function smears signal to both higher and lower frequencies, a
chromatographic detector is directional in nature. It may record material as
arriving later in time than the “true” signal, but not earlier.

In the above graph, chromatographic peaks have been deconvolved using an

exponential detector response function. The “true” signal consists of near
Gaussian peaks with greater amplitudes and narrower widths than those

Deconvolving anExponential Detector Response Function 1-7

PeakFit Concepts

within the original data. The one-sided exponential deconvolution removes

the directional smearing which shifts the observed signal to greater times.

The Deconvolve Exponential IRF option, found in the Prepare menu,

offers an automated FFT one-sided exponential deconvolution procedure
that can be used to remove detector response effects prior to fitting. If you
can successfully remove most of the peak skew using this one-sided
exponential deconvolution, you may be able to fit simple Gaussians, and
further you may be able to assume that these Gaussians represent the true
signal from the chromatographic separation, deriving theoretical quantities
defining column performance and capacity.

Convolution Model: Since the EMG (exponentially-modified Gaussian) function is the

EMG convolution of a Gaussian and an exponential decay, by fitting the EMG
model to peaks, you are using non-linear fitting to actually perform this
deconvolution for you. The parameters of the EMG function directly yield
the deconvolved Gaussian. Whereas Fourier domain deconvolution adds a
great deal of noise to data and requires extensive filtration, no noise is
introduced by fitting a convolution model.

Deconvolving Intrinsic Peak Skew

A chromatographic column can be viewed as a “convolution engine” which
produces a Gaussian peak shape through multiple sequential mathematical
convolutions. In practice, however, symmetric Gaussians are seldom
observed in chromatography. Part of this is due to the extracolumn
detector response effects mentioned in the previous section. An equally
important factor is that a variety of non-idealities within intracolumn
dynamics also introduce an intrinsic skew to peak profiles. Some of these
non-idealities can be viewed as smearing the peaks with a directional
constraint owing to the directional flow or to natural sequences, such as
adsorption necessarily preceding desorption.

Convolution Model: The simplest form of convolution model producing an asymmetric shape is
GMG the GMG (half-Gaussian modified Gaussian). It is the result of convolving
an unconstrained or full Gaussian with a directionally-constrained or
half-Gaussian.

1-8 Deconvolving Intrinsic Peak Skew

 PeakFit Concepts

In the above graph, chromatographic peaks have been deconvolved with a

one-sided Gaussian. The effect is similar to that of deconvolving an
exponential detector response. The peaks shrink in width, increase in
amplitude, and shift to earlier times.

The Deconvolve Gaussian IRF option, found in the Prepare menu, offers
an automated FFT one-sided Gaussian deconvolution procedure that can be
used to remove both Tailed and Fronted asymmetry prior to fitting. If you
can successfully remove most of the peak skew using this one-sided
Gaussian deconvolution, you may be able to fit simple Gaussians.

By fitting the GMG model to peaks, you are using non-linear fitting to
perform this deconvolution directly. The parameters of the GMG function
directly yield the deconvolved primary Gaussian.

Deconvolving Intrinsic Peak Skew 1-9

PeakFit Concepts

Non-Parametric Digital Filtering and Enhancement

Locally-Weighted Non-parametric fitting methods model data based on some form of
Least-Squares sequential subset of points. Locally weighted least squares fitting procedures
are possibly the most powerful and versatile types of non-parametric
algorithms. For any x value, an estimated y is produced from a weighted
least squares fit comprising nearby points. Weights are assigned based upon
the proximity to x. The best known procedure of this type is the Loess
algorithm. PeakFit’s non-parametric digital filter offers a variation of the
Loess algorithm which uses a Gaussian rather than tricube weighting
function. PeakFit offers both linear and quadratic fitting.

Any N1→Any N2 This form of algorithm not only offers one of the best forms of digital
filtration (reduction in data count), but also a very stable form of data
enhancement (increase in data count). The output stream will always have
uniformly spaced x-values. You can produce any size output data stream,
smaller or larger. In the graph below, DNA gel data consisting of 212
points (larger circles), is enhanced to 1000 points (smaller squares):

The Digital Filter option is found in PeakFit’s Data menu. This option is
particularly important if you need to restore uniform X-spacing.

1-10 Non-Parametric Digital Filtering and Enhancement

 PeakFit Concepts

Data Smoothing and Filtering

PeakFit offers effective smoothing algorithms, many of which are used
internally in the various AutoFit procedures. You may elect to use these
algorithms to smooth data prior to placement and fitting. There are two
points to consider. The first is whether or not you wish to pre-smooth the
data at all.

• By pre-smoothing data, you may remove local perturbations within the

data that the peak fitting algorithm would otherwise “lock” onto during
the fitting. For example, small peaks not well characterized by the input
data can “wander” about somewhat in the course of fitting iterations.
Such peaks may become fixed in any noise region where the data has
some local maximum, however small such may be.
• On the other hand, pre-smoothing alters the original data, possibly
introducing undesired effects. For example, time-domain smoothing
procedures tend to slightly broaden peak widths.
It is not necessary to pre-smooth data in order for PeakFit’s AutoFit
algorithms to properly detect peaks. All AutoFit procedures work with a
smoothed copy of the data for the purpose of peak placement, parameter
estimation, and baseline detection.

If you choose to pre-smooth data, then next issue is which algorithm to use.

• The FFT smoothing algorithm removes all noise in higher frequencies

with little to no distortion of signal frequencies. PeakFit’s Fourier
Domain Editing allows you to graphically set this frequency threshold.
• The Gaussian Convolution smoothing is also a frequency domain
procedure that smoothes locally by convolving a narrow width Gaussian,
and globally by automatically filtering out higher frequency channels. It
often produces the highest power of PeakFit’s smoothing methods.
• The Savitzky-Golay procedure is an excellent time domain procedure
that fits a fourth order polynomial in a moving window consisting of a
sizable number of data points.
• The Loess procedure is generally less effective and more prone to feature
distortion, but has the advantage of not-requiring uniform X-values

PeakFit’s Smooth and Fourier Domain Editing options are in the Prepare
menu.

Data Smoothing and Filtering 1-11

PeakFit Concepts

Sectioning and Uniformly Spaced X-Values

When data is “sectioned”, you isolate a particular section or partition for
fitting. Points are either active or turned on, meaning they will be included
in fitting, or they are inactive or excluded and will not be processed.

Global Sectioning PeakFit offers two levels of sectioning. Global sectioning affects the current
state of the main data table. The Section option in the Prepare menu is
used for global sectioning. You would normally use this global form of
sectioning to disable all elements of data that you would never wish to see
fitted under any circumstances.

Local Sectioning Local sectioning exists within each of the AutoFit Peaks options. Here,
only a temporary copy of the data is altered. You may wish to locally
section the data in order to isolate a single peak for testing the fit of several
models before fitting the full data set. It is also useful for breaking up a data
set into several sections where each is clearly separated by the baseline.

Sectioning allows both individual points and bands or regions of data to be

turned on and off. It is thus a simple matter to produce an active set of
points that lacks uniform x-spacing. This is especially true when toggling off
one or more points representing some form of artifact which should indeed
be removed prior to fitting.

Uniform X-Spacing If the original data lacks uniform x-spacing, or if such is true after
sectioning, there is a very good chance you will run into trouble with certain
of PeakFit’s operations. In particular, all FFT-based procedures and all
smoothing procedures except Loess require constant x-spacing in the data.
PeakFit does not limit processing options based on detecting this lack of
uniform x-spacing since the noise introduced by this non-constant spacing
may be less than that removed by a smoothing procedure.

There are two recommended courses to take for data lacking uniform
x-spacing:

• Use the Loess smoothing option within the AutoFit Peaks I Residuals
option. This approach does not require uniform X-spacing.
• Use the non-parametric Digital Filter option in the Data menu to create
a uniformly-spaced data set.

1-12 Sectioning and Uniformly Spaced X-Values

 PeakFit Concepts

Baseline Processing
PeakFit offers three options for managing baselines:

• Subtract a baseline contained in an external file from the imported data.

This is the Import and Subtract Baseline option in the Prepare menu.
• Fit the points representing the baseline within the current data to
parametric or non-parametric models, and subtract this baseline from the
data prior to placement and fitting. This is the AutoFit and Subtract
Baseline option in the AutoFit menu.
• Place a parametric baseline along with the peaks, and fit peaks and
baseline together in a single step. This can be done in any of the three
AutoFit Peaks options.

Removing a Baseline It is often possible to very accurately fit a baseline independently by

Prior to Fitting selecting points which represent the baseline. The AutoFit and Subtract
Baseline option seeks to do this automatically, and you are free to select
those specific points you feel should be included in such a baseline fit. If
your baseline has an unusual shape, this option also offers an effective
non-parametric fit. The disadvantage of fitting and subtracting the baseline
prior to peak placement and fitting is that this step represents an altering of
the original data that introduces an additional level of uncertainty, one that
will not be reflected in the fit statistics.

Fitting Baseline and Fitting baseline and peaks together preserves the original purity of the data,
Peaks Together and when all models are appropriate, can result in the finest quality of fit and
the most accurate fit statistics. The three AutoFit Peaks options use an
autoscan procedure that fits a specified parametric baseline to a copy of the
data and then subtracts such in order to make peak placement as convenient
as possible. This baseline pre-fit is simply used to set the estimates for the
baseline model in the peak fitting.

The problem with fitting baselines along with peaks is that the fitting
algorithm sometimes minimizes the merit function of fit by inappropriately
adjusting a baseline to compensate for inadequacies within the fit of peaks.
This frequently occurs when fitting many peaks with a model incapable of
modeling the specific skew in the peaks.

Baseline Processing 1-13

PeakFit Concepts

AutoScan
PeakFit’s three AutoFit Peaks options offer a unique autoscan algorithm
that detects and places peaks using a temporary copy of the data table.
Peaks are placed “as if” the original data had been pre-smoothed and any
baseline pre-subtracted. This enables you to alter baseline functions and
change smoothing levels in real-time as peaks are scanned and placed. The
data that is actually fitted is in no way altered.

The smoothing or deconvolution present in the AutoFit Peaks options is

strictly for the purpose of creating a data stream used for the detection of
local extrema. The data that will be fitted is not modified in any way.

Similarly, the baseline fitting present in the AutoFit Peaks options is solely
done to establish the initial starting estimates for the baseline model. No
baseline is subtracted from the data that is fitted.

Levels of Peak Placement

PeakFit offers three levels for placing peaks prior to fitting. The first level is
the autoscan procedure. While quite specific to each AutoFit Peaks option,
this first level is completely automated.

The second level involves graphical adjustments where you use the mouse
to adjust the amplitude, center, width, or asymmetry of a given peak. You
need not have any specific understanding of the peak model’s parameters.
You simply move the peak anchors graphically. You can also place peaks
simply by clicking at the desired location.

The third level involves adjusting the numeric parameters of the individual
peaks. In previous versions of PeakFit, this was the principal form of peak
placement. It is now the option of last resort. You should adjust numeric
parameters only when there is no other alternative. This is necessary for a
handful of special functions, such as the Gamma Ray and Compton
models, and possibly in those cases where peaks exhibit an extreme
asymmetry.

1-14 AutoScan
 PeakFit Concepts

Non-Linear Peak Fitting

PeakFit uses an enhanced version of Levenburg-Marquardt non-linear
minimization algorithm for peak fitting. All non-linear fitting algorithms are
limited in the sense that this minimization must proceed iteratively from
some initial set of parameter estimates.

Local Minima The overriding limitation of non-linear minimizations is their susceptibility

to what are known as local minima. This means the multi-dimensional
minimization converges to a set of parameter estimates that represent a
“local” best least-squares solution. A better set of parameter estimates may
exist, but unless the algorithm starts reasonably close to such estimates, this
“global” or “true” least-squares solution is never reported.

Peaks and Local Peaks are especially prone to local minima. All that is needed is to have a
Minima placed or fitted peak shift far enough left or right of that peak’s presence
within the data so that the opposite sides of function and data overlap at
some point near the half-maximum of each.

Constraints PeakFit’s fitting engine offers complete constraint control over fitted peaks.
By setting an a1 constraint to 5%, for example, you can insure that the fitted
center value of every built-in peak varies no more than 5% from its initial
placed position. PeakFit’s fitting algorithm can work its way through even
large numbers of constraint violations to find the best fit free of all
constraints.

Constraints are a most important element of peak fitting. A peak that is

unconstrained in all parameters can assume negative amplitudes, shift to
undesired locations, or diminish to effectively a zero width. This sometimes
occurs on iterations where a major peak moves much closer to the data, but
a very small nearly inconsequential peak shifts in some unfortunate way. If
no constraints are set, the fitting algorithm will adopt the new solution
because there is a net improvement in the fit’s merit function. Lacking
constraints, there is no way to alert the fitting algorithm that something
quite undesirable has occurred. Since a peak that wanders off into some
unfortunate position does so because there is very little data definition
associated with it, seldom will such a peak wander back to where it belongs.

Constraints are expressed as percents. This simplifies matters quite

considerably on one hand, but it also means that you may need to refine
these. For example, if your x-range varies from 1000 to 1100, a 5% a1

Non-Linear Peak Fitting 1-15

PeakFit Concepts

constraint would limit the center of a peak to approximately ±50 which is

essentially the whole of the x-range. On the other hand, with an x-range
varying from 0 to 1000, this ±50 may be too restrictive.

Robust Fitting PeakFit’s fitting algorithm includes three robust minimizations. Robust
fitting is also known as maximum likelihood or m-estimate fitting. Here a
merit function far less susceptible to outliers and the dynamic range of the y
variable is used. In peak fitting, the advantage lay in the fact that small
peaks will have a greater weight in the overall fitting process than they do in
least-squares. It is less likely that an unconstrained minor peak will lose its
initial positioning when using a robust minimization. Of the three included
in PeakFit, the Lorentzian minimization is especially recommended.

Sparse Curvature With each iteration, the fitting algorithm must evaluate each peak function
Matrix Processing at each x value and also build a curvature matrix. This matrix will be of size
n x n where n is the number of fitted parameters. Each element consists of
the product of two partial derivatives summed across all x values. These
computations represent a major part of the computational time. Sparse
curvature matrix processing determines when such computations are not
going to have any influence on the fit, that is, those x values where
evaluating a specific function will not impact the value of the overall model
and where its partial derivatives will not impact this curvature matrix.
Eliminating such unnecessary computations can significantly speed fitting.
PeakFit offers both a bi-directional limits testing and a root-finding
procedure for sparse curvature matrix processing.

Extent While you can reduce fitting time by using the Import Digital Filter option
to reduce a data set’s size upon import, and the Digital Filter option to
reduce a data set once the data has been imported, you may also set an
“extent” preference in the actual fitting. An extent specifies a degree of
exclusion, such as “Every Other Point”. This may be convenient if you
discover an existing fit proceeding too slowly.

1-16 Non-Linear Peak Fitting

 Quick Start Tour

2 Quick Start Tour

The brief tour contained in this chapter highlights the key elements
necessary to get you up and running with PeakFit. This tour focuses
primarily on PeakFit’s automated fitting.

Loading PeakFit
Click on the PeakFit icon or shortcut to launch the program.

Hidden Peak Data

This tour uses a data set designed specifically to illustrate the fitting of
hidden peaks with noisy data. The data consists of fifteen peaks total, four
of which are “hidden” since they produce no local maximum in the data.
This synthetic data set consists of peaks placed at 0.5 intervals from x=1.0
to 8.5, with the exception of there being no peak at x=4.5. Five percent
random Gaussian noise has been added to simulate particularly noisy data.

Importing SAMPLE.XLS
All PeakFit sample data is in a single Excel worksheet, SAMPLE.XLS. Each
data set is on its own sheet or page in the overall worksheet file. We will
import the data set in the first sheet.

Use the File menu’s Import option, or click on the first toolbar icon.
Change the file type to Excel and select sample.xls.

Loading PeakFit 2-1

Quick Start Tour

First click on column A!A to select the X variable, and then on A!B to select
the Y variable. PeakFit uses an Excel-like nomenclature to specify the
different sheets or pages of the worksheet.

Click on OK to accept the selections. The next dialog offers an opportunity

to modify the basic titles. Simply click on OK to accept the titles imported
from the spreadsheet.

Automated Fitting Using Residuals

Select the AutoFit menu’s AutoFit Peaks I Residuals option, or click on
the toolbar button with the peak and Roman numeral I.

Because of the high amount of noise within this sample set, the default
smoothing level and amplitude rejection thresholds are insufficient. Initially
you will see 100 placed Gaussian peaks.

Click on the AI Expert button next to the smoothing level. PeakFit

contains an algorithm that seeks to determine an optimum smoothing level.
For this data set, you will see that the Savitzky-Golay smoothing window
was increased almost five-fold.

2-2 Automated Fitting Using Residuals

 Quick Start Tour

The upper plot of the upper graph contains three elements. These consist
of the raw data minus the current estimate for the baseline, a smoothed
copy of this data, and the sum curve consisting of all peaks currently placed
and active. The lower plot of the upper graph contains the individual
component peaks.

The upper plot of the lower graph consists of residuals, the y-difference
between the smoothed data and sum curve. The lower plot consists of the
raw data with points turned on or off to reflect PeakFit’s automated
baseline fitting.

Setting the Amplitude Note that two peaks were detected in the noise at the two extremes of the
Rejection Threshold data. To clear these out, change the Amp% (amplitude threshold) to 8%.
You can enter this value directly, use the spin buttons, or right click the
field or spin buttons and select 8% from the popup menu. Note that the
amplitude threshold is represented by a dotted line in the components and
residuals plots.

Adding Hidden At this point, you should be looking at the 11 local maxima peaks (the
Residuals Peaks status display indicates peak count). The residuals graph should show four
very clear and quite substantial hidden peaks, well above this 8% threshold.
Check the Add Residuals box to automatically add these four peaks to the
overall model. You should now have 15 peaks total:

Automated Fitting Using Residuals 2-3

Quick Start Tour

For this data set, we will fit the default linear baseline, Gaussian amplitude
peaks, and accept the constant width setting. When Vary Widths is turned
off, all peaks will be fitted to single width.

Visual Fitting For this data set we will visually fit the data. This means there is a graphical
update for each iteration where there is an improvement in the overall fit
merit function.

Click on the Full Peak Fit with Graphical Update button located near the
bottom of the control panel. You will see the fit progress graphically. After
9 iterations, the fit will be complete with an r2 goodness of fit of 0.99087.
Click on Review Fit to proceed to PeakFit’s Review.

PeakFit’s Review uses a desktop metaphor. You may have the main peak fit
graph, the residuals graph, the numeric summary, and a data summary all
simultaneously displayed if you wish. At this stage, we will explore a few of
the Review options.

Fitted points are colored by standard error by default. Points that are red in
color are outside 3 standard errors and those outside 2 standard errors are
yellow.

2-4 Automated Fitting Using Residuals

 Quick Start Tour

Click on the Function Labels button

near the far right of the graph’s
toolbar.

Select Area Parameter for the label

type. Even though we fitted a
Gaussian amplitude model, an
analytic area is computed using the
parameters from the fit. You should
now see this analytic area reported
directly above each peak.

Click on the Show Confidence/Prediction Intervals button at the far right

of the toolbar. You should see the default 95% prediction intervals forming
a confidence band about the fitted curve.

The Set Confidence/Prediction Intervals, % Confidence button offers

the means to set either or both types of intervals, and 90, 95, or 99%
confidence levels.

After observing the intervals, simply click once again on the Show
Confidence/Prediction Intervals button to toggle off the intervals.

In peak-fitting, the numeric analysis is very often the primary objective.

PeakFit offers a comprehensive numeric summary. Click on the Numeric
button.

Automated Fitting Using Residuals 2-5

Quick Start Tour

This peak analysis summary can be printed, copied to the clipboard, saved
to file, or edited within PeakFit. Note that the Options menu enables you
to turn on and off specific sections of this summary.

We will explore additional Review options at the conclusion of the next fit.
Click on Numeric or close the Numeric Summary window directly. Click
on OK to close the Review. If your own data has appreciable noise, click on
the green check mark (OK) button in the AutoFit option. This saves the
current settings. Otherwise click on the red X (Cancel) button.

Automated Fitting Using the Second Derivative

Select the AutoFit menu’s AutoFit Peaks II Second Derivative option, or
click on the main toolbar button with the peak and Roman numeral II.

Again, due to the high amount of noise within this sample set, the default
smoothing level and amplitude rejection thresholds are insufficient. Here
PeakFit’s limit of 100 peaks is depleted even earlier in the data. The
smoothed second derivative is in the upper plot of the lower graph.

Click on the AI Expert button next to the smoothing level. Note that AI
Expert smoothing level produces a smooth second derivative curve and a
smooth raw data stream in the uppermost plot.

Set the Amp% value to 10% to clear out the peaks appearing in the noise
close to the baseline.

A second derivative produces minima at peak locations. Very often, peaks

which evidence no local maxima in the original data, do show up as local
minima in a smoothed second derivative.

Here you will note that we directly have the 15 peaks we seek to fit. The
second derivative procedure directly found the 11 local maxima peaks as
well as the 4 hidden peaks.

The peaks should appear as follows:

2-6 Automated Fitting Using the Second Derivative

 Quick Start Tour

Numeric Fitting For this data set we will numerically fit the data. This is a very fast way to
complete a fit when you can be reasonably assured nothing will go wrong.
PeakFit offers optional constraints for the built-in functions which can go a
long way toward this assurance. Click the Fast Peak Fit with Numerical
Update button. The fit should take the same 9 iterations. Click on Review
Fit when the fit is completed.

If you would like to

create a printed
graph, click on the
Print button in the
graph’s toolbar.
Print preview is
automatic and font
scaling can be set
prior to printing.

Automated Fitting Using the Second Derivative 2-7

Quick Start Tour

In least-squares fitting, the standard errors and confidence limits for

parameters can only be judged valid when the assumption of normal errors
is verified. This means that the residuals profile must be Gaussian.

Click on Residuals to open a standard residuals plot.

Click on the Display Residuals Distribution button to see a histogram of

the residuals. You will note it is Gaussian in appearance.

Click on the Display Residuals in Stabilized Normal Probability Plot

button to see a definitive test for normality. The four lines on each side of
the data represent 90, 95, 99, and 99.9% critical limits. A 99% critical limit
means that in only 1 out of 100 data sets should even a single point fall
outside this limit. For this fit, all values are safely within the 90% critical
limits.

Close the Residuals window directly or click on the Residuals button. Click
on OK to close the Review. Again, if your own data will have noise of this
magnitude, click on the green check mark button in the AutoFit option to
preserve the current settings. Otherwise click on the red X button.

2-8 Automated Fitting Using the Second Derivative

 Quick Start Tour

Automated Fitting Using Deconvolution

Select the AutoFit menu’s AutoFit Peaks III Deconvolution option, or
click on the main toolbar button with the peak and Roman numeral III.

When data is deconvolved with a Gaussian response function, it is

“sharpened” in the sense that peaks become narrower and of greater
amplitude. Deconvolution often produces local maxima in what were
previously hidden peaks. The deconvolved data is in the upper plot of the
lower graph. Deconvolution is extremely sensitive to the width of the
response function (it must always be less than the smallest peak width) and
to the level of Fourier domain noise filtration.

Here PeakFit’s initial estimate for a response function width and the default
noise filtration produce the 15 desired peaks as well as a some small ones
within the baseline noise.

Click on the AI Expert button next to the filter level. Note that even this
small change produces a significant change in the deconvolved data.

Set the Amp% value to 8% to clear out the peaks appearing in the noise
close to the baseline.

Note that the 11 local maxima and 4 hidden peaks in the original data are
quite clearly 15 local maxima peaks in the deconvolved data stream.

Placement Levels Beyond automated placement, PeakFit offers a graphical or visual

placement capability as well as a numerical parameter adjustment.

Visual Adjustment At present, each peak has a single primary adjustment “anchor” which
defines its amplitude and center. A peak can thus be dragged to any location
desired simply by clicking and holding down the left mouse button while
dragging the peak to a new position. Left clicking a primary peak anchor
without motion toggles a peak on and off. At this point, experiment with
moving and toggling peaks.

A peak is added simply by left clicking, with no movement, where there is

currently no peak. Move the mouse to the gap in the peaks (to about
4.5,150) and left click the mouse. A new peak appears.

Zoom-in is simply a matter of left clicking (other than on a peak anchor)

with movement. The standard Windows type of zoom-in band is shown.

Automated Fitting Using Deconvolution 2-9

Quick Start Tour

Zoom-in on the peak just added. Afterwards, left click the mouse anywhere
except on a peak anchor. This restores default scaling. You can also use the
Reset Scaling toolbar button.

At this point, check the Vary Widths checkbox. Until now, we have fit a
single width for all peaks. By checking this box, PeakFit’s autoscan offers
one additional degree of peak refinement and additional anchors are now
shown on each peak. These are used to adjust peak widths. Use the
half-height anchors to adjust one or more peak widths.

Right clicking a primary anchor opens a function popup dialog. This dialog
can remain up when selecting different peaks. When it is up, peaks can be
selected by left or right clicking the primary peak anchors. This dialog is
used to delete a peak, to select a different function for this particular peak,
or to individually adjust parameters and their states. A locked parameter is
fixed at its current value and is not fitted. A shared parameter is shared with
all other parameters at this particular parameter position who also have a
shared state.

Left click the peak that was added. Change this peak to a Lorentz Amp
model. Adjust the peak graphically and note the update in parameter values.
Afterward, click on the Delete Peak icon to permanently remove this peak.

To restore the autoscan conditions, click on the Reset AutoScan button. It

is among the buttons at the top of the control panel.

Click on the Full Peak Fit with Graphical Update button located near the
bottom of the control panel. The fit will progress graphically. Click on
Review Fit once the fit has converged.

The Copy button is used

to copy the graph to
either clipboard or file
based metafiles or
bitmaps. You can also
copy all of the numeric
information used for the
graph to a large
spreadsheet block..

2-10 Automated Fitting Using Deconvolution

 Quick Start Tour

The Data option offers a point by point data summary with predicted
values, residuals, confidence limits, and prediction limits. To explore this
option, click on the Data button. When finished, either click again on this
button or close the Data window directly.

The Eval option offers an interactive evaluation of the fit model. In

addition to function evaluations, this option also offers first and second
derivatives, cumulative and region areas, and the means to generate tables
based upon the fitted model.

The Export option is used to

create ASCII, Excel, Lotus,
Quattro, or SigmaPlot files
containing the raw data as well
as the predicted values for the
overall model and each
component peak. You may
specify the actual range and
increments of the X values used
for generating the model and
component values.

If you wish to create an exported file, click on Export, select the file and
type, and click on OK and then enter the desired file name.

Click on OK to close the Review and then click on the green check mark
button in the AutoFit option to preserve the current settings. Exit PeakFit
by closing its main window or via the File menu’s Exit command.

This concludes a brief look at PeakFit’s automated fitting and Review

options. For a more in-depth understanding of these aspects of PeakFit as
well as for pre-fit options such as smoothing, digital filtering, Fourier
domain editing, automated baseline fitting, and response function
deconvolution, please refer to the reference sections which follow.

Automated Fitting Using Deconvolution 2-11

Quick Start Tour

Notes

2-12 Automated Fitting Using Deconvolution

 PeakFit Common Elements

3 PeakFit Common Elements

This chapter covers elements of PeakFit common to multiple options within

the program. These include:

• PeakFit Graphs
• PeakFit Text List Windows
• PeakFit ASCII Editor
• Function Insert Help
• Function Evaluation Procedure

PeakFit Graphs
All graphs within PeakFit use a common interface. Each graph contains its
own tool bar and status bar. All graph options are accessed from the tool
bar.

• Prints the current graph to an output device.

• Copies the current graph or its numeric information to the clipboard.

• Sets Process Points mode. This mode restricts the mouse to data point
positions.

• Sets Sectioning mode. This mode is used to toggle data regions and individual
points on and off.

• Sets Zoom mode. This button does not usually appear since zoom-in is
always available for most graphs. When it is not, this mode must be selected in
order to zoom-in with the mouse.

• Toggles XY hints on and off.

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• Toggles status bar on and off.

• Sets fixed size graph layout.

• Sets floating size graph layout.

• Sets maximized size graph layout.

• Splits Y and Y2 plots.

• Inverts Y and Y2 plots.

• Hides Y2 plot.

• Hides Y plot.

• Opens Graph Scaling dialog.

• Resets default scaling.

• Sets common Y,Y2 autoscaling.

• Sets log X axis.

• Sets log Y axis.

• Sets log Y2 axis.

• Includes inactive points in autoscaling.

• Sets 2D view (layout, split, states of titles, labels, and grids).

• Sets point formats.

• Sets custom titles and title and label font sizes.

• Font selection for current graph. Graphs are limited to a single typeface.

• Select color scheme and customize colors.

• Select peak labels.

• Toggles peak anchors on and off. Peak anchors are used for on-screen
graphical adjustment of peaks in the AutoFit Peaks options.

• Toggles amplitude thresholds on and off. These thresholds are used for
accepting and rejecting peaks in the AutoFit Peaks options.

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• Opens Confidence Intervals dialog.

• Toggle confidence/prediction intervals.

• Display actual residuals.

• Display residuals as % of Y.

• Display residuals as fraction of SE (standardized).

• Display residuals distribution (histogram).

• Display delta SNP (stabilized normal probability) plot.

Most PeakFit graphs allow a standard zoom-in using the left mouse button
and a restoration of default scaling by right clicking the mouse anywhere
within the graph region except on a data point or peak anchor.

Printing Graphs
PeakFit graphs offer a full print preview Print option. You can use this
option to print a half-page, full-page, or custom size graph. The dialog will
contain a preview image of the graph, rendering the image to be printed as
accurately as possible.

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This preview graph will be updated as orientations are changed, options are
selected, margins are set, and font scalings changed.

Color or Black and You may print the graph in color by checking the Use Color item. Note that
White black and white graphs can be printed regardless of the current color
scheme. PeakFit internally uses the Page White BW color scheme for
monochrome printing.

Bounding Frame You may elect to print a bounding frame around the graph by checking the
Frame Graph option.

Margins If you wish to print a custom portrait or landscape graph, you must specify
the four margins which define the boundaries of the graph.

Font Scaling In some instances, PeakFit may use a larger screen font to maintain legibility
in graphs. If this Print Preview indicates the need to increase or decrease the
overall font scaling, you may enter a different value in the Font % field. You
may also use the spin buttons to change this value or you may select the
value from a popup menu activated by holding the right mouse button down
when over the field or spin buttons.

Save Custom Settings You may Save a custom print file containing the current information. This
custom print graph file will have a default [CPG] extension.

Read Custom Settings Use the Read option to import a custom print settings from a custom print
[CPG] file.

Printer Setup To select a given printer, use the Setup button. When you select a printer,
you will also have the option to modify the printer driver’s own dialog, as in
the case of setting printer resolution, paper size, and printing multiple
copies.

Initiating the Print Simply press OK to begin the printing, or Cancel to abandon printing. Once
printing begins, there will be a brief time where a print job can be canceled
during the Windows spooling operation. After that time, a print job must be
canceled from Windows Print Manager.

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Driver Problems For some printer and plotter drivers, particularly those from HP, you may
need to set the orientation for the plot within the printer setup dialog.

You may also find that the HP Plotter driver does not support rotation for
True Type and Adobe Type Manager fonts. You may need to use the
Modern, Roman, or MS San Serif font in order to have the proper rotation
for the the Y-axis title.

PeakFit Graphs use a special procedure for producing enhanced grid lines
which look very good with digital devices. For plotter output, you will
probably need to set a Grid Level of 0 ( a pure line) in order for grids to
appear. This zero grid level may also be necessary with certain versions of
the Deskjet driver which only partially render grid lines or which do so with
incorrect colors.

Graph Copy
The Copy Graph to Clipboard option in PeakFit graphs offers a variety of
methods for copying the current graph or its numeric contents to the
clipboard or to file.

Clipboard destination options are as follows:

• Windows Bitmap to Clipboard

• Windows Metafile to Clipboard
• Windows Enhanced Metafile to Clipboard
• All Numeric Info as Spreadsheet Block to Clipboard

File destination options include:

• Windows Device Independent Bitmap (DIB) To File

• Windows Metafile to File in Aldus Placeable Format
• Windows Enhanced Metafile to File

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Controls consist of:

• Use Color
• Bounding Frame
• Preserve Superscripts, Subscripts, and Symbols in Metafiles

The All Numeric Info as Spreadsheet Block to Clipboard is the only

option that does not export the graph image. This option copies all of the
numeric information used to create the graph to the clipboard in a
spreadsheet format. It is easily pasted into Excel, Lotus 123, Quattro Pro
Windows, SigmaPlot and many other Windows applications. Note that for
large graphs with many component peaks, this may be a very large data
block with hundreds of rows and dozens of columns.

All other options export either a bitmap or a metafile. Most Windows word
processing, spreadsheet, and graphics software can paste in Windows
Bitmaps and Windows Metafiles copied to the clipboard by PeakFit.

Bitmaps A Windows bitmap is a raster image consisting of pixel-type information. A

bitmap will accurately retain the positioning of all elements within the graph
and its titles. The main drawback to using a bitmap image is the significant
loss of resolution that occurs when the program importing the graph
stretches, shrinks, or otherwise scales the image. This is especially true for an
aspect ratio that differs significantly from the original image.

You will almost always be successful pasting a clipboard bitmap into an

application. Although the Device Independent Bitmap name suggests a
similar success in importing these DIB files, Windows applications are not
consistent in their handling of DIBs. You may find wide variations in DIB
handling even amongst different software applications from a single
company. PeakFit always writes the common 8-bit 256 color DIB, even if
you are using a different video resolution.

Metafiles Windows metafiles consist of a series of Windows graphics vector

instructions which are intended to reproduce the original image. Windows
Metafiles offer superior resolution and scaling, although the program
importing the image may have difficulties in properly rendering text
elements such as superscripts, subscripts, and symbols in their proper
positions.

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Metafiles are highly sensitive to the individual applications importing them.

If you are unsuccessful in pasting a clipboard metafile image into your
software, first check the Windows Clipboard Viewer to insure the image is
actually there. If so, and if your software supports WMF file import, you
should try importing a disk-based metafile. The Windows Metafile to File
in Aldus Placeable Format option produces a disk-based Windows Metafile
that is “placeable” and which is based upon the Aldus specification. This is
the Windows Metafile format preferred and sometimes required by high-end
desktop publishing and drawing software. The default file extension for
Metafiles is [.WMF].

You may also wish to try the 32-bit Enhanced Metafile formats if your
importing application is a 32-bit Win32 program which supports this new
format.

Windows has traditionally offered no intrinsic support for subscripts,

superscripts, or changing to a symbol font in the middle of a text string. As
such, a title containing formatted text has required multiple text output
instructions. When a metafile image is placed and scaled, the software
importing the file may not preserve the proper character spacing across the
separate text segments. If this occurs, try initially placing the image in the
aspect ratio of your video display. If this fails, you should uncheck the
Preserve Superscripts, Subscripts, and Symbols in Metafiles box.

Color and Bounding To copy the graph as a Page-White image, simply uncheck the Use Color
Frame item.

When the Bounding Frame box is checked, there will be a thin-box at the
boundaries of the image.

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Graph 2D View
The 2D View Options dialog for a PeakFit graph itself incorporates a copy
of the current graph.

Changes made in any of the view options are immediately reflected in the
graph. This option is used to control a graph’s layout, the state of its status
bar and XY hints, how the Y and Y2 axes are to be arranged, the state of all
titles and labels, and the state and size of grids.

Primary Layout The Fixed Frame option uses a fixed graph area that never varies relative to
its position within the overall frame. This option would normally only be
used for final output. Sufficient room exists to handle the font sizes of most
titles. This option is directly available in the tool bar of most graphs.

The Floating Frame option seeks to create the largest possible graph area
while accommodating all titles and labels. This layout seeks to preserve
aesthetics, and yet at the same time, provide a large canvas for the graph
information. The actual graph frame floats in size, depending on the space
requirement of the various elements outside the graph frame. This option is
also directly available in the tool bar of most graphs.

The Maximum Area option discards all external titles and labeling, using
only small internal X and Y labels. This option offers the greatest possible

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graph area. It was designed to maximize the amount of peak information. It

is the default for non-output graphs in PeakFit. This option is likewise
available in the tool bar of most graphs.

The Border option controls the state of a graph’s outermost bounding

frame. When copying graphs to the clipboard or saving graphs as image files
for importing into other applications, you may not wish to have an external
border. This option is disabled on maximized area graphs since they do not
have an external frame.

The Status option controls whether or not the graph’s status bar should be
displayed.

The XY Hints option controls whether or not the X,Y data hints appear
below the cursor when moving it across graphs.

Y - Y2 Layout These options will be available if a given graph has both a Y and a Y2 plot,
and it is possible to separate the two plots.

The Split item controls whether the Y2 graph is to appear separately, above
the Y graph, or whether the Y and Y2 plots are to be plotted within the
same area.

The Invert item enables the Y2 graph to appear on the bottom and the Y
graph to appear on top. This option is only available when the Y and Y2
plots are split.

The 50:50, 67:33, and 75:25 options specify the relative areas of split plots.

Titles You may set the state for any title in a PeakFit Graph. It is usually simpler to
turn a title off than to use the Custom Titles option to clear it. When All is
clicked on, all titles are toggled on. When it is clicked off, all titles are turned
off. Titles 1-5 are those above the graph. The Y and Y2 titles on both the
left and right sides of the graph can be independently controlled.

Labels You may also set the state for any axis label in a PeakFit Graph. Axis labels
are the numeric values plotted along the X and Y axes. You may also set the
label precision (Prec) from a low of 3 significant digits to a high of 8 digits.
The font size of the axes labels is set in the Custom Titles dialog.

Grids The state of the grids used in the graph are similarly set. You may also set a
grid Level from 0 to 10. A grid level of zero draws a simple line. For all

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other levels, the grid is drawn as a series of dots. This enhances the
appearance of the grid in printed output. A level of 1 skips every other pixel.
An level of 10 skips 10 pixels for each one drawn. As such, the grid intensity
decreases as the level value increases.

When printing, you may encounter a problem with some printers with the
dot-type grids. In this instance, you will need to set a level of zero and use
simple line grids.

The intermediate grids with log scaling are automatic. If a logarithmic axis
has a large number of decades, however, the intermediate log grids will not
be drawn.

Saving a 2D View The Save item is used to save the current 2D View to a disk file. These are
binary files with [V2D] extensions. Since each graph in PeakFit contains its
own settings, you may wish to save a 2D View to disk if you wish to use it
for a number of the program’s graphs.

Importing a 2D View The Read item is used to update the current graph with the settings in a
previously saved [V2D] custom view file.

Reset The Reset button restores the current 2D View to the state existing when
the dialog was opened.

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Graph Scaling
The Scaling dialog for a PeakFit graph itself incorporates a copy of the
current graph. All changes made in any of the scaling options are
immediately reflected in the graph.

This custom scaling option is used for special scaling needs. For zooming in
any given graph, it is much easier to use the left mouse button, starting at
one corner of the desired region and pressing and holding the button while
moving the mouse to the opposite corner of the desired region. Default
scaling can then be restored simply by right clicking the mouse anywhere
within the graph region, except upon a point or function anchor.

The X, Y, and Y2 (if present) axes are treated independently. The Y entry
fields serve either the Y or Y2 axis, depending on whether Y or Y2 is
selected. The Y2 options will not be available if the graph contains a single
plot.

The Automatic option produces PeakFit’s default scaling for the particular
axis. To manually scale an axis, this Automatic box must be unchecked.

The Log option will produce a logarithmic axis. No errors are reported for
any data or functions which have negative values. The points or function

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simply will not be drawn. Some graphs, such as Residuals Graphs, use an
absolute value when a log scale is used.

The Reverse option creates a descending rather than ascending axis. This
may be of value in spectral plots where the X axis is Wave Number.

The Common Y,Y2 option specifies a common axis be used for both Y and
Y2 plots. The ranges are simply adjusted to accommodate all information in
both plots.

The Priority option allows you to have Y axes scaled for either Data or Fn
(Functions). With a Data priority, the minimum and maximum in the
primary data determine the scaling. With a Fn priority, the minimum and
maximum of all functions plotted determine the scaling limits. If a user
function is wholly out of control (nowhere to be seen amongst the data),
changing from a Data to Fn priority may be of value.

The Zoom option enables an axis to be automatically zoomed out in ten

increments. These steps will be linear for linear scaling. When the scale is
logarithmic, alternating log units are incrementally added to each side of the
scale.

The Min, Max, and Divs fields are active when manual scaling is used
(Automatic is off). The values you enter are used exactly as the limits of the
scale. The divisions field specifies the number of divisions, not the number
of interior grid positions (the interior grid count will be one less than the
number of divisions). For logarithmic scaling, you are free to set non-integer
powers of 10, and subsidiary logarithmic grids will be drawn, although they
will lack the traditional meaning.

Grids are specified in the Graph 2D View option.

The Reset option restores default scaling. Default Scaling can also be reset
within a PeakFit Graph by clicking the Reset Default Scaling button, or right
clicking the mouse anywhere in the graph area except on a point or function
anchor.

The Save option will save a binary custom scaling [SCL] file to disk for
future recall and use. You should save any custom scaling that you may wish
to use for a number of replicate or similar data sets.

The Read option imports a binary custom scaling [SCL] file previously
saved.

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To accept the custom scaling shown in the dialog and graph, simply click on
OK. To abandon any modifications, click on Cancel.

Graph Point Format

The Data Points dialog for a PeakFit Graph itself incorporates a copy of the
current graph. All changes made in any of the point format options are
immediately reflected in the graph.

The entry fields serve either the Y or Y2 axis, depending on whether Y Axis
or Y2 Axis is selected. The Y2 options will not be available if the graph
contains a single plot. This option sets only the principal data points of each
Y axis. Reference data sets, if present, use a preset format.

Size A graph’s data point size may be set from zero to ten. Size 0 points are
always pixels (non-bar shapes) or lines (bar shapes), regardless of the device.
If the graph specifies a bar shape, this option affects the width of the bar.

Shape A graph’s points may be set as square, circle, diamond, triangle down,
triangle up, or plus symbols. You may also choose a bar graph where the
bar’s base is zero, or the Ymin or Ymax of the plot.

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Fill You may choose to leave the points unfilled or fill them with one of the four
point colors within PeakFit. For Review Graphs, you may also choose Color
by Residuals. Here the points are colored based upon the absolute value of
the number of standard errors from the Y-predicted value. The default color
schemes employ a coloring based upon ascending wavelength. Points less
than 1 standard error are shown in blue, those between 1 and 2 standard
errors are green, those between 2 and 3 standard errors are yellow, and those
beyond three standard errors are red. Points that are beyond 2 standard
errors may represent outliers which may be adversely impacting the overall
peak fit.

Options • You may elect to have inactive points filled or unfilled.

• Points may be hidden entirely.
• A graph’s points may be connected with lines.

Graph Custom Titles

The Basic XY Titles, which consist of a main, X, and Y title, are used within
the default titles provided in the various graphs. The Custom Titles option
offers the means to extensively customize the titles used within a PeakFit
Graph.

Note that titles and labels can be toggled off in the 2D View option. They
need not be cleared in this custom titles option.

This option offers the means to add or edit up to five titles as well as the X,
Y, and Y2 titles. This option also offers the means to control the sizes of
these titles. You will need to use this option if you wish to add superscripts,
subscripts, or special upper ASCII or symbol characters to the titles. This
option offers the means to save the custom titles to disk for subsequent use.

Titles Entry You may enter up to 5 titles for display above the graph. The titles presented
will either be the default titles for the graph or the custom titles currently
active. To reset the defaults, use the Reset button. If you will be reading in a
custom title file, you may wish to use the Cut or Copy button to place one
of the titles in the clipboard before the previous titles are read. You would
then use the Paste button to paste this over the previous title not pertinent
to the present graph.

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Subscripts, To create a subscript, superscript, or to make a portion of a title bold or

Superscripts, Bold, italic, first enter the text and then highlight the portion of text you wish to
Italic superscript or subscript. When you press the subscript, superscript, bold, or
italic button, the appropriate control codes are inserted into the text. Nested
superscripts or subscripts are ignored.

Special Characters If you wish to add a character from the upper 128 characters of the current
font, or from the Symbol font, place the cursor at the desired location where
you wish to have the special character inserted. When you press the Symbol
(µ) button , you will be presented with a display of the available characters.
Select the character desired and it will be inserted, along with the appropriate
control codes, into the title text.

~ Character The ~ character is used internally by PeakFit for formatting subscripts,

superscripts, and special characters. In order to use the ~ character in a title,
you must enter the ~ character twice.

Sizes You may set the font size of the main title, the secondary title, the third-fifth
titles, the X,Y,Y2 titles, and the X,Y labels. The font sizes appear to the right
of the titles. Simply enter the point sizes desired, or set the values using the
spin buttons, or select the values from the popup menu activated by right
clicking and holding the mouse on the field or spin buttons.

Titles will not appear when the graph layout is maximized. If a fixed size
layout is selected, one or more of the titles may not be displayed if the font
sizes are too large.

Saving Titles All of the information in the Custom Titles screen can be saved to disk.
These files have default TTL extensions. These are binary files which can
only be generated from within the program. Custom titles are not saved
across sessions. You must explicitly Save a set of custom titles if you wish to
use them again.

Recalling Titles The Read item will read previously saved custom title information into the
current graph.

Review Graph titles contain peak information and goodness of fit statistics.
You must be careful not to import titles with incorrect information from
some other fit.

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Graph Colors
The Select Color Scheme and Customize Colors item in PeakFit Graphs is
used to set the colors for the current graph to one of the predefined color
schemes or to select or revise a custom color scheme for the type of graph
displayed. This dialog incorporates a copy of the current graph. Any color
scheme selection is immediately reflected in the graph.

Note that you may print a Black and White graph from any color scheme
(PeakFit internally changes to the Page White BW color scheme for printing
a monochrome graph). The most recent custom color scheme for each
graph in the program is saved across sessions. Saved also for each graph is
the most recent customized color set.

Color Selection The various graphs within PeakFit are preassigned a default color scheme. If
you dislike a given color scheme, you can select one of the ten predefined
schemes or build your own custom scheme.

Note that many of the color schemes are not immediately apparent if the
graph layout is currently maximized. If you wish to modify or create a
custom color scheme, you may wish to first change to one of the other
graph layouts containing titles and a standard background.

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Selecting or Reading a A custom color scheme for each graph is automatically saved across
Custom Color Scheme sessions. This color scheme is listed as the User Customized option.
Initially, this user color scheme will contain the Page White Color scheme,
the one you would most likely want to modify for color printing. To first
create a custom color scheme, first select the one existing color scheme
closest to the desired colors and then select the Customize Colors button.
To read a custom colors [CLR] file from this initial color selection dialog,
simply select the Read option.

Creating a Custom The Customize Colors button opens a Custom Colors dialog. Like the
Color Scheme color scheme dialog, it too contains a copy of the current graph which will
be updated with each color modification.

This dialog will display the graph colors used in the program and to the left
of each you will see the current colors. Simply click on the color you wish to
customize and then select one of the 16 pure Windows colors in the
selection palette. The graph will immediately change to reflect your choice if
that element exists within the current graph.

These 16 colors are those which are guaranteed to be present in all Windows
applications, regardless of palette settings.

Saving a Custom The most recent custom color scheme for a given type of graph is
Color Scheme automatically saved across sessions. If you wish to have more than one
custom color scheme for a given type of graph, such as plotter and color
laser printer color schemes for peak fit graphs, you will have to Save the
custom colors to a disk file and use the read item to recall them prior to
printing.

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You will also wish to save the custom color scheme to disk if you wish to
use these same colors in other program graphs. Colors files have [CLR]
default extensions. These are binary files which can only be generated from
within the program.

Reading a Custom The Custom colors option also has a Read option to import custom color
Color Scheme [CLR] files.

Reset The Reset option restores colors to the state existing when the dialog was
opened.

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Peak Labels
A PeakFit graph offers the means to add Peak Function Labels. The
following labels are available:

• None - do not display any labels

• X at Ymax Scanned - scan each peak data stream for the X value location
of the global Y maximum
• Ymax Scanned - scan each peak data stream for the global Y maximum
• X at Ymin Scanned - scan each peak data stream for the X value location
of the global Y minimum
• Ymin Scanned - scan each peak data stream for the global Y minimum
• Center Parameter - label each peak with the center parameter currently set
in the AutoFit scan
• Amplitude Parameter - label each peak with the center parameter
currently set in the AutoFit scan
• Area Parameter - label each peak with the analytic area currently
computed in the AutoFit scan

Labels are centered above each peak and written in the color used for that
particular peak. Positions are automatically adjusted upward or downward so
that no labels overlap.

The four scanned options do not use any external peak scan information.
Each component peak to be plotted consists of an X,Y table containing one
entry for each X pixel in the graph. These numeric values are scanned for X
and Y minima and maxima to produce these labels. As such, the resolution
of these labels will be somewhat limited.

In the AutoFit scan procedures, the three parameter options will display the
auto scan values. These will only be as accurate as the scan estimates. The
area is particularly likely to be inaccurate. In the Review, the actual values
derived from the fit are displayed as labels. At this stage, these values will
have the accuracy indicated in the Numeric Summary. Note that you will
need to fit an area parametrization of a given peak model to have a fit
uncertainty available for the area of each peak.

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PeakFit Text List Windows

PeakFit text list windows appears throughout the program for displaying,
modifying, and printing text information.

Considerations The text view window is used in the:

• ASCII List option

• Evaluation Procedure in the Inspect Function(X) option
• Data Summary in Fourier Domain Editing option
• List Peak Estimates in the three AutoFit Peaks options
• Numeric Summary in the Review
• Data Summary in the Review
• Evaluation Procedure in the Review

The viewer can display up to 16384 lines of information. You may use either
the cursor keys or the scroll bars to move about the text. If the contents of
the view window have been generated by PeakFit, the text will contain
columnar formatting to properly align all fonts, there will be a predefined
color formatting, and there will be support for displaying and printing
subscripts, superscripts, and symbols.

File Menu The viewer’s File menu contains options to Save the information in the
window as an ASCII text file, as a WK1 Lotus worksheet, as a WK3 Lotus
worksheet, or as a formatted file preserving superscripts and subscripts.
Although any text can be converted into a WK1 or WK3 spreadsheet, the
primary use will be with numeric columns of data. The formatted file may be
useful for importing information with subscripts and superscripts into
various word processing or desktop publishing software. When doing so,
you may need to specify the format as “Wordstar”.

The Printer Setup item is used to select and optionally configure a printer
for use by PeakFit. Note that the orientation option in the printer driver’s
own configuration dialog will be overridden by the orientation set in the
Print Text dialog.

The Print item opens the Print Text dialog, allowing a header to be
specified, the date and page number to be included, the orientation to be
chosen, and the margins to be set. You may also elect to center the longest
line and to print either all information or only that portion currently visible

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 PeakFit Common Elements

in the window. If you have a color printer, you may also elect to print the
information in color.

Edit Menu The Copy option copies the text in the window to the Windows clipboard
as both ASCII text and in the Lotus WK1 format. The text format is space
delimited and is used when you to paste the information into a text-based
program, such as a word processor. The Lotus WK1 format is used when
pasting the information into a spreadsheet such as Excel, Lotus for
Windows, Quattro Pro Windows, or SigmaPlot Windows. The most
practical use of this option will be with text containing columns of numeric
information. This Copy option is limited only by available memory.

The ASCII Editor option copies the contents of the viewer into the PeakFit
ASCII editor. On Win32S systems, the contents of the view window must
be less than 64K in size in order to use the editor. You will need to use the
ASCII Editor option to copy selected portions of the information to the
clipboard.

Style Menu The text windows each use their own specific font. If you select a different
font using the Font Select item, it will be used only for the current type of
text window. Since PeakFit’s internal windows are preformatted, you may
freely select variable pitch fonts, although you may wish to use a fixed width
font for the ASCII List and ASCII Editor options, especially when working
with files containing numeric columns of data.

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The Color item simply toggles the text between its predefined color format
and the default black text on white. the font and color setting are saved
across sessions for each type of window.

ASCII Text Editor

The PeakFit ASCII Editor is a NotePad-like editor available in many of the
program’s menus. The ASCII Editor option in the Edit menu within the
program’s main menu is used primarily for editing the X-Y data table since it
offers an update of the X-Y data table with the numeric contents when
closing the editor.

Most of PeakFit’s principal text windows offer an ASCII Editor option

which will place the contents of the window into the editor. If you wish to
edit a non data table text file from the main menu, you may select the ASCII
List option in the main File menu, select the file desired, and then select the
ASCII Editor option from the list window’s Edit menu.

Considerations There should be no size limitations if you are running Windows NT.
Otherwise, Windows limits edit controls to 64K. Except in Windows NT,
when reading files larger than 32K, the ASCII Editor automatically strips all
whitespace except that necessary to delimit the information. The ASCII
Editor is available for the:

• ASCII XY Data Table Editor option

• ASCII List option
• Evaluation Procedure in the Inspect Function(X) option
• Data Option in Fourier Domain Editing option
• List Peak Estimates in the three AutoFit Peaks options
• Numeric Summary in the Review
• Data Summary in the Review
• Evaluation Procedure in the Review

Columnar formatting, if present, can be preserved in the editor using a fixed

width font, such as Courier. Special characters such as subscripts,
superscripts, and symbols will not be preserved.

3-22 ASCII Text Editor

 PeakFit Common Elements

File Menu The standard items to create a New file, to Open an existing file, to Save the
current file, to Save As a different file, and to Exit are here.

The contents of the editor may be saved as an ASCII text file, a Lotus WK1
file, or a Lotus WK3 file. Although a WK1 or WK3 can be created
regardless of the editor contents, such files generally make sense only for
columns of numeric information.

The Printer Setup item is used to select and optionally configure a printer
for use by PeakFit. Note that the orientation option in the printer driver’s
own configuration dialog will be overridden by the orientation set in the
Print Text dialog.

The Print item opens the Print Text dialog, allowing a header, date, and
page number to be included, the orientation to be chosen, and the margins
to be set. When printing from the editor, it is not possible to center the
longest line nor to print only the visible portion of the editor.

Edit Menu The standard items to Cut to, Copy to, and Paste from the clipboard are
here as well as a single step Undo, the Delete of selected text, and a Select
All for selecting all text.

The Copy option will copy the selected or highlighted text to the clipboard
in a text format, and if at least a full line is selected, also in a WK1
spreadsheet format suitable for a very rapid paste by Excel, Lotus Windows,
Quattro Pro Windows, or SigmaPlot Windows.

Search Menu The standard items consist of a Find, a Find and Replace, and a Next Item
which will continue an existing Find or Find and Replace operation.

Style Menu The editor always loads with a standard Courier font. The Font Select
option will allow any other font to be used. For editing the content of
PeakFit text windows, a fixed width font is recommended.

There are four formatting items in the Style menu. The Comma to Decimal
item is for converting European format numeric entries to the decimal type
required by PeakFit. This option simply converts all comma characters
within the file to the period character, as in [123,45] being converted to
[123.45]. You may enter data with commas as decimal separators and then
use the Comma To Decimal option to convert to it to the format required
by PeakFit.

ASCII Text Editor 3-23

PeakFit Common Elements

The Space Delimited option will strip all whitespace from the program
except for a single space delimiter between the entries. The Comma
Delimited option accomplishes the same task, except that the single
delimiter will be the comma character. In the Tab Delimited option, the tab
character serves as the delimiter.

Function Insert Feature

A special help exists for accessing the various functions used within user
entered expressions and equations. This feature is found within the
Calculation entry, Inspect Function(X) entry, and User-Defined Function
(UDF) entry procedures.

You simply click on the type of function you are seeking and then on the
specific function of interest. The function is automatically inserted into the
calculation, Inspect Function(X), or user-defined function string at the
current cursor position. You may need to modify the symbols used as
arguments in the function to match the variable intended for the expression.

The General, Trigonometric, Statistical, Bessel, XY Data Table, Peak

Fns, and Supplemental functions will always be available. The Calculus
functions which include derivatives, sums, and integrals are available only in
UDFs. The conditional IF terms are in the General group.

The built-in peak functions are in the Peak Fns group. A built-in peak
function has two forms.

In the first form, X is implicit and assumed to be the X in the data table, as
in GAUSS(A0,A1,A2).

In the second, the X in the function must be explicitly specified as in

_GAUSS(X,A0,A1,A2). Adding the underscore to the beginning of the
function name results in the explicit function.

One use of the explicit functions is for building functions where X is the
variable of integration ($) rather than the X in the data stream. An example
would be _GAUSS($,A0,A1,A2).

3-24 Function Insert Feature

 PeakFit Common Elements

Evaluation Procedure
The Evaluation procedure offers extensive numerical evaluation of Inspect
Function(X) and Peak-Fit models. Use this option to find function
evaluations, roots, derivatives and cumulative areas. This option’s Generate
Table feature enables the creation of any data table based upon the equation
being evaluated.

Single Data Input

This aspect of the Evaluation requires individual data input from the
keyboard. Enter a value in the X, Y, or X2 field and then click on the
desired function operation. The result of the Y=F(X), X=Root(Y),
Y=dF(X)/dY, Y=d2F(X)/dY2, Y=Area(Xmin,X) or Y=Area(X,X2) is placed
in the Evaluation Table.

Evaluation Table This table is a PeakFit text view window that accumulates all manually or
automatically generated data, up to 16384 total entries. The table lists the
data element, the X value, the Y value, and the type of function operation
that produced the data. When entering data manually, you may conclude an
entry with Enter if you wish a Y=F(X) evaluation.

Roots If there are multiple roots within the X range, these various roots will be
added to the evaluation table. Note that the bracketing for the roots is
limited to 10 partitions. In the case of peak data, this means only a portion
of the roots may be found.

Evaluation Procedure 3-25

PeakFit Common Elements

Confidence or If you are evaluating a peak fit, you may also check the ConfLim or
Prediction Limits PredLim items. In these instances, the confidence or prediction limits will
be computed for the fitted model at the X,Y listed and these will be included
in two additional columns. The confidence level currently set in the Review
is used.

Areas The cumulative area computations use a Gaussian Quadrature procedure.

The precision attained is shown in brackets. A minimum fractional error of
1E-8 is sought.

Saving the Evaluation To save only the X and Y columns of the Evaluation table, use the Save XY
Table Data button in the button panel. You may save the data to a 15 digit precision
ASCII text file, or to a full binary precision Lotus WK1 or WK3 file. These
files can be subsequently imported into PeakFit’s data table.

To save a file containing the Evaluation table as it appears in the window (all
columns of information), use the Save As item in the Evaluation window’s
File menu.

Copying the Evaluation To copy only the X and Y columns of the Evaluation table to the Windows
Table Data to the clipboard, use the Copy XY button in the button panel. This copies the X-Y
Clipboard data as 15 digit precision space-delimited ASCII text, in a full binary
precision Lotus WK1 and Lotus WK3 format.

To copy the Evaluation table as it appears in the window (all columns of

information), use the Copy item in the Evaluation window’s Edit menu.

Clearing the Evaluation To clear the contents of the Evaluation table, use the Clear Table button in
Table the button panel.

Generating Data The Generate Table option automatically creates either the evaluation table
or a file from the function being evaluated.

You have the option of generating the input values by specifying the starting
value, increment, and ending values, or you may read the input values as a
column from any ASCII, Lotus, Excel, SigmaPlot, Quattro Pro, DIF, or
dBase file.

The computed values can be Y=F(X), X=Root(Y), Y=dF(X)/dY,

Y=d2F(X)/dY2, or Y=Area(Xmin,X).

3-26 Evaluation Procedure

 PeakFit Common Elements

In the generated option, only the first detected root is reported. You may
use this option to generate a file or to fill the evaluation table, up to a limit
of 16384 entries.

Table Format Use the Font Select item in the Style menu of the Evaluation Table window
to change the text font. The Evaluation Table is internally formatted to
work properly with proportional fonts. You may choose to disable the color
text display if you wish.

Printing the Evaluation You may select a printer using the Printer Setup option in the Evaluation
Table Table’s File menu. To print the table, use the Print item. You may print only
the currently displayed contents of the window or the entirety of the
window.

Evaluation Procedure 3-27

PeakFit Common Elements

Notes

3-28 Evaluation Procedure

 Data Import and Entry

4 Data Import and Entry

This section covers the file and data entry operations in PeakFit. These items
are contained in PeakFit’s File and Edit menus.

File Menu
PeakFit’s main File menu consists of options to:

• Import or append the data table from a file

• Import or append using an averaging digital filter

• Import or append the data table from the clipboard

• Save the data table to an ASCII or spreadsheet file

• Display, print, or edit any ASCII file

• Reset all program defaults

• Exit PeakFit

• Recall previously imported files

• Import dragged and dropped files

Edit Menu
PeakFit’s main Edit menu consists of options to:

• Edit the data table using a spreadsheet-like internal editor

• Edit the data table using an ASCII editor

File Menu 4-1

Data Import and Entry

• Copy the data table to the clipboard in ASCII and spreadsheet formats

• Toggle on and off the main window’s tool bar and status bar

PeakFit Import Options

The Import option is used to read the PeakFit data table from a file where
every valid data point is added to the table.

To digitally filter data from a file into PeakFit’s data table, use the Import
Digital Filter option.

To import the data table from the Windows Clipboard, use the Import
Clipboard option.

The Import Digital Filter option is identical to the Import option except
that you additionally specify a point count and starting position for the
averaging digital filter.

The Append, Append Digital Filter, and Append Clipboard options

function similarly except existing data is preserved. The newly imported data
is simply appended.

PeakFit’s maximum data table size is 16384 points.

Supported File Formats

• Importing ASCII files that contain alternating X-Y values, or those
containing only Y values
• Selecting an X and Y column, and optionally a weights column, from a
multi-column ASCII file
• Selecting an X and Y column, and optionally a weights column, from any
page or sheet in an Excel, Lotus-123, or Quattro Pro Windows file
• Importing chromatographic data from AIA net CDF files
• Selecting an X and Y column, and optionally a weights column, from a
SigmaPlot worksheet file

4-2 PeakFit Import Options

 Data Import and Entry

• Importing DIF files that contain alternating X-Y values, or those

containing only Y-values
• Selecting an X and Y column, and optionally a weights column, from a
multi-column DIF file
• Selecting an X and Y field, and optionally a weights field, from a dBase
III+, dBase IV, or compatible database file

Importing File Data

File Selection The Import, Import Digital Filter , Append, and Append Digital Filter
options open a standard file selection dialog. The List Files of Type drop
down box will contain all of PeakFit’s supported file formats. Simply select
the file desired.

When Data Table The Append, and Append Digital Filter options will be available only if an
Already Exists existing data table is already present. If you have a calculation currently
active, you will also be asked if you wish to have the input data filtered
through this calculation. If you are not appending to an existing data table,
and if you have not saved the current data, you will be offered an option to
save it before the read operation proceeds.

Standard Data Import The Import and Append options will read every data point and place each
within the PeakFit data table.

Digitally Filtering Input Digitally filtering incoming data is one way to produce a smaller data table
Data that can be more rapidly fitted. This is an averaging digital feature, so it
should be used only with very large data sets that can readily tolerate the
impacts of averaging. In all other instances, the non-parametric Digital
Filter option in the Data menu is recommended.

After file selection, the Import Digital Filter and Append Digital Filter
options open a dialog which requests a value for n, the number of input
stream points to be averaged before placing a value in the PeakFit data table.
This value can be anywhere from 1 to 999.

By using the highest value of n, you can filter data sets of up to 16.4 million
points into the 16384 point maximum limit of PeakFit. Both X and Y values
will be averaged. You must also supply the point at which PeakFit should
begin sampling data. Usually, you will begin at the first position.

Importing File Data 4-3

Data Import and Entry

Note that setting n to 1 and starting in the first position is the same as using
the Import or Append options.

Handling of Zero Since all dBase numeric fields in all existing records contain zero values until
Values in dBase a different entry is placed within them, PeakFit will automatically mark all
zero-containing points as inactive when reading DBF files. For all other
formats, PeakFit assumes that zero values are both legitimate and intended.

Active and Inactive With the exception of this instance of zero values in dBase files, every data
Data Points point is active by default, meaning that it will be included in the peak fit
processing. Any individual data point or band of points can easily be made
active or inactive within the Section option in the Prepare menu. Points can
also be individually excluded using the PeakFit Editor in the Edit menu.

If an ASCII file is generated, or translated from some proprietary format, it

is possible invalid data points will be present if empty data positions are filled
with zero values by some conversion routine. ASCII file extracts from
spreadsheets and databases sometimes add such zero values to maintain
row/column and record positions. Also, in copy operations involving
formulas, spreadsheets will sometimes write zero values in destination cells
corresponding with empty source cells. In such instances, these points must
be excluded prior to fitting.

Importing Worksheet Data Into PeakFit

Column Selection After selecting the file name to be read, PeakFit verifies the file format. If the
file consists of a supported spreadsheet, a column selection dialog will
follow. You must select a column for the X and Y values, and you may
optionally select a Weights column.

These X, Y, and Weights columns can come from any page or sheet within
the worksheet file. The column selection list will show both the sheet and
column in an Excel-like nomenclature. For example, the column designated
as C D is column D in the third sheet.

The column identifier and the strings present in the first 100 rows are used
to aid in selecting the columns desired. For a given column to be offered for
selection there must be at least one cell entry present for that column within the first
rows. Across all sheets or pages, the maximum number of columns available
for selection is 512.

4-4 Importing Worksheet Data Into PeakFit

 Data Import and Entry

The first column clicked will automatically become the X-column and the
second column clicked the Y-column. You may double click the Y column
to immediately close the dialog with the X and Y values selected. A weights
column or a revision of the X or Y column must be explicitly entered by
clicking on the X-Values, Y-Values, or Weights buttons after highlighting
the column of interest.

Lotus 123 PeakFit supports the Lotus 123 for Windows WK4 format, as well as the
WK3, WK1, and WKS formats. Most spreadsheets can save at least one of
these standard Lotus formats.

Excel The Excel XLS spreadsheet format is supported for versions 3 and above.
For v5 and above, OLE functionality is required since the files are saved as
OLE compound documents. 32-bit OLE functionality is provided by
Windows NT v3.5+, Windows 95, and by Win32S v1.20 and above. For
these systems, PeakFit can import Excel files even if they are currently open
and locked for exclusive use within Excel.

For Windows NT v3.1, and for versions of Win32S prior to v1.20, PeakFit
executes a separate 16-bit program XLS16.EXE to access the necessary
16-bit OLE DLLs to process the XLS file. You will need to have Excel (or
another 16-bit OLE2 application) installed in order for these DLLs to be
available. PeakFit does not supply them. For this 16-bit workaround to
succeed, the file must not be locked for exclusive use. Since Microsoft OLE
applications such as Excel or Word appear to exclusively lock the files

Importing Worksheet Data Into PeakFit 4-5

Data Import and Entry

actively being edited, you will need to close the target file within Excel or
Word before attempting to open it within PeakFit.

Quattro Pro PeakFit supports the Quattro Pro Windows WB2 and WB1 formats as well
as the original WQ1 and WKQ formats.

Import Guidelines • All data to be imported must be in numeric or formula format. Numbers
saved as strings will not be read. Strings may appear anywhere within the
columns.
• PeakFit will seek to read a main title from the first row of the sheet or
page. In order to have this title available, the first row must contain only
one string entry. It can be in any column.
• PeakFit will offer the first string in each selected column as a title.
• Valid data entries must be present in the columns in order for that row’s
entry to be added to the PeakFit table. If for a given row, the column
specified for one variable has a numeric value and the column for another
is empty or is filled with a string, no entry will be made to the table.
• All instances of a zero value in an X or Y column will be included in the
data imported, and these data pairs will be marked as active. Such points,
if invalid, can be made inactive using the Prepare menu’s Section option
or by using the PeakFit Editor.
• You will not be able to select the same column for more than one
variable.
• To insure proper X-Y correspondence, all columns are required to reside
on a single page or sheet.
When the read operation is completed, you will be shown the number of
points read and you will be offered the option of using the titles detected in
the spreadsheet as the main, X, and Y titles for the data set.

Using Previous Titles If you press Previous Titles in the titles entry form, the titles present prior
to the read, if any, are restored.

4-6 Importing Worksheet Data Into PeakFit

 Data Import and Entry

Importing Data From SigmaPlot Files

PeakFit can build its data table from the numeric values stored in SigmaPlot
SPW, SP5, and SPG graph files. The import process is very similar to that of
the spreadsheets. There are a few differences:

• The SigmaPlot column selection list consists of the column labels

assigned within SigmaPlot rather than the strings in the initial columns.
• Only numeric entries are read. Embedded strings and missing values are
not imported.

Importing AIA Chromatography Files

Net CDF data files which comply with the AIA Chromatography standard
can be imported.

The sample_name and experiment_title fields are used for the main title, the
retention_unit is used for the X title, and the detector_unit is used for the Y
title.

If the actual_delay_time is non-zero, the chromatographic dead time

transformation is automatically made:
time
x = −1
actual _ delay _ time

If this field is zero and you have an experimentally determined dead time for
the column, the calculation X=X/T0-1 should be applied as a Calculation,
where T0 is this time.

The default extension for AIA chromatography files is CDF.

Importing Data From dBase Files

PeakFit can build its data table from the numeric information in dBase
databases. Both dBase III+ and dBase IV type DBF files are supported. The
import process is similar to that of a spreadsheet except numeric fields are
selected for the X and Y variables. The following rules apply:

Importing Data From SigmaPlot Files 4-7

Data Import and Entry

• Only numeric fields (type N in III+ and types N and F in IV) are offered
for selection.
• If a dBase record is marked for deletion, the data from the selected fields
will still be added to the PeakFit table, but this X-Y pair will be marked as
inactive and will not be included in fitting.
• All instances of a zero value in one or both of the fields will be included
in the data imported, but these data pairs will be marked as inactive. Such
points, if representing meaningful data, can be made active by using the
Prepare menu’s Section option or by using the PeakFit Editor.

dBase database files do not use a binary format for storing numeric values.
Numeric storage is generally in fixed decimal position ASCII strings. This
limits the range and precision of floating point values. The PeakFit import
procedure will preserve whatever precision exists within the original dBase
numeric field.

Under certain conditions, versions III+ and IV can both store ASCII
scientific notation. Type N and F fields containing scientific notation will be
correctly read by PeakFit.

When selecting columns from a dBase file, you are selecting database field
names for the X and Y, and optionally the Weight, variables.

Importing Data From ASCII Files

PeakFit’s Standard X-Y PeakFit X-Y ASCII data files meet the following conditions:
ASCII Format
• The X and Y-values must be sequential. The X-value must come first,
followed by the Y-value and there must be some form of delimiter
between the values. The first valid number is read into X1, the second into
Y1, the third into X2, the fourth into Y2, and so on. The delimiter can be
any whitespace character (space, new line, tab, etc.), a comma, or most any
other non-numeric character.
• The first three lines of the file can optionally contain a description of the
data, an X-title, and a Y-title (if these begin with a numeric character, the
string must be delimited by double quotes).
• Comment strings can exist in the data area, provided these are on a
separate line or at the end of a line or numeric data. If a string begins with

4-8 Importing Data From ASCII Files

 Data Import and Entry

a number, it must be delimited by double quotes. Do not start a comment

with the asterisk (*) or the W character.
• A concluding asterisk (*), alone or with comment text, marks the X-Y
data pair as inactive. Such marked points are not included in the PeakFit
processing.
• A concluding W, followed by a number, is read as the weight for the data
point. The numbers must be between 1E-30 and 1E+30. All weights are
1.0 by default.

Typically, the standard PeakFit X-Y ASCII file will look like:
Main description
X-title
Y-title
1.234 2.345
3.456 4.567
5.678 6.789
7.890 8.901
9.012 10.123

or:
1.234,2.345
3.456,4.567
5.679,6.789
7.890,8.901
9.012,10.123

Scientific Notation All numeric values must be between 1E-30 and 1E+30. Scientific notation
uses the standard format. For example, 1.2345E+12, 1.2345e+12, and
.12345E+13 are valid entries. Space should not exist anywhere within the
characters.

An ASCII data file can be created in any ASCII editor. The ASCII extracts
of databases, spreadsheets, and other programs usually require no
modification provided only two columns or fields of data are involved.

The main problem with extracted or generated ASCII files will usually be
how empty or numeric overflow values are handled. If such an X-value or
Y-value is represented by a string, PeakFit will skip over it and the proper
X-Y-X-Y... sequence will be lost. If a value is stored as 0.0, which is
frequently the case, PeakFit will register this point as an active data pair, with

Importing Data From ASCII Files 4-9

Data Import and Entry

possibly unfavorable consequences on the resulting peak-fit processing. Such

files must be pre-edited.

PeakFit considers .PRN, .DAT, and .TXT to be default ASCII file

extensions.

Single And Multi-Column ASCII Files

Single Column ASCII Frequently, the X-data will begin at a given point and thereafter vary with a
Files constant increment. Many instruments, therefore, tend to generate single
vector data files for space reasons. If you cannot save an X-Y file directly,
PeakFit can read the single column file. If the fourth line of the data file
contains only a single numeric value, you will be asked if you wish to do a
Single-Column Read. If you choose this option, you must specify the starting
X-value and the X-increment so that the X-values can be automatically
generated.

Multi-Column ASCII If the fourth line in an ASCII data file contains three or more numeric
Files entries, you are presented with an option to do a Multi-Column Read. If you
choose this option, you will be offered a column selection based upon all
entries in the column, regardless of whether these are strings or numeric
values.

Just as with the spreadsheet options, you simply select a column for the
X-variable, for the Y-variable, and optionally for the Weights.

For PeakFit to read a multi-column ASCII file successfully, it must be in one

of two formats, both of which require all row entries for a given sequence of
columns to be on a single line which concludes with a carriage return:

• Comma Delimited Format - Here all values are separated by commas.

You may have any number of spaces within the data, but each column
entry, including empty positions, must be separated from other entries by
a comma. With this format, ASCII character strings do not have to be
delimited with quotes. You will have to delimit a string with quotes,
however, if it contains a comma.
• Space and Quote Delimited Format - In this case, there are no commas
separating the different column entries. Rather PeakFit relies on a space
or spaces between the entries of numeric values, and on double quotes

4-10 Single And Multi-Column ASCII Files

 Data Import and Entry

beginning and concluding every character string. With this format, quoted
character strings are essential. You must also use a pair of empty quotes to
represent empty column positions. The following two examples illustrate
both formats and represent valid multi-column ASCII files:
Column 1,Column 2,Column 3,Column 4,Column 5
1.234,2.345,Data Missing,7.890,8.901
3.456,4.567,6.789,9.012,10.123
5.678,6.789,7.890,11.234,12.345
7.890,8.901,,13.456,14.567

"Column 1" "Column 2" "Column 3" "Column 4" "Column 5"
1.234 2.345 "Data Missing" 7.890 8.901
3.456 4.567 6.789 9.012 10.123
5.678 6.789 7.890 11.234 12.345
7.890 8.901 "" 13.456 14.567

It is always a good idea to carefully inspect the input data from multi-column
files whose formats may not be fully compatible. The PeakFit multi-column
reader should read most multi-column ASCII files generated by commercial
software. “

Data Interchange Format (DIF) Files

Single Column DIF If a DIF file contains only one column of numeric data, a single-column read
Files is offered. You must specify the starting X-value and the X-increment so
that the X-values can be automatically generated as the Y values are read.

XY DIF Files If a DIF file contains two columns of numeric data, the file is automatically
read assuming the first column to contain the X-values and the second
column the Y-values. If the variables need to be reversed, this can readily be
done within the PeakFit Editor.

Multi-Column DIF Files When a DIF file contains more than two columns of numeric information, a
multi-column read is offered. You will be offered a column selection based
upon a column ID and any title assigned to this column. Just as with the
spreadsheet options, you simply select a column for the X-variable, for the
Y-variable, and optionally for the Weights.

Data Interchange Format (DIF) Files 4-11

Data Import and Entry

Importing Data From The Windows Clipboard

The Import Clipboard and Append Clipboard options are used to process
clipboard information for use by PeakFit. Three clipboard formats are
supported. If the data was copied from a Lotus Windows, Excel, Quattro
Pro Windows, or SigmaPlot worksheet, a WK3 or WK1 format will be used.
PeakFit will save the information to either a CLIPBRD.WK3 or
CLIPBRD.WK1 disk file. If the Lotus formats are not available, PeakFit will
seek to access the clipboard contents as simple text, saving the numeric
values to a CLIPBRD.PRN ASCII file.

If the data is copied from a spreadsheet, it must consist of at least 2 columns

of data. In all cases, a column selection will follow.

If it is copied in a text format, it can consist of a single column of data, two

columns of X-Y data, or multiple columns. PeakFit will automatically offer a
single column or multi-column read option if an alternating X-Y format is
not found.

Zero Values in In a clipboard import, zero values are considered valid and are added to the
Clipboard Data data table as active points. If your clipboard data contains zero values in
place of missing values, numeric errors, or strings, you will need to use the
Prepare menu’s Section option or the PeakFit Editor to exclude these
points from the fitting.

Digitally Filtering When data is read from the clipboard, the digital filter option is not directly
Clipboard Data available. If you wish to digitally filter data that you have in the clipboard,
first import the data normally via the Import Clipboard or Append
Clipboard options.

At this point, the non-parametric digital filter can be used. This is the Digital
Filter option in the Data menu.

Alternately, if the data set is amenable to an averaging type of digital filter,

you can note the file source in the PeakFit status window, and then begin the
import anew using the Import Digital Filter option and selecting this file. It
will have a CLIPBRD prefix and a WK3, WK1, or PRN extension.

4-12 Importing Data From The Windows Clipboard

 Data Import and Entry

Saving X-Y Data

PeakFit’s data table can be saved to disk in either an ASCII or Lotus format
using the Save As option in the main File menu.

ASCII Format The default format is a 15 digit precision ASCII file with a PRN extension.

These ASCII data files can be modified with any ASCII editor, including the
ASCII Editor available from within the program. Saved files are constructed
according to the following rules and conventions:

• The main title, X-title, and Y-title will be on separate lines in the first
three lines of the file. A title will be saved with double quotes if it begins
• with a number.
• Each X and Y-value pair will be on a single line with the X-value
preceding the Y-value. The values will be delimited by spaces. The default
extension [.PRN] is added if none is supplied.
• If you assign weights to a given data set, an uppercase W will follow the X
and Y values and it will be immediately followed (no spaces) with the
weight value assigned.
• An asterisk (∗) will follow the X and Y values if the X-Y pair has been
excluded from the table (marked as inactive).
• Data is saved by ascending X-values.

A W and a weight value will conclude the data line if weighting is present (at
least one weight differs from 1.0)

Lotus WK1 When you save the data table as a Lotus WK1 or Lotus WK3 spreadsheet
Lotus WK3 file, the numeric values are saved in a binary format. These files will preserve
full binary floating point precision.

The first row of the spreadsheet will contain the main title. The X and Y data
titles will be in row 2. The data begins in row 3. The X values will be in the
first column, the Y values in the second, and if weighting is used, the weights
will appear in the third column. The flags for active and inactive points are
not preserved when saving a WK1 or WK3 file.

Saving X-Y Data 4-13

Data Import and Entry

ASCII List
The main File menu’s ASCII List option is used to view any ASCII file. The
limit of file length is 16384 lines. This option can be used to view ASCII data
files intended for import into PeakFit. You may scan the file with using the
mouse or cursor keys.

This option uses a PeakFit text view window which appears throughout the
program for displaying, modifying, and printing text information. If the
contents of the view window have been generated by PeakFit, the text will
contain columnar formatting to properly align all fonts, there will be a
predefined color formatting, and there will be support for displaying and
printing subscripts, superscripts, and symbols.

The File menu in the ASCII List window offers an Open option to read
sequential files and an Append option to add file information to the
information already present in the Window.

PeakFit text windows are covered in detail in Chapter 3.

The Edit menu in the ASCII List window contains an ASCII Editor option
which copies the contents of the viewer into the PeakFit ASCII editor.

PeakFit’s ASCII Editor is also covered in Chapter 3.

Reset All Defaults

PeakFit is designed to be customized with use. All changes made in graphs
such as layouts, fonts, color schemes, points, labels, grids, as well as autoscan
settings, fit preferences, window positions and sizes, session data files and
UDFs, and other program parameters are automatically saved at the end of
each session.

The File menu’s Reset All Defaults option will restore all of the program’s
original defaults.

4-14 ASCII List

 Data Import and Entry

Previous Files
PeakFit automatically remembers the last five files read into the program and
places these files at the bottom of the File menu. These are saved across
sessions. You may select any one of these files to immediately load the file
listed into the program.

Use this option to reaccess the columns of a spreadsheet, multi-column

ASCII file, SigmaPlot file, DIF file, or dBase file containing data for multiple
peak-fits.

Drag And Drop Files

Drag and Drop You may drag and drop any file into the program. If you drag a single file
Single File and drop it either into the main window or into the status window, this file
will be immediately read.

Drag and Drop If you drag and drop multiple files, select no more than five total files. The
Multiple Files last file selected will be immediately read. The remainder will be added to the
Previous Files list at the bottom of the File menu.

Previous Files 4-15

Data Import and Entry

The PeakFit Editor

The Edit menu’s PeakFit Editor is a spreadsheet-like procedure for the
input or editing of the program’s data table. The PeakFit editor offers such
features as automatic X entry, the application of a calculation as values are
entered, and the option of viewing a graph of the data as it is being entered.

Editing or Input Simply place the cursor in the numeric field of interest and enter or modify
the entry. You may enter any numeric value, expression, or equation up to 80
characters in length. A numeric entry can be concluded by pressing Enter, by
clicking in any other field, by clicking on the Next button, or by clicking on
any of the arrow movement controls.

Moving Between Data To move about the data table, you may use the arrow controls in
Fields conjunction with the table’s scroll bar. You may also use the standard
keyboard cursor keys.

Active and Inactive An active data point is included in the peak-fitting. An inactive data point is
Data Points excluded from the fit. The Ex column of the PeakFit editor contains

4-16 The PeakFit Editor

 Data Import and Entry

checkboxes for excluding individual points. A point is inactive or excluded

when the box is checked.

You may also graphically toggle points between active and inactive using the
Prepare menu’s Section option.

Copy, Cut, Paste Use these buttons to Copy, Cut, or Paste a single numeric entry. You will
use this most often to copy a numeric value that is repeated rather than
having to re-enter the value.

Note: These are local options limited to a single cell or field in the editor.
The Paste option cannot be used to paste multiple values into the data table.
To append to the data table from the clipboard, use the Append Clipboard
option from the program’s main File menu.

Deleting Rows A data row can be deleted by pressing the Delete button or by pressing
Ctl-Y. You must be certain the cursor is in the row you actually want to
delete.

Inserting Rows To insert a row, place the cursor in the row that will follow the inserted
entry. Click on the Insert button or press the Insert key or Ctl-N. You must
provide both an X and a Y value before you can move to any other data row.
The program’s data is automatically sorted by ascending X upon exit from
the editor, and as such, you may simply wish to input inserted data at the end
of the table.

AutoEntry Use these options if you wish to have the editor automatically input a given
value. This is useful for constant increment X or Y values. You may have
either AutoEntry X or AutoEntry Y on at any given time. This option will
determine the value to enter based upon the two previous values. The
position in the editor is automatically advanced.

The AutoEntry W for the weights is not based upon previous values, but
rather upon the default value of 1.0 as the weight for each data point. You
will wish to leave AutoEntry W on unless you will be specifically entering
weights for the individual data points.

Entering a Calculation The Calculation button is used to enter a calculation inside the editor. The
calculation is applied to each additional X-Y pair at the time it is entered.

Applying the After entering a calculation in the editor, you are offered an option to Apply
Calculation the Calculation to all existing entries in the data table. It is best to restrict X

The PeakFit Editor 4-17

Data Import and Entry

and Y calculations used in the editor to functions of the respective variable

being entered. For more information on Calculations, please refer to the
Data Preparation chapter.

Toggling a Calculation A calculation is applied only to new data that is being entered at the bottom
of the data table. It is also applied only if the Apply Calc item is currently
checked.

Viewing a Graph of the The Graph button is used to open a separate window which displays the
Data As It Is Entered data as it is entered in the editor. This option requires at least three data
points be present. The graph is automatically rescaled as new data is entered
or when old data is deleted. You may wish to use this option to insure your
data is being entered properly.

Saving the Data Table The Save option saves the data table to an ASCII file. This is the program’s
default data format. Unlike the Save option in the File Menu, this option
preserves the actual ordering of the data points.

Clearing the Data The Clear button is used to clear the data table of all entries.

4-18 The PeakFit Editor

 Data Import and Entry

Basic Titles Use the Titles button to reassign the main, X, and Y titles for the data table.
These titles can be overridden using the custom titles option in all of
PeakFit’s graphs.

Sorting the Data Table The Sort Table option can sort the data table by ascending X, descending X,
ascending Y, or descending Y. You must save the sorted table inside the
editor in order to preserve this ordering. The data table is automatically
sorted by ascending X values in the prescan which occurs when exiting the
editor. All data tables saved from the program’s main menu will have this
ascending X ordering.

Reversing X and Y The Reverse X,Y button will interchange the X and Y data.
Values
Valid Operators and Editor entries can consist of any numeric expression. Only the evaluated
Functions result is saved and stored in the table. You can use full expression evaluation
to take the place of a calculator, although you will wish to use a calculation
for repetitive computations.

Appending File or To append file or clipboard data to data entered in the PeakFit editor, you
Clipboard Data must use the Append option in the program’s main File menu.

Restoring Original Data If you exit the editor with the Cancel button, you are asked if you wish to
lose all changes made in the editor. If you answer Yes, the original data is
fully restored, even if revisions were saved to file within the editor. If you
respond No, the changes are incorporated into the data table.

The PeakFit Editor 4-19

Data Import and Entry

Weighting Data
The data table can be weighted in one of two ways. You may specify a
weights column in any multi-column import operation, including
multi-column ASCII files, Lotus 123 files, Excel files, Quattro Pro files,
SigmaPlot files, DIF files, and dBase files. This is done using the File menu’s
Import or Append options. You may also use the PeakFit Editor to enter
weights for individual data points. By default all data points have a floating
point weight of 1.0.

Values for weights may be anywhere from 1E-30 to 1E+30. A point with a
weight of 100 will factor in 100 times more strongly than a point with a
weight of 1 in the fitting.

Weights as Inverse The data table weights are true floating point multipliers. When each data
Variances pair consists of an average of Y observations at a given x, you may wish to
set that pair’s weight value to the inverse square of the standard deviation. If
your weights are true standard deviations, apply a 1/σ^2 conversion in your
spreadsheet or enter the weight value followed by ^-2 in the editor. If you
are entering a significant number of data points, you may wish to enter the
weight values as standard deviations and apply a W=1/W^2 calculation using
the Data menu’s Enter Calculation operation.

Normalization of The weights used in PeakFit are normalized so that the sum of the weights
Weights equals the number of active data points. This conserves the degree of
freedom relative to unweighted data, and results in coefficient standard
errors which better reflect the impact expected from a true floating point
weighting scheme. This type of weighting differs from statistical programs
which use integer weights to specify the number of identical X,Y pairs. If
you are entering identical X,Y pairs and need to see the degree of freedom
increased by one for each identical pair, you will have to enter each identical
pair separately.

4-20 Weighting Data

 Data Import and Entry

ASCII Editor
The Edit menu’s ASCII Editor option opens a special PeakFit ASCII Editor
window designed specifically for editing the program’s X-Y data table as an
ASCII text file. For editing and copying contents of the various text
windows within PeakFit, there is an ASCII Editor item in the menu of the
specific text window of concern. This data table ASCII Editor option is
provided for those who prefer to input or edit a data table as a simple ASCII
file using a familiar NotePad-like editor.

Size Limitations In Windows NT, this editor should be able to handle any size data table up
to PeakFit’s 16384 maximum. In 32-bit systems limited to 16-bit edit
controls, such as Windows 3.1 with Win32S or Win95, the editor is limited
to a 64K buffer. On such systems, files imported into the editor larger than
32K are automatically stripped of all whitespace except the necessary
delimiter in order to maximize editor capacity. This allows up to about
1500-2000 points.

Automatic Update When the ASCII Editor is opened, its contents will reflect the current data
table. You may edit the data or you may open and edit any ASCII file. When
you exit the editor, you will be offered the option of updating the PeakFit
data table with the current contents of the editor. If you choose not to save
revisions made to the data table, the modified data is automatically saved to
PEAKFIT.PRN and this file will be listed as the data table’s file source in the
PeakFit status window.

Editing Guidelines While it is not required, it will be easiest if you keep the X and Y values for a
given data pair on a single line. If you wish to exclude a data pair, enter a
space and an asterisk (*) character after the Y value. If you wish to assign a
weight, enter a space and an uppercase W followed by the numeric weight
value. You may insert comments anywhere in the file so long as the
comment strings do not begin with *, W, or a number. All information on a
line following the detection of a comment is disregarded.

PeakFit’s ASCII Editor is used throughout the program’s text windows. The
general ASCII Editor options are covered in Chapter 3.

ASCII Editor 4-21

Data Import and Entry

Copying Data Table To Clipboard

The Edit menu’s Copy option copies the current data table to the Windows
clipboard as ASCII text, in the Lotus WK1 format, and in the Lotus WK3
format. The ASCII data will contain 15 digits of precision, approximately
equal to the binary precision in the WK1 and WK3 formats.

The text format separates the numeric columns with white space rather than
tabs. It is used when you paste the data table into a text-based program, such
as a word processor. When you paste the data table into Excel or SigmaPlot
Windows, the WK1 format is used. When either Lotus Windows or Quattro
Pro Windows is the destination, the WK3 format is used.

This Copy option is limited only by available memory and can copy up to
PeakFit’s 16384 data table maximum.

4-22 Copying Data Table To Clipboard

 Data Import and Entry

Tool Bar
The Tool Bar Edit menu item toggles the main PeakFit tool bar on and off.
The tool bar consists of the following options:
• Import XY Data Table from File
• Import XY Data Table from Current Contents of Clipboard

• Save Current XY Data Table to File

• Enter or Edit XY Data Table via PeakFit Editor
• Enter a Calculation to Modify XY Data Table
• Graphically Compare Data with Imported Reference
• Non-Parametric Digital Filter
• Graphically Section Data
• Smooth Data
• Fourier Domain Editing
• Deconvolution of Gaussian Instrument Response Function
• Deconvolution of Exponential Detector Response Function
• Automatically Fit and Subtract Baseline
• Automatic Peak Detection and Fitting, I - Residuals
• Automatic Peak Detection and Fitting, II - Second Derivatives
• Automatic Peak Detection and Fitting, III - Gaussian Deconvolution
• Enter or Edit Any or All of PeakFit’s 15 UDFs

Status Bar
The Status Bar Edit menu option toggles the main PeakFit window’s status
bar on and off.

Tool Bar 4-23

Data Import and Entry

Notes

4-24 Status Bar

 Preparing Data

5 Preparing Data

This section covers the operations which prepare or modify data prior to
fitting. These items are contained in PeakFit’s Data and Prepare menus.

Data Menu
• The Compare with Reference option offers the means to graphically
compare the current data with any imported reference.
• The New XY Titles option is used to edit the principal titles for the
current data set.
• The Enter Calculation option is used to enter one or more equations for
data transformation.
• The Apply to X-Y Table is used to apply such a calculation.
• The Cancel Calculation option clears the calculation from memory.
• The Zero Negative Data is used to zero the Y value of all data points
whose Y is less than zero.
• The Area Normalize will normalize the Y values so that the overall data
area is unity.
• The Cumulative Area option integrates the current data to produce
cumulative information.
• The Digital Filter option is a special graphical non-parametric fitting
option which enables any size data table to be translated to any other size
table. The size can be diminished or augmented by any count desired,
with excellent preservation of data detail.
• The Clear Inactive Points option is used to discard all points currently
marked as inactive from the data table.
• The Clear XY Data is used to completely remove all entries from the
data table.

Data Menu 5-1

Preparing Data

Prepare Menu
• The Section option is used to activate or deactivate regions of data, or
specific data points, prior to fitting. Sectioning can be done both
graphically and numerically.
• The Smooth option offers four different methods to remove noise from
data. The option can also be used to generate smooth first through sixth
derivatives.
• The Fourier Domain Editing option allows interactive zeroing of
frequency channels. Carefully done, this can be one of the most effective
methods for noise reduction.
• The Deconvolve Gaussian IRF function can be used to undo the
smearing caused by spectrometer optics.
• The Deconvolve Exponential IRF can be used to undo the tailing
introduced by chromatographic detectors.
• The Import and Subtract Baseline option is used to remove a baseline
that has been externally generated.
• The Inspect 2nd Derivative and Inspect 4th Derivative options assist
in the exploration for hidden peaks.
• The Inspect Function(X) option will graph up to five different
functions of x.

5-2 Prepare Menu

 Preparing Data

Compare with Reference

The Data menu’s Compare with Reference option is useful if you wish to
compare the current data set with an external reference data set. A reference
can come from any of the supported import formats:

• Single, X-Y, and Multi-column ASCII files

• Excel (XLS v3, v4, and v5+)
• Lotus 123 (WK4, WK3, WK1, WKS)
• Quattro Pro (WB2, WB1, WQ1, WKQ)
• SigmaPlot (SPW, SP5, SPG)
• AIA Chromatography Files (CDF)
• Single, X-Y, and Multi-column DIF files
• dBase III+ and dBase IV (DBF)

A reference can represent a single data file or a set of appended data files
produced by repeated Append operations followed by a final Save As. A
reference can be generated from a peak fit using the Review’s Eval or
Export options. Such a fit can likewise be based on a single set, or any
number of appended sets intended to produce a standard.

Compare with Reference 5-3

Preparing Data

Both data sets are displayed in a PeakFit graph. The current data table will
be plotted on the Y-axis, the reference on the Y2-axis.

New X-Y Titles

There are two levels of title information. The New X-Y Titles option
applies only to the basic main, X, and Y titles used for the main data table.
This New X-Y Titles option is used to create or edit these default titles.
These titles are saved with the data table.

The main title can be 80 characters in length. The X and Y titles can be up
to 40 characters long.

A much more extensive custom titles option is available in all of the

program’s principal graphs. To use symbols or upper ASCII characters in
the main, X, or Y titles appearing within graphs, it is simplest to use this
custom titles option. For more information on custom titles, please refer to
the PeakFit Graphs section of Chapter 3.

5-4 New X-Y Titles

 Preparing Data

Enter Calculation
A calculation consists of mathematical equations used to transform one or
more of the variables within the X-Y data table. In a calculation, the X, Y, or
weights data table values are treated as vectors and simply referenced as X,
Y, and W.

Some of the most commonly used PeakFit calculations are:

X=(X/t0)-1.0 (where t0 is the dead time of the chromatographic column)

This time transformation is required in order to obtain capacity factors (k’
values) from the fits of chromatographic models.

X=(1/X)*1E7 (for nm to inverse cm)

X=(1/X)*1E8 (for Angstroms to inverse cm)
X=(1/X)*1E4 (for micrometers to inverse cm).
These calculations transform X as a wavelength to wave number in inverse
centimeters. Spectral fitting requires a wavenumber, energy, or frequency X
scale.

Y=LOG(100/Y)
This calculation converts Y as % transmission to the absorbance required
for quantitative fitting.

Y=Y+GNOISE(5)
This calculation adds 5% random Gaussian noise to a data set. It can be
used to see if a particular peak model holds up well when noise is present.

Enter Calculation 5-5

Preparing Data

Entering a PeakFit A PeakFit calculation consists of an X= expression, a Y= expression, and a

Calculation W= expression. Only one expression need be present in a calculation. You
may enter any numeric calculation. All of the functions and constants
available within PeakFit can be accessed via a special Function Insert help.
This Function Insert help feature is covered in Chapter 3.

The X=, Y=, and W= prefixes are automatically supplied. If you enter a
calculation with multiple cross references to X and Y, bear in mind that the
X calculation will be applied first, then the Y, and finally the weight
calculation.

Copy, Cut, Paste Use the Cut, Copy, and Paste buttons to paste in expressions from the
clipboard, or to modify the text across the various fields.

Read Calculations that have been saved to disk are read by using the Read
button. It is a good idea to save a calculation that will likely be used on
future data sets. Calculation files are binary files with [CLC] extensions.
They can be created only within the program.

Save The calculation active at the close of any given session is not automatically
saved across sessions. You must explicitly save a calculation to disk using
the Save button in order to have it available in a future session.

Validation The calculation will be validated when you exit the Calculation entry screen.
If there is an error in any of the expressions, an error message will report a
specific parser or math error and you will be shown where within the
expression the evaluation failed. The cursor will also be placed at this
position.

Note that a calculation must be defined at the Xmean and/or Ymean of the
data table or at values of 1.0 if the data table is empty. Calculations are
compiled for a very rapid processing with large data sets.

Option to Apply You will be given an option to immediately apply the calculation you have
entered to the current data table. You will wish to answer no if the
calculation is to be applied only to additional data that will be appended to
the current table.

5-6 Enter Calculation

 Preparing Data

Apply to X-Y Table

The Apply to X-Y Table option applies the current calculation to the full
X-Y data table. The X-calculation, if present, is first applied to the full data
table. Then the Y-calculation, if present, is applied to the full table. Finally
the weight calculation is applied, if present, again to the full table.

If you have an X calculation that references Y and a Y calculation that

references X, note that the X calculation will use the original Y and the Y
calculation will use the modified X.

Undo After the calculation is applied, you are presented with a PeakFit graph
which graphically renders the data after the calculation has been applied.
You may choose to undo the calculation if you are not satisfied with the
results or if you wish to experiment with different calculations. If you
choose Undo, the original data will be restored. Simply choose OK to
accept the modifications.

Cancel Calculation
Use this Cancel Calculation selection to delete the current calculation from
memory. If you wish to use this same calculation in some future session, you
may wish to first save it to disk by using the Save item in the Enter
Calculation option.

Apply to X-Y Table 5-7

Preparing Data

Zero Negative Data

The Zero Negative Data option automatically performs the equivalent of a
Y=IF(Y<0,0,Y) calculation where all negative Y values are set to zero.

There are two reasons you may wish to zero all Y values which are slightly
negative. The simplest is that PeakFit’s autoscaling will set a minimum Y
that is negative in order to accommodate these points. Some find such
graphs to be aesthetically lacking.

Aside from the convenience of having the autoscaling produce the aesthetic
zero-based Y scale, a much more important reason to zero negative Y
values is that such points may not be properly accounted for by a baseline
being subtracted or fitted, or perhaps you are choosing not to subtract or fit
any baseline. In these cases, the negative values may adversely impact a fit.

Widths or shapes of particularly vulnerable (small area) peaks may change in

an undesirable fashion as the fit minimization seeks to compensate for
these negative values. If all constraints are off, it is even possible that you
will see one or more negative amplitude peaks, the result of the fit seeking
to account for these negative points.

The zero-processed data is displayed in a PeakFit graph. You may accept

the zeroing by using the OK button or fully restore the original data table
by using the Undo button.

Note that the AutoFit and Subtract Baseline option offers the means to
zero negative points as a part of the baseline subtraction procedure. It is far
more scientifically sound to zero negative points as part of an effective
baseline fitting.

5-8 Zero Negative Data

 Preparing Data

Area Normalize
The Area Normalize option modifies the Y-values of the data table so that
the area below the X-Y data from Xmin to Xmax is 1.0. When this
normalization is done, the area of all fitted peaks plus any fitted background
should be very close to unity. The normalization is based upon a simple
rectangular integration of the data.

The normalized data is displayed in a PeakFit graph. You may accept the
normalization by using the OK button or fully restore the original data table
by using the Undo button.

Cumulative Area
The Cumulative Area option integrates the current data to produce
cumulative information. The data is integrated from the first active data
point through the last active data point, using a simple rectangular
integration procedure.

If the data table consists of relatively few points, and you wish the most
accurate cumulative possible, it is recommended that you first use the Data
menu’s Non-Parametric Digital Filter to significantly increase the size of
the data table.

The cumulative data is displayed in a PeakFit graph. You may accept the
integration by using the OK button or fully restore the original data table by
using the Undo button.

Note that while a cumulative curve consisting of multiple transitions may be

much smoother than the constituent peaks from which it was built, it is
much more difficult to extract accurate parameters from fitting sums of
transition functions. Note also that PeakFit provides an exceedingly broad
range of peak functions, but only a modest number of cumulative or
transition models. If your data comes to you in an integrated form, it is
recommended that you use the Savitzky-Golay Smoothing procedure to
produce a good quality first derivative that can be processed as conventional
peak-type data.

Area Normalize 5-9

Preparing Data

Non-Parametric Digital Filter

The Data menu’s Digital Filter option offers a very effective way to
decrease or increase the size of the data table without distorting the features
of peaks within the data.

The PeakFit non-parametric digital filter is really a fitting and smoothing

algorithm similar in function to that of the Loess smoothing and
non-parametric estimation procedure. Both the PeakFit algorithm and the
Loess algorithm perform a weighted least squares fit for each estimation.
The PeakFit algorithm uses a Gaussian weighting function rather than the
tricube weighting function used in Loess.

The only controls for the non-parametric digital filter are the final desired
number of data points, the size of the processing window, and whether a
linear or quadratic model is fit.

Final Data Count You may set any number of points for the output data, up to a maximum of
8192. The output set will have the same initial and final X values, and will
consist of an equidistant X spacing. Within reason, you can thus reduce a
data set to a smaller size while preserving the quality of the input data

5-10 Non-Parametric Digital Filter

 Preparing Data

stream. You can conversely augment a data set with too few sampled points.

Data with Deleted The equal X-spacing of the output data stream is important if you find it
Artifacts, Disabled necessary to delete artifacts from a spectrum or chromatogram. The loss of a
Points, or constant X-spacing seriously hampers or altogether ruins the effectiveness
Non-Uniform X of a number of PeakFit’s processing options. For example, all FFT based
Spacing procedures are likely to suffer badly. This includes the FFT smoothing,
Gaussian convolution smoothing, Fourier domain editing, the Gaussian
response function deconvolution, the exponential detector response
function deconvolution, and the AutoFit procedure based upon Gaussian
deconvolution. Similarly, the Savitzky-Golay smoothing requires uniformly
spaced X values. It is central to the AutoFit procedure based upon second
derivatives.

While it is possible to effectively fit data with unequal X spacing using the
AutoFit procedure based upon residuals in conjunction with the Loess
smoothing procedure, you will probably be much happier if you simply use
this non-parametric digital filter to produce a uniformly spaced data set.

Window Point Count The default Window Point Cnt of 4 is recommended since it incurs the
minimum of peak amplitude attenuation. Higher window counts, however,
can be used to smooth the data. The power of smoothing will be roughly
comparable to the Loess procedure. In general, you can expect the most
favorable results from using the digital filter with the minimum 4 data point
window, and then smoothing the new data with the Savitzky-Golay or one
of the Fourier domain smoothing or filtering options.

Model You have a choice of fitting either a Linear or Quadratic model. For peak
type data, the quadratic model is probably the better choice. It tends to
produce less peak amplitude attenuation, and for a given window point
count, less smoothing as well.

Real-Time Graphical As you adjust the final data count, window size, or model, you will see a
Update PeakFit graph where the new points appear in reference to the original data.
This enables you to effectively judge the density of the final data as well as
any distortion of peak features occurring at higher window counts. Simply
select OK to accept the modified data or Cancel to retain the original data.

Non-Parametric Digital Filter 5-11

Preparing Data

Clear Inactive Points

The Clear Inactive Points option will clear all points in the current X-Y
data set that have been excluded from fitting. Such points are marked as
inactive either in the PeakFit Editor or ASCII Editor or by sectioning of the
X-Y Data. After discarding the inactive points, you may wish to use the File
menu’s Save As option to save the data absent these excluded points.

Note that all three AutoFit Peaks options process only the active points
and even offer a local sectioning which disables points only for the duration
of the specific fitting. As such, there is no need to clear inactive points in
order to facilitate the peak fitting.

Clear X-Y Data

The Clear XY Data option will clear all X-Y values from the current data
table.

Note that the current data table is automatically replaced with new data
when any of the Import options are used.

5-12 Clear Inactive Points

 Preparing Data

Section
The Prepare menu’s Section option is used to section the data into active
and inactive regions, or to disable or enable specific data points. Only active
regions are are scanned for peaks, and only active points are fitted.

The option will present a dual coupled PeakFit graph. The upper graph will
be scaled for only the active points, those used within peak fitting. The
lower graph will be scaled for all data points.

You may graphically section either graph, and the changes will be reflected
in both. Inactive points are shown in the current inactive point color. You
may section for active or included bands, or inactive or excluded bands, in
one of two ways.

Numeric Sectioning When you know exactly which ranges to include or exclude, numeric
sectioning may be simplest. The sectioning can be set to either Include or
Exclude the points in the specified region. A sectioning region consists of
2D box defined by an initial Xi, a final Xf, an initial Yi, and a final Yf.

If no entries are made for a given variable, the program assumes the full
range of the variable when including points, and no range on the variable
when excluding points. To apply the numeric sectioning to the data, you

Section 5-13
Preparing Data

must select either the Apply Existing or Apply New button. The Apply
Existing applies the given region to the current status of the points whereas
the Apply New option resets all points prior to the sectioning.

Graphical Sectioning For visual sectioning, be sure the PeakFit graph is in Section mode. You
may graphically section either graph. To exclude points, simply place the
mouse cursor at the initial point to be excluded, click and hold the left
mouse button down, and slide the mouse to the right. As you do so, the
points will be grayed (indicating that they have been deactivated or
excluded). Once the left mouse button is released, both graphs will reflect
the sectioning.

To reinstate a band of points, simply start at the rightmost point desired,

click and hold the left mouse button down, and slide the mouse to the left.
As you do so, the points will be restored to an active state.

Graphical Toggling of To toggle a single point between active and inactive, the PeakFit graph
Points must be in either Section or Process Points mode. In Process Points mode,
the cursor will track points, and only horizontal mouse movement is
processed. In either mode, simply place the mouse cursor on the point
desired. The mouse cursor will change to a crosshair with a point in its
center. If you have the hints active, the hint color will change and indicate
the point under the cursor. If the graph’s status bar is active, you will see
the point under the cursor also indicated there. Simply click the left mouse
button without moving the mouse to toggle the point on or off.

Moving Data Points While not normally recommended, PeakFit allows you to move any point
you wish to a different Y-value. This must be done in Process Points mode.
Moving a single data point may be one way to deal with single channel
artifacts. Simply place the mouse cursor on the point desired, hold the left
mouse button down, and move the cursor up or down to the new
Y-location. The first time this is done, you will be asked to confirm that you
genuinely wish to alter the value of the data point in this manner.

5-14 Section
 Preparing Data

Zoom-Mode In cases of highly concentrated data, you may wish to zoom-in a given
region prior to sectioning. The zoom-mode is provided for this purpose.
Either graph can be zoomed in this mode by simply clicking and holding
down the left mouse button while enclosing the zoom-in region desired.

To restore normal scaling, simply click the right mouse button anywhere
within the graph region.

Reset The Reset button restores all data points to an active state. Points that have
been moved to a new Y-location are not reset.

Accepting the Exit the dialog with OK to accept the sectioning shown in the graph, or
Sectioning Cancel to exit without adopting any modifications.

Temporary Sectioning Note that PeakFit offers two levels of sectioning. The sectioning in the
AutoFit Peaks options is a temporary one which uses a copy of the main
data table. The altered active or inactive states persist only for the duration
of the AutoFit procedure. This may be the most convenient way, for
example, to isolate and fit a single peak in order to first determine the most
appropriate model.

The Section option in the main Prepare menu modifies the actual PeakFit
data table. In this instance, to restore sectioned data to its original state, you
must either re-section or re-import the data.

Section 5-15
Preparing Data

Smooth
The Smooth option in the Prepare menu is used to smooth noisy data or to
create a smooth derivative directly from the raw data. The raw and
smoothed data are shown in a PeakFit graph for rapid determination of the
effectiveness of smoothing.

There are four methods of smoothing available:

• FFT Filtering
• Loess
• Gaussian Convolution
• Savitzky-Golay

The Savitzky-Golay algorithm can also be used to create a smooth first

through sixth derivative directly from the raw data.

Level The % smoothing for each of the algorithms defines the breadth of a
smoothing or filtering window.

5-16 Smooth
 Preparing Data

For the FFT Filtering, the % smoothing controls the channels that are
zeroed in the frequency domain. A 10% smoothing level zeroes the upper
1/2 of the frequency channels. A value of 20% zeroes the upper 3/4 of the
channels. A value of 50 zeroes the upper 9/10 of the channels. For
peak-type data, the optimum smoothing level is generally between 25 and
40%.

For the Loess and Savitzky-Golay procedures, the % smoothing determines

the actual smoothing window. A 10% smoothing level results in a
smoothing window containing 10% of the data points.

For the Gaussian Convolution smoothing, the % smoothing corresponds

with the Gaussian FWHM being convolved with the data. A 5% smoothing
levels convolves a Gaussian having a FWHM 5 times the average X spacing
(sampling interval) of the data set. The Gaussian convolution also includes
an intrinsic frequency domain filtration which is automatically determined.

You can modify the % smoothing by entering the desired value, by changing
the value with the up or down spin buttons, or you may right click and hold
down the right mouse button on the edit field and spin controls, selecting
the level from the popup menu. After a brief delay, the smoothed data will
be reflected within the graph.

AI Expert In order to make the determination of smoothing levels as automatic as

possible, PeakFit offers an AI Expert option which seeks to automatically
determine the optimum smoothing level. This is that level of smoothing
which offers the greatest possible noise reduction without adversely
affecting features within the data.

Depending on the algorithm, the AI Expert is based upon either a first

derivative of the overall standard error (between raw and smoothed) relative
to smoothing power, or a first derivative of the noise reduction relative to
smoothing power.

The AI Expert option will work very well for most cases. It is most easily
fooled on data sets where relatively few points determine an individual peak,
and on data sets where the noise is not consistent across the X range of the
data.

FFT Filtering The FFT Filtering option is essentially the automated version of what you
can do manually in the Fourier Domain Editing option. This option
removes any linear trend which might appear as a low frequency component

Smooth 5-17
Preparing Data

in the FFT, assumes the data is linearly spaced, performs a forward FFT,
zeroes the higher frequency components, performs an inverse FFT, restores
the linear trend, and presents the smoothed data, all in a single automated
step.

Because the signal tends to appear only at low frequencies, PeakFit uses the
aforementioned non-linear smoothing scale.

Linearly-spaced X values are recommended but not always required for this
algorithm to offer a very effective smoothing. Any non-uniformity of X
values does introduce noise, but such may be small compared to the overall
noise reduction achieved by the filtering.

Because of the effectiveness of the FFT, this algorithm is quite fast with
large data sets.

In lowpass filtering, any low frequency harmonic oscillations that are

present will pass without being filtered. If you see unwanted sinusoidal
behavior within the baseline areas, you should probably use a time domain
procedure (Savitzky-Golay or Loess).

Loess The Loess procedure is a locally weighted regression smoothing algorithm

that performs a full least-squares fit for each data point. Its strength lay in
its ability to effectively manage data with non-uniformly spaced X values.
This is most numerically intense of the smoothing algorithms and quite
slow with large data sets.

The PeakFit implementation of Loess fits a linear model. In general, this

means a tendency to see some attenuation of the peaks at higher smoothing
levels. As such, you will probably find the Loess procedure the least
attractive when you are working with data having equally spaced X-values.
If you are working with data containing non-uniform X-values, and you do
not wish to create a uniform set using PeakFit’s non-parametric Digital
Filter, Loess may be the only algorithm that produces acceptable results.

Gaussian Convolution This algorithm is unique to PeakFit. It uses an automatic FFT Filtering for
global smoothing, and convolves a narrow width Gaussian for local
smoothing.

In the Gaussian Convolution smoothing, any linear trend in the data is

removed, the data is transformed to the frequency domain, and there a
Gaussian response function is convolved with the data. The frequency

5-18 Smooth
 Preparing Data

domain data is then automatically filtered, the inverse FFT is made, and any
linear trend is restored.

The frequency domain filtering is automatically determined with this

algorithm, and it will have some dependency on the Gaussian width used in
the convolution.

The % smoothing specifies only the FWHM of the Gaussian convolving the
data, this as a multiple of the average X-spacing in the data set. As such, this
algorithm will generally require % smoothing levels between 2 and 5, this
corresponding with Gaussian response function FWHM of 2x to 5x the
average sampling interval.

Anything greater is likely to produce unwanted attenuation of the peaks. A

response function convolution procedure does conserve peak area, however,
so such attenuation is not as harmful as might be the case in a time domain
procedure.

This algorithm will probably produce the highest degree of noise reduction
of the four methods, but bears the same limitations as the FFT Filtering. As
such, this algorithm is recommended only for uniformly spaced X-data,
although very satisfactory results can sometimes be achieved lacking such.
Since the FFT procedures are quite fast, the algorithm is efficient with large
data sets, though somewhat slower than the simple FFT Filtering.

Savitzky-Golay The Savitzky-Golay procedure is a time-domain method of smoothing

based on least squares quartic polynomial fitting across a moving window
within the data. The method was originally designed to preserve the higher
moments within spectral data.

The algorithm has been modified by PeakFit and will offer a higher level
smoothing than that traditionally associated with Savitzky-Golay. The
PeakFit implementation of the Savitzky-Golay algorithm employs sequential
internal smoothing passes to improve overall noise reduction.

For the fifth derivative, a quintic smoothing must be used. For the sixth
derivative, a sixth order polynomial is fitted. All derivatives also use
sequential smoothing passes. The number of such passes is automatically
determined.

The higher order of polynomial makes it possible to achieve a high level of

smoothing without peak attenuation. Because this algorithm relies on the

Smooth 5-19
Preparing Data

linearity of an unweighted polynomial model, it is also quite efficient with

large data sets.

Unlike the FFT procedures which have some tolerance for non-uniformly
spaced X-values, the Savitzky-Golay procedure does require a constant
X-spacing within the data. If you wish to use the Savitzky-Golay procedure
to smooth your data, or if you wish to find hidden peaks using the AutoFit
Peaks II Second Derivatives option, you should use PeakFit’s
non-parametric Digital Filter to create data with a constant X spacing.

PeakFit uses the Savitzky-Golay procedure exclusively to produce a

smoothed second derivative in the AutoFit Peaks II Second Derivative
option.

Equivalent Noise % PeakFit contains an algorithm which estimates the measure of uniform
random noise present within a given data set. Reported are the equivalent
noise levels for both the raw and smoothed data as well as an overall %
noise reduction.

This estimate only determines how smooth given data is, and does not
serve as an indicator for oversmoothing. In general, oversmoothing is easily
observed visually in the attenuation of peak amplitudes.

Modifying the Simply click on OK to accept the smoothing shown in the PeakFit graph.
Primary PeakFit Data The primary PeakFit data table is updated with the smoothed data. To
Table retain the current data table, simply click on the Cancel button.

Note that PeakFit offers an Inspect Second Derivative option whose

purpose is to produce an enhanced and smoothed second derivative. It is
the preferred option for inspecting data for hidden peaks since it does not
change the data table, and offers two levels of smoothing to produce the
cleanest possible second derivative.

5-20 Smooth
 Preparing Data

Fourier Domain Editing

The Fourier Domain Editing option presents an automated means for
achieving the best possible frequency domain noise reduction for a given
data set. Here you graphically set the threshold for zeroing of the frequency
channels.

Removing Linear You must first decide if you wish to remove the linear trend from the data.
Trend in Data This should probably be done when you are working with non-waveform
data and the purpose of the frequency domain editing is for noise reduction.
You will probably wish to retain the linear trend if you are processing
waveforms. A linear trend in non-waveform data appears in the low
frequency channels, often adversely impacting the FFT in the threshold
range where frequencies are zeroed for noise reduction.

Real-Time PeakFit will present you with dual PeakFit graphs. The upper graph contains
Frequency-Time the frequency domain plot for the data. The lower graph contains the raw
Domain Graphing data (Y2 axis) and also the time domain inverse of the frequency spectrum
(Y axis).

The initial defaults will show the raw data as discrete points, the FFT inverse
as connected lines, these sharing a common plot and scaling. As channels

Fourier Domain Editing 5-21

Preparing Data

are sectioned or zeroed in the frequency domain, the effect is immediately

reflected in the time domain graph. This allows for directly perceiving the
relationship between frequency domain channels and the state of noise and
signal within the time domain.

If the input data is determined to have uniformly spaced X-values, the

X-scale will be an absolute frequency. If uniformly spaced X-values are not
present, the the data is assumed to have equally spaced X-values for the
purpose of constructing frequency channels with a spacing of unity. Here
the X scale will be an FFT frequency bin number, starting at zero. A real
FFT is symmetric, so you will see only half the original number of data
points, adjusted upward to an integer power of two.

While the magnitude-squared or power spectrum plot is the more common

format for frequency domain data, you will probably find editing simplest
using the magnitude plot. For peak-type data, whether you are looking at a
magnitude or a magnitude-squared spectrum, the signal should be present
in only the lower channels. It should decay very rapidly to a point where it
intersects what is often a rather constant level of noise.

Optimum Threshold The art to Fourier domain editing is simply selecting this one point in the
for Zeroing Channels spectrum to where the signal has decayed to where it intersects this noise
level. By simply sectioning or zeroing the channels beyond this threshold,
you are able to discard much of the noise, while retaining essentially all of
the signal. The graph on the previous page illustrates zeroing from this
threshold upwards.

In general, there is little benefit in peak-type data of having anything other

than this simple truncation type of lowpass filter. The frequency domain
graph will have a log Y axis by default, and it should be apparent that the
noise is significantly less than the signal in the lower channels, usually by
some orders of magnitude. An optimal filter which also removed the noise
in the signal channels would produce little change in the overall time
domain inverse.

Zeroing Bands or To zero channels from this critical Signal=Noise threshold upward, simply
Channels or place the mouse cursor at this desired frequency channel, click and hold the
Individual Channels left mouse button down, and slide the mouse to the right. As you do so, the
channels will be grayed (indicating that they have been zeroed). The time
domain graph will reflect the effect of zeroing these channels once the left
mouse button is released.

5-22 Fourier Domain Editing

 Preparing Data

To zero an individual channel, simply place the mouse cursor on the point
desired (or at the top of a bar). The mouse cursor will change to a crosshair
with a point in its center. If you have the hints active, the hint color will
change and indicate the frequency channel under the cursor. If the graph’s
status bar is active, you will see the frequency channel under the cursor also
indicated there. Simply click the left mouse button without moving the
mouse to toggle the channel on or off.

To reinstate a band of channels, simply start at the rightmost channel

desired, click and hold the left mouse button down, and slide the mouse to
the left. As you do so, the channels will be restored to their original values,
and the non-zeroed color will again be indicated.

Equivalent Noise % PeakFit contains an algorithm which estimates the measure of uniform
random noise present within a given data set. Reported are the equivalent
noise levels for both the input and the FFT-processed data as well as an
overall % noise reduction. Note that this estimate does not serve as an
indicator for signal deterioration. In general, such is easily observed visually
in the attenuation of peak amplitudes and in a sinusoidal appearance in the
baseline areas.

The FFT Data Option The Data button opens a text window which lists the numeric FFT data.
This will always contain columns containing channel number, magnitude,
and the real and complex components. If the FFT graph has an absolute
frequency X-scale, as occurs when uniformly-spaced X-values are detected,
the data option will also contain frequency, wavelength, amplitude, and
phase columns. You may save this data to text or spreadsheet files, copy it to
the clipboard in both text and spreadsheet format, or edit it using PeakFit’s
ASCII editor.

Modifying the Primary Simply click on OK to accept the FFT-processing shown in the lower
PeakFit Data Table PeakFit Graph. The primary PeakFit data table is updated with the
processed data. To retain the current data table, simply click on the Cancel
button.

Fourier Domain Editing 5-23

Preparing Data

Deconvolve Gaussian IRF

The Deconvolve Gaussian IRF option offers a Fourier domain procedure
which seeks to deconvolve a Gaussian instrument response function from
the current PeakFit data. Most spectral instruments will smear a spectral
line into a Gaussian peak, or smear a Lorentzian peak into a wider Voigt
peak, the Voigt being the convolution of a Lorentzian and Gaussian. The
smearing occurs as a function of the non-idealities in the optics, electronics,
and detectors. This option is primarily offered as a means to potentially
undo this smearing prior to peak analysis.

In this option, the incoming and processed data are shown in a PeakFit
graph. The deconvolved or convolved data will be shown in the Y-axis plot,
and the incoming data will be plotted on the Y2 axis. The initial defaults
show the input data as points, the deconvolved or convolved data as
connected lines, all on a common scaled graph.

Gaussian deconvolution is often effective in revealing hidden peaks within

the data. It is the basis of the AutoFit Peaks III Deconvolution option.

The following Gaussian deconvolution “sharpens” the peaks in the

spectrum, revealing four hidden peaks. Note that the deconvolved peaks
have narrower widths and larger amplitudes. Note also that the overall area
is conserved.

5-24 Deconvolve Gaussian IRF

 Preparing Data

Convolution and In simplest terms, convolution is the smearing of a data set by a given
Deconvolution instrument response function. Deconvolution is the procedure of undoing
that smearing in an effort to see what the data would look like had the
instrument perfectly rendered it.

Whereas convolving data with a Gaussian response function produces

smoother data, the deconvolution process of undoing the smearing
produces exactly the opposite effect. Noise is introduced, usually in drastic
amounts, and especially at higher frequencies. PeakFit’s Gaussian
deconvolution procedure seeks to deal with this major growth in noise by
adding a Fourier domain filter which removes most of the noise added in
the deconvolution procedure. When the data has not yet been smoothed,
this deconvolution procedure can actually result in reduced noise.

Non-Linear Nature of The nature of Gaussian convolution and deconvolution is not linear. Rather
Convolution and when a Gaussian is smeared by a Gaussian instrument response function,
Deconvolution the result is also a Gaussian with a variance equal to the sum of the
individual variances. If a Gaussian with a standard deviation of 3 is
convolved with a Gaussian instrument response function having an equal
standard deviation of 3, the resulting peak will have a standard deviation of

3 + 3 = 42
.

In other words, the price of smearing an equal width peak is not a peak with
twice the width, but a peak only 1.4x as wide. And as a further mathematical
blessing, the convolved peak will have the same area as the initial peak. The
amplitude simply drops as the width becomes greater.

In deconvolution, the opposite holds true. That 4.2 SD peak, deconvolved

with an SD 3.0 Gaussian, will yield an SD 3.0 Gaussian. The area is again
conserved, as the amplitude increases to offset the diminished width. In
other words, even though you can deconvolve a very large instrument
response function, the resulting peak will still have quite a significant width.

Deconvolution Pitfalls Deconvolution is fraught with many pitfalls, and the Fourier domain
procedure, while simplest, is not always successful. If you attempt to
deconvolve a response function close to the width of a peak or exceeding
such, the result is generally nonsense. Even with effective frequency domain
noise filtration, noise present at lower frequencies can produce something
other than a smooth deconvolution. When the width of the instrument
response function approaches the width of peaks, the deconvolved data will
sometimes contain negative values or small sinusoidal components.

Deconvolve Gaussian IRF 5-25

Preparing Data

Note that it is not possible to use deconvolution to regenerate a pure

spectral line. If you achieve peaks with 30% less width than the original
peaks, and the data looks respectable, then you have done well indeed. That
is about the extent to which you can hope to “sharpen” peaks using Fourier
deconvolution.

Direction You can choose to Convolve or Deconvolve the Gaussian instrument

response function. While the Deconvolve option will be primarily of
interest, the Convolve option may be useful to see the impact on data if the
Gaussian IRF were to become broader due to some deterioration in the
optics or detector. While the Convolve function can be used for smoothing,
a separate Gaussian Convolution smoothing is available in the program’s
Smooth option.

Response Function When this option is used for the first time, PeakFit will supply an initial
Width response function width based upon the range and midpoint of the data.
Depending on the number of peaks within the data, this value may be quite
low or quite high, perhaps significantly greater than the width of the peaks.
While you are offered a wide range of entry, you should consider restricting
the response width to no more than 75% of the widths of the peaks in the
data.

The Gaussian response function width can be specified as either a standard

deviation (SD) or as a full width at half maxima (FWHM).

This response function width will be saved across sessions in order to

preserve the entry of a carefully measured or inferred instrument response
width. Note that if you import data from a different instrument, or if the
data consists of peaks with larger or smaller widths, this previous width may
not represent even a viable starting point.

Symmetry For spectral data, you will almost universally wish to deconvolve a
Symmetric response function. If you are dealing with chromatographic
data, and have fronted or tailed peaks, you may wish to try the Tailed or
Fronted options. In the fortuitous event that a half-Gaussian convolution
describes the column non-idealities responsible for the tailed or fronted
state of the peaks, a one-sided Gaussian response deconvolution may result
in Gaussian or near Gaussian peaks, enabling you to fit simple Gaussians
rather than the more complex chromatographic models.

5-26 Deconvolve Gaussian IRF

 Preparing Data

Note that PeakFit offers a separate Deconvolve Exponential IRF

procedure to deal with the response function of chromatographic detectors.

Filter You may enter any value between 1 and 99 % for the Filter setting, or you
may set it by using the up and down spin buttons, or by clicking and holding
the right mouse button down while on the edit field or spin buttons, and
then selecting the value from the popup menu. The filter is a simple Fourier
domain truncation filter and it uses the same non-linear scale as the FFT
Smoothing. Here though, you will have to use much higher settings, as noise
will tend to definitely encroach, sometimes significantly, upon the signal
channels. Typically, values between 65 and 85% will be required.

A little exploration will quickly reveal the significance of the increased noise
from the Fourier deconvolution procedure and the vast importance of
effective filtration. At low filter settings, with the IRF width a significant
percentage of peak widths, the deconvolution is likely to produce nonsense.
Also you may note that small differences in the filter setting can significantly
affect the resulting deconvolution.

AI Expert The AI Expert option will seek to determine the optimum filter for the
deconvolution. Using sophisticated scanning and smoothing techniques
within the frequency data, an estimate is made for the optimum frequency
threshold between signal and noise, that is, the point where the two are
equal. The procedure is not foolproof, although it works quite well until
response function widths start to approach the widths of the peaks.

Equivalent Noise % PeakFit contains an algorithm which estimates the measure of uniform
random noise present within a given data set. Reported are the equivalent
noise levels for both the input and the deconvolved data as well as an overall
% noise change.

Modifying the Primary Simply click on OK to accept the processing shown in the PeakFit graph.
PeakFit Data Table The primary PeakFit data table is updated with the processed data. To retain
the current data table, simply click on the Cancel button.

Deconvolve Gaussian IRF 5-27

Preparing Data

Deconvolve Exponential IRF

The Deconvolve Exponential IRF option offers a Fourier domain
procedure which seeks to deconvolve an exponential detector instrument
response function from the current PeakFit data.

This assumes the chromatographic detector has a first order response rate.
This exponential response will shift the center of a peak to a slightly greater
time, and the peak will have a tailed or right-shifted asymmetry.

EMG Model The EMG (Exponentially-Modified Gaussian) chromatographic model is a

convolution of a Gaussian with an exponential response function. Rather
than fit EMG functions to chromatographic data to deal with this
extracolumn effect, it may be possible to undo this exponential smearing
using Fourier deconvolution prior to peak analysis. It may then be possible
to fit simple Gaussians to the data. This deconvolution procedure works
best with input data that has very little noise, and where peaks would be
perfect Gaussians save the impact of the detector.

In this option, the incoming and processed data are shown in a PeakFit
graph. The deconvolved data will be shown in the Y-axis plot, and the
incoming data will be plotted on the Y2 axis. The initial defaults show the
input data as points, the deconvolved data as connected lines, all on a
common scaled graph. The following graph shows the exponential
deconvolution of some chromatography peaks:

5-28 Deconvolve Exponential IRF

 Preparing Data

Note that the deconvolved peaks are shifted slightly to lower times, have
slightly higher amplitudes, and are much more Gaussian in appearance (most
of the tailing is gone). Note also that the overall area is conserved in the
deconvolution.

Convolution and In simplest terms, convolution is the smearing of a data set by a given
Deconvolution instrument response function. Deconvolution is the procedure of undoing
that smearing in an effort to see what the data would look like had the
instrument perfectly rendered it.

Whereas convolving data with a response function produces smoother data,

the deconvolution process of undoing the smearing produces exactly the
opposite effect. Noise is introduced, usually in drastic amounts, and
especially at higher frequencies. PeakFit’s exponential deconvolution
procedure seeks to deal with this major growth in noise by adding a Fourier
domain filter which removes most of the noise added in the deconvolution
procedure. When the data has not yet been smoothed, this deconvolution
procedure can actually result in reduced noise.

Deconvolution Pitfalls Deconvolution is fraught with many pitfalls, and the Fourier domain
procedure, while simplest, is not always successful. If you attempt to
deconvolve a response function with an unreasonably large time constant,
the result is generally nonsense. Even with effective frequency domain noise
filtration, noise present at lower frequencies can produce something other
than a smooth deconvolution. When the exponential time constant is of the
same order at the width of peaks, the deconvolved data will sometimes
contain negative values or small sinusoidal components.

If the right asymmetry is due to anything other than a one-sided exponential

smearing, the deconvolution is not likely to produce symmetric Gaussian
peaks.

Chromatographic Peak In general, chromatography peaks can be tailed due to both extracolumn
Considerations (detector response) and intracolumn (grouped column non-idealities) effects.
The exponential deconvolution is likely to address only the tailing having an
extracolumn origin. If tailing is also due to intracolumn effects, and the
intracolumn portion is not well described by a one-sided exponential
convolution, the exponential deconvolution is unlikely to resolve peaks to
symmetric Gaussians. You may see some trace of the tailing persist, even as
the overall peak begins to show a fronted appearance. The peak may even
begin to evidence a bimodal appearance.

Deconvolve Exponential IRF 5-29

Preparing Data

In this instance, you may wish to try a combination of exponential response

deconvolution and a Gaussian Response Function Deconvolution, using
the latter with its Tailed symmetry setting. A peak amenable to exponential
response deconvolution should be well fitted by the EMG
(Exponentially-Modified Gaussian) model. A peak amenable to a one-sided
Gaussian deconvolution should be well fitted by the GMG (Half-Gaussian
Modified Gaussian). To resolve two Gaussians peaks that are well fitted by
any of PeakFit’s models which combine the EMG and GMG, you may
need to first deconvolve the exponential response, and then deconvolve the
directionally-constrained Gaussian response.

Time Constant When this option is used for the first time, PeakFit will supply an initial
response function time constant based upon the range and midpoint of the
data. Depending on the number of peaks within the data, this value may be
quite low or quite high, perhaps out of line with the width of the peaks.
While you are offered a wide range of entry, you should consider restricting
the time constant to no more than perhaps half of the FWHM of the peaks
in the data.

This response function time constant will be saved across sessions. Note
that if you import data from a different instrument, or if the data consists of
peaks with larger or smaller widths, this previous time constant may not
represent even a viable starting point.

As you increase the time constant, the right-shifted or tailed peaks will give
way to peaks with a left-shifted or fronted appearance. An inferred time
constant is one that produces the least tailing or fronting, the most
Gaussian-like symmetry. Such an inference assumes the tailing is due to
wholly extracolumn effects.

Filter You may enter any value between 1 and 99 % for the Filter setting, or you
may set it by using the up and down spin buttons, or by clicking and
holding the right mouse button down while on the edit field or spin
buttons, and then selecting the value from the popup menu.

The filter is a simple Fourier domain truncation filter and it uses the same
non-linear scale as the FFT Smoothing. Here though, you will have to use
much higher settings, as noise will tend to definitely encroach, sometimes
significantly, upon the signal channels. Typically, values between 55 and
80% will be required.

5-30 Deconvolve Exponential IRF

 Preparing Data

A little exploration will quickly reveal the significance of the increased noise
from the Fourier deconvolution procedure and the vast importance of
effective filtration. At low filter settings, with the time constant a significant
portion of the peak widths, the deconvolution is likely to produce nonsense.
Also you may note that small differences in the filter setting can significantly
affect the resulting deconvolution.

AI Expert The AI Expert option will seek to determine the optimum filter for the
deconvolution. Using sophisticated scanning and smoothing techniques
within the frequency data, an estimate is made for the optimum frequency
threshold between signal and noise, that is, the point where the two are
equal. The procedure is not foolproof, although it works quite well until the
time constants start to become unreasonable.

Equivalent Noise % PeakFit contains an algorithm which estimates the measure of uniform
random noise present within a given data set. Reported are the equivalent
noise levels for both the input and the deconvolved data as well as an overall
% noise change.

Modifying the Primary Simply click on OK to accept the processing shown in the PeakFit graph.
PeakFit Data Table The primary PeakFit data table is updated with the processed data. To retain
the current data table, simply click on the Cancel button.

Deconvolve Exponential IRF 5-31

Preparing Data

Import and Subtract Baseline

In some cases, it may be possible to get a valid baseline from running a
solvent blank or in some other manner generating an external baseline. In
such an instance, PeakFit offers the Import and Subtract Baseline option.
Note that some instruments will produce a different baseline depending on
whether a substance to be analyzed is present or absent.

A baseline can be imported from any of PeakFit’s supported file formats:

• Single, X-Y, and Multi-column ASCII files

• Excel (XLS v3, v4, and v5+)
• Lotus 123 (WK4, WK3, WK1, WKS)
• Quattro Pro (WB2, WB1, WQ1, WKQ)
• SigmaPlot (SPW, SP5, SPG)
• AIA Chromatography Files (CDF)
• Single, X-Y, and Multi-column DIF files
• dBase III+ and dBase IV (DBF)

The baseline is subtracted after input and you are shown a PeakFit graph
containing the baseline corrected data. Simply click on OK to accept the
correction and modify the primary PeakFit data table, or click on Undo to
restore the original uncorrected data.

5-32 Import and Subtract Baseline

 Preparing Data

Inspect 2nd Derivative

Although PeakFit offers the AutoFit Peaks II Second Derivative option in
order to find and fit hidden peaks, this Inspect 2nd Derivative option is
provided for preliminary exploration of hidden peaks.

This option offers two levels of smoothing, the second consisting of the
Savitzky-Golay smoothing to a second derivative.

Second Derivatives The smoothed second derivative of data will show local minima at peak
and Hidden Peaks positions. Peaks with local maxima in the input data will produce second
derivative local minima with values that fall below zero. Hidden peaks which
evidence no local maxima in the input data will, if found by second
derivatives, produce a second derivative local minima with values that tend
to be above or near zero.

In the graph below, the 11 local maxima peaks apparent in the input data
(upper plot) appear as second derivative local minima with significant
negative values (lower plot). The four hidden peaks in the input data appear
as smaller second derivative local minima.

This is the principle used to find hidden peaks in the AutoFit Peaks II
Second Derivative option.

Inspect 2nd Derivative 5-33

Preparing Data

Smoothing Levels The first level of smoothing is optional. If a Level 1 Algorithm is selected,
and Algorithms the Level 1 input field will specify its level of smoothing. The second level
of smoothing is the Savitzky-Golay second derivative procedure. Its level of
smoothing is specified in the Level 2 field.

These levels and algorithms are exactly as described in the Smooth option.
In most cases, a Level 1 algorithm will not be needed to reveal the hidden
peaks. Particularly difficult cases may benefit from a Level 1 Fourier
domain procedure such as the FFT Filtering or Gaussian Convolution.

AI Expert The AI Expert option seeks to automatically determine the optimum

smoothing levels, those which offer the greatest possible noise reduction
without adversely affecting features within the data. In general, this criterion
also works very well for producing clean second derivative data This AI
Expert works identically to the Smoothing AI Expert. If both levels of
smoothing are active, the AI Expert will find an optimum for both
algorithms. The AI Expert option will work very well for most data. Still,
whether or not a hidden peak is revealed by second derivative is quite
dependent on the quality of the input data and the level and quality of the
smoothing.

Equivalent Noise % PeakFit contains an algorithm which estimates the measure of uniform
random noise present within a given data set. Reported are the equivalent
noise levels for both the raw and smoothed second derivative data as well
as an overall % noise reduction.

Closing the Dialog This feature does not modify the PeakFit data table. To preserve the
current settings for future inspections, simply click on OK. To disregard
any settings from a current inspection, simply click on the Cancel button.

5-34 Inspect 2nd Derivative

 Preparing Data

Inspect 4th Derivative

This Inspect 4th Derivative option is provided for preliminary exploration
of hidden peaks. This option offers two levels of smoothing, the second
consisting of the Savitzky-Golay smoothing to a fourth derivative.

Fourth Derivatives and The smoothed fourth derivative of data will show local maxima at peak
Hidden Peaks positions. Peaks with local maxima in the input data will produce fourth
derivative local maxima with values above zero. Hidden peaks which
evidence no local maxima in the input data will, if found by fourth
derivatives, produce a fourth derivative local maxima with values that tend
to be less than or near zero.

In the graph below, the 11 local maxima peaks apparent in the input data
(upper plot) appear as major fourth derivative local maxima with significant
positive values (lower plot). The four hidden peaks in the input data appear
as smaller fourth derivative local maxima.

Smoothing Levels and The first level of smoothing is optional. If a Level 1 Algorithm is selected,
Algorithms the Level 1 input field will specify its level of smoothing. The second level
of smoothing is the Savitzky-Golay fourth derivative procedure. Its level of
smoothing is specified in the Level 2 field.

Inspect 4th Derivative 5-35

Preparing Data

These levels and algorithms are exactly as described in the Smooth option.
In most cases, a Level 1 algorithm will not be needed to reveal the hidden
peaks. Particularly difficult cases may benefit from a Level 1 Fourier
domain procedure such as the FFT Filtering or Gaussian Convolution.

AI Expert The AI Expert option seeks to automatically determine the optimum

smoothing levels, those which offer the greatest possible noise reduction
without adversely affecting features within the data. In general, this criterion
also works very well for producing clean fourth derivative data This AI
Expert works identically to the Smoothing AI Expert. If both levels of
smoothing are active, the AI Expert will find an optimum for both
algorithms.

The AI Expert option will work very well for most data. Still, whether or
not a hidden peak is revealed by fourth derivative is quite dependent on the
quality of the input data and the level and quality of the smoothing.

Equivalent Noise % PeakFit contains an algorithm which estimates the measure of uniform
random noise present within a given data set. Reported are the equivalent
noise levels for both the raw and smoothed fourth derivative data as well as
an overall % noise reduction.

Closing the Dialog This feature does not modify the PeakFit data table. To preserve the
current settings for future inspections, simply click on OK. To disregard
any settings from a current inspection, simply click on the Cancel button.

5-36 Inspect 4th Derivative

 Preparing Data

Inspect Function(X)
The Inspect Function(X) option is used to graph up to five functions of X
across the X range of your choice. The graph can be scaled, formatted,
printed, or copied to the clipboard. You may also plot the derivatives or
cumulative area of the functions and you may numerically evaluate a
function you have entered.

Inspect Function(X) Enter up to five functions of X you wish to graph. You may enter any
Entry numeric calculation. All of the functions and constants available within
PeakFit can be accessed via a special Function Insert help. The Y= prefixes
are automatically supplied.

The Function(X) functions are limited to single line expressions. Use the
Copy, Cut, and Paste buttons to paste in expressions from the clipboard, or
to copy or modify the function text. If the functions are not successfully
validated, the math or parser error causing the failure will be shown and the
cursor will be placed at the location in the function where the error was
detected.

There are four options for displaying the Function(X) functions:

• Fn - displays the functions in a single plot graph

• Fn, D1 - displays the functions (Y axis) and first derivatives (Y2 axis) in a
dual plot graph
• Fn, D2 - displays the functions (Y axis) and second derivatives (Y2 axis)
in a dual plot graph

Inspect Function(X) 5-37

Preparing Data

• Fn, Int - displays the functions (Y axis) and cumulative areas (Y2 axis) in
a dual plot graph

Simply select one of the four options to create the desired PeakFit graph.

Note that cumulative areas require the evaluation of an integral for every
pixel drawn in the plot. With a number of functions, with mathematically
complex functions, or with large high-resolution graphs, these integrations
may require some time. The integrations are done from a lower limit which
corresponds with the lower limit of the displayed graph. By changing the
lower X of the graph, you are thus changing this lower limit of integration.

The functions will be plotted in colors which correspond with the

component colors. These colors will also be used for the main titles of the
PeakFit graph.

Undo Use the Undo item to return to the Function entry screen in order to revise
the functions or X range.

Zoom Ten levels of zoom-out are available by using the scrollbar in the graph’s
control panel. If this is a Fn,Int graph which displays cumulative areas,
there may be an appreciable delay in updating the graph following a change
in the zoom setting.

Evaluation The Eval item is used to perform a full-featured numeric evaluation of one
of the functions. If two or more functions are present, you must choose
one for this numeric evaluation.

5-38 Inspect Function(X)

 Automated Peak Fitting

6 Automated Peak Fitting

This chapter covers PeakFit automated fitting options as well as the

installation of user-defined functions. These items are in PeakFit’s AutoFit
menu.

AutoFit Baseline
This is PeakFit’s automated baseline fitting option. In PeakFit, you can deal
with an intrinsic baseline in two ways. You can choose to fit the baseline
along with the peaks, or you may elect to fit and subtract the baseline prior
to peak fitting. This AutoFit Baseline option is used to remove the baseline
prior to peak placement and fitting.

When fitting relatively few peaks, it may be simpler and more efficient to
skip this separate baseline processing step and include the baseline in the
overall model being fitted. On the other hand, when fitting a considerable
number of peaks, or when you are dealing with very unusual baselines, it
may be best to remove the baseline prior to fitting.

Fitting the Baseline The primary advantage of fitting baseline and peaks together is that no
with the Peaks additional error is introduced by a separate baseline subtraction. The model
simply describes all of the data, peaks and baseline, and in that sense it is
complete from a statistical perspective. Life is also a bit simpler since the
peak analysis can be done directly, without this added step of data
preparation. All of the AutoFit Peaks options allow for easy inclusion of the
baseline into the fitted model.

Fitting the Baseline The primary advantage of fitting a baseline separately and subtracting it prior
Separately to fitting rests within the stability such a step adds to the process. When
fitting many peaks, baselines will often shift away from an obvious baseline
position in order to compensate for very small peaks near the baseline that

AutoFit Baseline 6-1

Automated Peak Fitting

are not being fitted or for the discrepancy between the peak shape of the
data and that which the peak model is actually capable of delivering.

For example, when fitting a symmetric peak model to data whose peaks
have some skew, even a linear baseline will sometimes shift in an
unfavorable direction to compensate the inability of the peak model to deal
with that skew.

Another advantage of pre-fitting the baseline is the ability to use a

non-parametric baseline estimation procedure, one that can describe
virtually any shape of baseline. Such is not possible in conjunction with
parametric peak fitting.

A final advantage is that pre-subtracting the baseline simplifies the fit

somewhat and in reducing the degree of freedom, the fit statistics may be
slightly improved.

This AutoFit Baseline option offers three principal algorithms for

autodetection of points comprising the baseline.

Initial,Final Lin This is the simplest option. It simply selects the first and last active points,
and constructs a line between them. The Numeric option is not available
when this simple baseline is chosen since there are no fit statistics to report.

Progressive Lin This option is appropriate for those data sets where it is known that a clear
baseline exists at each end of the active data. PeakFit begins at the two limits
and progressively fits a linear model to increasingly more points until the
goodness of fit begins to deteriorate, usually signaling the first appearance of
a peak.

This progressive linear fitting algorithm was developed for PeakFit. There
are two progressive fits, one that works in an ascending direction, and one in
a descending direction. The direction offering the better goodness of fit is
automatically chosen. The Numeric option will display the fit statistics. For
certain data sets, the ascending and descending statistics may be identical.

2nd Deriv Zero This is the advanced baseline processing offered by PeakFit. Its algorithm
and methods are unique to PeakFit. The general principle is that baseline
points tend to exist where the second derivative of the data is both constant
and zero. For this 2nd Deriv Zero algorithm, you must select one of 10
options:

6-2 AutoFit Baseline

 Automated Peak Fitting

• Best (Best of the 8 Parametric models)

• Cnst (Constant, y=a0)
• Lin (Linear, y=a0+a1x)
• Quad (Quadratic, y=a0+a1x+a2x2)
• Cubic (Cubic, y=a0+a1x+a2x2+a3x3)
• Log (Logarithmic, y=a0+a1lnx)
• Exp (Exponential, y=a0+a1exp(-x/a2))
• Power (Power, y=a0+a1xa2)
• Hyp (Hyperbolic, y=a0+a1a2/(a2+x))
• NParm (Non-Parametric)

The exponential, power, and hyperbolic baselines are non-linear models and
are fitted iteratively using the same Levenburg-Marquardt algorithm used in
peak fitting. The constant, linear, quadratic, cubic, and logarithmic baselines
are linear and are fitted in a single step matrix solution. All fits use
least-squares minimizations.

The Best option fits and selects from all eight of these parametric models.

The Non-Parametric option uses a PeakFit-specific non-parametric

algorithm similar to Loess, but employing a Gaussian weighting function.
This algorithm can fit any shape baseline if defined by sufficient points. It is
the non-parametric algorithm used in PeakFit’s non-parametric Digital
Filter. Here the algorithm’s data window is fixed at 4 points and a linear
model is used.

The 2nd Deriv Zero fits are a multiple step process. In the first step, an
automatically smoothed second derivative is analyzed for constant zero
values. The Tol % specifies a second derivative tolerance whereby points are
judged to comprise the baseline. Typical tolerances should be from 0.5 to
5%. Generally, the greater the noise in the data, the greater will be the
required tolerance.

The AI Expert will seek to select an optimum tolerance based on detected

noise in the data.

In the second step, these points are fitted to the model indicated, or in the
case of Best or Non-Parametric, to all eight of the parametric models. For

AutoFit Baseline 6-3

Automated Peak Fitting

the Best and Non-Parametric options, the model with the best goodness of
fit (F-statistic) is selected.

The tolerance is then used again in a third step which involves the residuals
(the difference between the fitted curve and the data). Residuals within this
tolerance of the fitted curve are now marked as the active baseline points.
These are the active points shown in the graph.

The fourth and final step involves fitting this new set of baseline points to
the specified model, or in the case of Best, to all of the parametric models.
In the case of the Best option, it is at this stage the final model is
determined, again by F-statistic. This model may not be the same model as
that selected in the second step to produce the preliminary points.

The Numeric option can be used to view the fit statistics for the various
options. When all parametric models are fitted, this list may be helpful if the
selection of a different baseline is desired.

Baseline Graph The baseline produced is shown in a PeakFit graph. The graph will use a
temporary copy of the current PeakFit data. Points used for determining the
baseline will be active. All others will be grayed or in the inactive color,
indicating that they were not used in the baseline fit.

6-4 AutoFit Baseline

 Automated Peak Fitting

The Include Inactive Scaling button is particularly important for baselines

since this option must be on to see all of the points, and off for a scaling
based on only those points used to determine the baseline.

Tolerance If you see too many points present, and some are obviously within the decay
of peaks, you should decrease the tolerance. Simply enter the desired value,
set the value by the spin buttons, or right click and hold the mouse when
over the entry field or spin buttons selecting the desired value from the
popup menu.

If the baseline fit fails, or if you feel that too few points are included in the
baseline, you should increase the tolerance.

AI Expert The AI Expert will seek to set an optimum tolerance based on the measure
of noise detected within the data. Since a single point being fitted can rather
dramatically alter a baseline, this selected tolerance may not represent an
optimum.

Real-Time Point While the constant second derivative procedure is often very effective at
Selection and Fitting determining where a baseline exists, it is often easily improved upon by
human visual perception. You may see points that you know should be
included in the baseline, and points that you know should not be included.

PeakFit does not allow you to create an arbitrary visual baseline, but you are
able to toggle points on and off at will, and as you do so, the fit will be
automatically updated. Simply place the mouse cursor on the point desired
and click with the left mouse button.

You may also enable or disable a band of points using the same procedure as
that used in PeakFit’s Section option. To exclude active points, simply place
the mouse cursor at the initial point to be excluded, click and hold the left
mouse button down, and slide the mouse to the right. As you do so, the
points will be grayed (indicating that they have been deactivated or
excluded). Once the left mouse button is released, the new fit will be made
and the graph will be updated with the new baseline.

To reinstate a band of points, simply start at the rightmost point desired,

click and hold the left mouse button down, and slide the mouse to the left.
As you do so, the points will be changed to an active state and used in the
baseline determination.

AutoFit Baseline 6-5

Automated Peak Fitting

Zero Negative Subtracting a fitted baseline usually results in some small number of slightly
negative points.

These may adversely impact a subsequent fit, as very small peaks are
unfavorably altered in width or shape to compensate for a small negative
area. Also, there is the lack of aesthetics from the graph autoscaling which
will produce a negative Ymin in order to render these slightly negative points.
The Zero Negative option assures that all points are positive or zero. The
negative points are simply set to zero.

Subtracting the Once you are satisfied with the fitted baseline, simply click on OK. The
Baseline PeakFit data table will now reflect the removal of the fitted baseline, and the
zeroing of negative values, if selected. To abort the baseline procedure,
simply click on Cancel.

6-6 AutoFit Baseline

 Automated Peak Fitting

AutoFit Peaks Overview

PeakFit offers three AutoFit options for identifying and fitting peaks, each
of which is capable of finding hidden peaks. PeakFit defines hidden peaks as
those which do not evidence a local maximum within the data stream.

The AutoFit Peaks I Residuals option uncovers hidden peaks by looking at

residuals, the difference between the peaks identified by local maxima and
smoothed data.

The AutoFit Peaks II Second Derivative reveals hidden peaks by looking

for local minima in smoothed second derivative data.

The AutoFit Peaks III Deconvolution option “sharpens” the data by

Gaussian deconvolution and frequency domain filtering. Such refinement
often uncovers hidden peaks.

The PeakFit built-in functions are covered in detail in Chapter 7. These can
be generally categorized as:

• Spectroscopy Functions
• Chromatography Functions
• Statistical Functions
• Miscellaneous Peak Functions
• Transition Functions

Automated Placement The primary peak placement is fully automatic. Peaks are placed by one of
the aforementioned autoplacement criteria. Only a single peak model can be
automatically placed for all peaks. Whether all peak widths and shapes are
constant or varying is a simple matter of whether the Vary Widths and Vary
Shape options are enabled.

When the width or shape is allowed to vary, a peak refinement algorithm is

used to characterize each peak as exactly as possible from the raw data. Most
data sets require no more than adjusting the few controls which affect the
automated detection.

Graphical The secondary placement operation is fully graphical. Peaks are simply
Placement added by left clicking the mouse at the desired center and amplitude. Each
peak will have a primary anchor defining its amplitude and center.
Additional anchors will be present if widths or shapes are allowed to vary.

AutoFit Peaks Overview 6-7

Automated Peak Fitting

By clicking and holding down the left mouse button when on an anchor, a
peak is easily moved to a new X position, increased or decreased in
amplitude, changed in width, or modified in asymmetry or shape.

By simply left clicking on a peak’s primary anchor without movement, a

peak is toggled on and off.

A right click on the primary anchor opens a popup dialog where that specific
peak can be changed to a different function or deleted. Here you can also
access supplemental functions, such as transition equations, which can also
be graphically placed and adjusted. This secondary level of placement
requires no numerical knowledge of functions or function parameters.

Numerical Placement The tertiary peak placement is for special cases that cannot be graphically
placed and adjusted, such as UDFs and unusual models such as the Gamma
Ray and Compton functions. Here placement is by numerically adjusting the
parameter values.

A right click on the primary peak anchor opens the peak popup dialog. In
addition to offering every model available in PeakFit, this dialog offers the
means to adjust the numerical parameters of the model, and also to lock a
given parameter (its value will not change during fitting), or to share a given
parameter (its value is shared with all others at this parameter position also
marked as shared). PeakFit also offers a common parameters dialog which
allows a given parameter to be simultaneously set for all peaks.

Recommendations If each peak in the data is readily identified as having a local maximum in the
data stream (there are no hidden peaks), you should use the AutoFit Peaks I
Residuals option. In this instance the Add Residuals checkbox should be
unchecked.

When hidden peaks are present, it is difficult to say which algorithm will be
most successful in automatically identifying them. In many cases, all three
will successfully uncover the hidden peaks, and the choice is largely a matter
of taste. Still, the procedures are numbered in accordance with their
simplicity and how forgiving they are in their use.

The AutoFit Peaks I Residuals option is the most basic but often the most
effective of the algorithms. It is the only procedure that is unaffected by data
lacking uniformly spaced X-values, provided the Loess procedure is used for
smoothing. It is also quite forgiving in terms of smoothing levels. The
algorithm adjusts the widths of peaks so that the overall area of the placed

6-8 AutoFit Peaks Overview

 Automated Peak Fitting

peaks is equal to that of the raw data. When hidden peaks are present, this
tends to produce negative residuals where peaks have been found, and
positive residuals where hidden peaks exist. Residuals peaks that exceed a
simple user-adjustable amplitude threshold are automatically added when the
Add Residuals is checked.

The AutoFit Peaks II Second Derivative option uses the Savitzky-Golay

algorithm to produce a smooth second derivative. As such, a constant X
spacing is required. If data lacks this uniform X-spacing, or if artifacts or
outliers have been excluded, this option can still be used, provided the data
is first processed with PeakFit’s non-parametric Digital Filter. This
procedure is more demanding of smoothing levels, since second derivative
peaks (local minima) can be “washed out” by excessive smoothing, and since
an ideally smoothed second derivative will require a greater smoothing
power than the raw data. This algorithm locates peaks exclusively from local
minima in the smoothed second derivative data.

The AutoFit Peaks III Deconvolution option is the least forgiving of the
procedures. Since the deconvolution and filtration take place in the
frequency domain, this algorithm also requires a uniform X-spacing. Lacking
this constant X-spacing, or if artifacts or outliers have been excluded from
the data, uniform spacing can be restored with PeakFit’s non-parametric
Digital Filter. This procedure requires a response function width that is
sufficient to produce a significant sharpening of the peaks so as to uncover
hidden peaks. Such widths will also push the deconvolution to the limits of
numeric stability, since deconvolution introduces a great deal of noise into
the data. Fourier domain filtration is critical, and this algorithm determines
such automatically. This automatic filter level determination is subject to
failure when response function widths start to approach the FWHM of the
peaks. This option locates peaks exclusively from local maxima in the
deconvolved data. A hidden peak in the input data stream must become a
peak with a local maximum in the deconvolved data in order to be
automatically detected.

AutoFit Peaks Overview 6-9

Automated Peak Fitting

AutoFit Peaks I Residuals

The AutoFit Peaks I Residuals option is one of three AutoFit Peaks
options offered by PeakFit.

This is the most basic of the options since it first identifies peaks by
identifying local maxima in a smoothed data stream. This first step finds
only those peaks which are visible peaks. PeakFit defines a visible peak as
one which produces a local maximum in the data stream. A hidden peak is
thus defined as one which fails to produce this local maximum. Note that
peaks which lack this local maximum may range from being quite apparent
in visual inspection to those which are altogether indiscernible.

Hierarchy of The control panel in the dialog contains the automated options. It is
Processing arranged in a hierarchical order. The change to any option will affect all
options below it. For example, changing the baseline results in baseline
processing, resmoothing, rescanning, and discarding all custom graphical or
numerical adjustments. Changing a scan setting results only in a rescan and
the discarding of any custom adjustments.

Note that changing any of the baseline, smoothing, function, or scanning

options in the control panel, or importing a file-based scan setup, results in a
complete rebuilding of the autoscan. All custom adjustments and placements
are lost. If you spend any time creating a custom placement, you should
explicitly save it as soon as you are satisfied with it. While all of the AutoFit
control panel’s settings are saved across sessions, custom placements must
be explicitly saved.

As If Processing Even though this option will present smooth baseline-subtracted data, what
you see is only for the purpose of placing peaks. The data that is fitted is
always the PeakFit data table present when this option is invoked.

The only exception is the local sectioning option which permits points or
bands of points to be temporarily enabled or disabled. Even in this case,
data values remain unchanged as only the active state of points is affected.

Baseline Fitting This option does not subtract a baseline prior to fitting. Any baseline selected in this
option is added to the overall fitted model. While this option does pre-fit a
baseline (if one is selected), the parameters derived from this baseline fitting
are used as initial parameter estimates for the baseline in the peak fitting.
These parameters are also used to create a baseline-subtracted temporary

6-10 AutoFit Peaks I Residuals

 Automated Peak Fitting

copy of the input data. This exists only to make the task of placing peaks
simpler.

To subtract a baseline from the data prior to fitting, you must use either the
AutoFit and Subtract Baseline for fitting the baseline in the current data,
or the Import and Subtract Baseline for removing a baseline that has been
externally generated.

Smoothing This option does not smooth data prior to fitting. All of the AutoFit options require
smooth curves for the detection of local minima or maxima. As such, the
smoothing observed in this option bears an important role in peak detection.
However, only a copy of the input data is smoothed, and only for the
purpose of detecting and placing peaks. The data to be fitted is not
smoothed in any way.

To smooth data prior to fitting, you must use the Smooth or Fourier
Domain Editing options.

Graphical Layout This option will display two PeakFit graphs. Each will have both a Y and Y2
axis.

AutoFit Peaks I Residuals 6-11

Automated Peak Fitting

The upper graph is common to all three AutoFit Peaks options. Its Y axis
will contain the component peaks and a copy of the input data,
baseline-subtracted using the current parameter estimates for the baseline. If
no baseline is being fitted, this data will consist of the input data. The Y2
axis will contain this same data along with the smoothed version of such.
The defaults will show the unsmoothed data as points and the smoothed
data as connected lines. The Y2 graph also displays the sum curve for the
constituent peaks. The baseline is not included in the sum curve.

The lower graph is unique to this AutoFit Peaks I Residuals option. The
Y2 axis will contain the residuals, the difference between the smoothed data
in the upper graph and the sum curve. The defaults display the data in bar
graph form. The Y axis will consist of the raw input data, modified to reflect
the active states of points used for the baseline estimation. The Y axis will
also display the baseline. If no baseline is fitted, the Y axis will plot the raw
and smoothed data.

In the graphs on the prior page, the Add Residuals option was off, and as
such the upper graph reflects only those peaks with local maxima. The
residuals plot in the lower graph similarly reflects the state where no residual
peaks have yet been added. In this instance, the residuals clearly reveal four
hidden peaks. To add the residuals peaks with magnitudes greater than the
Amp % threshold, simply check the Add Residuals option. The dotted line
in each graph reflects this amplitude threshold for peak acceptance.

Zoom-In As with most PeakFit graphs, zoom-in is a simple matter of left clicking the
mouse at one corner of the desired zoom region, holding the mouse button
down, and moving to the opposite corner of the desired region. If you are
zooming in from the constituent peak plot, you must not begin the zoom on
a function anchor. To restore default scaling, you can click on the Reset
Default Scaling button, or you can right click anywhere within the plot area,
except on a function anchor.

Level 1 of Peak The primary peak placement is fully automatic. This may be the only
Placement - Automatic placement you need concern yourself with if:

• You are fitting a common peak model to all peaks.

• You are fitting one of the built-in baseline models, or no baseline at all.

• You are satisfied that the automatic detection has correctly identified all
visible and hidden peaks present in the data.

6-12 AutoFit Peaks I Residuals

 Automated Peak Fitting

• You are willing to allow all peak widths to vary in the fitting or wish to
have all peak widths constant (shared).
• You are willing to allow all peak shape parameters to vary in the fitting or
wish to have all peak shape parameters constant (shared).

To have all peak widths (the a2 parameter of each model) held constant, the
Vary Widths should be unchecked. Frequently in chromatography and
spectroscopy, peak widths will increase with increasing time or wave
number. In such instances, and where peak widths are obviously not
constant, you should check the Vary Widths checkbox.

For the parameters which determine a peak’s shape (parameters a3 and

above), the choice may not be as simple, since it is not always visually
apparent whether or not peaks share a constant shape (asymmetry,
Gaussian-Lorentzian character, etc. ). Ideally, one would like a model and
data where the Vary Shape can remain unchecked. This simplifies the fit
considerably, resulting in faster fitting, more stable fitting, and usually better
confidence limits for the fitted parameters. If the F-statistic for a fit with the
Vary Shape on is higher than that with the Vary Shape off, there is good
indication that the peak shape is not constant. For very noisy data with
hidden or overlapping peaks, it is also unlikely that sufficient information
exists within the data to allow these shape parameters to vary.

Steps for Automated 1. Select a baseline to be fitted. If you have subtracted a baseline or wish to
Fitting fit no baseline, choose the No Baseline selection. The baseline options are
identical to those offered in the AutoFit and Subtract Baseline option.
Note that if you select No Baseline and some positive baseline is present,
the scan algorithm may detect a large number of small amplitude peaks
between zero and this baseline.

2. If you have selected a baseline to be fitted, adjust the baseline Tol % so

that the active points in the Y axis of the lower graph represent a meaningful
baseline. Note that this need not be particularly accurate, merely close
enough for peak detection. You may enter the baseline tolerance percent
directly, use the spin buttons, or right click and hold on the field or spin
buttons and select the value desired.

The Include Inactive Scaling button is particularly important for baselines

since this option must be on to see all of the points, and off for a scaling
based on only those points used to determine the baseline.

AutoFit Peaks I Residuals 6-13

Automated Peak Fitting

3. Select a smoothing algorithm. Unless you have non-uniform data and

need the Loess procedure, the default Savitzky-Golay procedure should be
more than adequate. As with the baseline estimation, the smoothing need
only be sufficient for peak identification and estimation. The smoothing
options are the same as those in PeakFit’s Smooth option.

4. Set the smoothing level. You may enter the smoothing level directly, use
the spin buttons, or right click and hold on the field or spin buttons and
select the value desired.

You may also click on the AI Expert option to have it select an optimum
smoothing level for you.

The peak scan algorithm does not use a width rejection criteria. Rather are
all local maxima detected as peaks. It is thus important to smooth
sufficiently to remove all noise perturbations which would appear as peaks,
but not so much as to wash out data features where peaks are present. The
smoothing is graphically shown in the Y2 plot of the upper graph (and also
in the Y plot of the lower graph when no baseline is being fitted). Generally,
there is a rather wide range of acceptable smoothing levels.

5. Select the peak function family desired. Spectroscopy, Chromatography,

Statistical, Miscellaneous, and All Peak Functions are options. The last
option will contain all of the peak functions that can be automatically placed.
The functions are covered in detail in Chapter 7.

6. Select the peak type desired. PeakFit offers over 70 built-in peak
functions which can be automatically and graphically placed. The sum of
squared residuals (SSE) may be of some value in peak selection. Note that
this value reflects only PeakFit’s ability to place the function, and not what
the function is capable of achieving in a least-squares fit.

The Peak Labels button offers the means to display center, amplitude, area,
or scan extrema above the individual peaks. Such labels are often very useful
in identifying peaks of small amplitude.

7. Select the amplitude rejection threshold (Amp % ) desired. This should

be high enough to discard the local maxima peaks that appear within and
about the baseline, but low enough to include the smallest of desired peaks.
You may enter the percent directly, use the spin buttons, or right click and
hold on the field or spin buttons and select the value desired.

6-14 AutoFit Peaks I Residuals

 Automated Peak Fitting

The amplitude thresholds appear as dotted lines in the graphs. These can be
toggled off if they are not desired.

8. For hidden peaks, check Add Residuals to add all residuals peaks above
the amplitude threshold limit.

9. To vary the widths in fitting, check the Vary Widths checkbox. An

additional level of autoscan will seek to carefully refine the widths for each
of the peaks where a FWHM (full width at half-maximum) is clearly defined.
Functions will have anchors at the half-maxima positions for graphical
adjustment of widths.

10. To vary the shape parameter(s) in fitting, if available, check the Vary
Shape checkbox. An additional level of autoscan will seek to carefully refine
the shape for each of the peaks where either a half-maxima asymmetry or
FW25 (full width at 25% maximum) is clearly defined. Asymmetric
functions will now have anchors at the half-maxima positions for graphical
adjustment of asymmetry. Symmetric functions will have additional anchors
at 25% half maxima for graphical shape adjustment.

11. If a shape parameter is available within the selected peak mode, and if
you are unsatisfied with the shape of the placed peaks, you may wish to
check the Refine Shape option. This performs a one-dimensional
minimization where a single graphical shape (asymmetry or FW25/FWHM)
is set for all peaks and varied until a minimum chi-square is found. The
algorithm uses a least-absolute deviation minimization.

12. If satisfied with the automatic placement, you can begin the PeakFit
automated fitting:

• Peak Fit with full graphical update

• Peak Fit with fast numerical update

If you are unable to achieve the desired placement automatically, or if the fit
produced unacceptable results, you can then go on to the secondary level of
peak placement.

In the case of an unsatisfactory fit, before resorting to graphical adjustment,

you may wish to explore the Peak Fit Preferences. PeakFit’s defaults use
somewhat rigid constraints on the initial three parameters to insure fitted
peaks remain very close to where they were initially placed. This is especially
critical when fitting a large number of peaks. If you are having difficulty with

AutoFit Peaks I Residuals 6-15

Automated Peak Fitting

a given fit, and constraints are shown as being violated during the fit, you
may wish to widen these constraints, or disable them altogether.

Level 2 of Peak The secondary placement operation is fully graphical. This is done within the
Placement - Graphical components plot of the upper graph. This secondary level of placement
requires no numerical knowledge of functions or function parameters.

Each peak will have a primary anchor defining its amplitude and center.
Additional anchors will be present if widths or shapes are allowed to vary.

Note that Peak Anchors must be selected in order for these anchors to be
displayed.

Graphically Modifying A peak is added simply by left clicking the mouse at the desired center and
Peaks amplitude. This must be done in the components plot of the upper graph.

A single residuals peak can be added by left clicking the mouse in the
residuals graph on or near the residuals peak desired. A residuals peak that is
added in this way bypasses the amplitude constraint.

A peak is deleted by right clicking on the peak’s principal anchor and then
clicking on the Delete Peak button in the Peak Popup dialog.

A peak is toggled on and off by left clicking the mouse on a peak’s principal
anchor. An inactive peak is not used in fitting. Toggling a peak on and off
may be a good way to see its approximate contribution to the overall data.

A peak is graphically adjusted by clicking and holding down the left mouse
button when on an anchor and moving it to a new position. Moving the
primary anchor results in a new center and amplitude. Moving a
half-maxima anchor on a symmetric peak changes the overall peak width.
On an asymmetric peak, moving a half-maxima anchor changes both the
width and asymmetry. On symmetric peaks with shape parameters, the
quarter maxima anchors adjust the FW25 (full width at 25% maximum),
thus changing the peak shape.

An individual peak can be changed to a different function by right clicking

on the principal anchor and selecting a different function from the popup
dialog. here you can also access supplemental functions, such as transition
equations, which can also be graphically placed and adjusted.

6-16 AutoFit Peaks I Residuals

 Automated Peak Fitting

Level 3 of Peak Numerical peak placement is for special cases that cannot be graphically
Placement - Numerical placed and adjusted, such as UDFs and unusual models such as the Gamma
Ray and Compton functions. Here placement is by numerically adjusting the
parameter values.

To modify an existing function, simply right click on the primary anchor. If

desired, a different function can be selected from the popup dialog.

To create a new function, add a peak at a desired anchor position by clicking

the mouse at this X,Y position. This position need not bear any relation to
the peak or data feature it represents, but it is where you must go to access
numerical adjustment for the function. The function will initially be the peak
model currently selected in the dialog’s control panel. Right click the new
anchor, and select the desired function from the popup dialog.

Certain functions that can only be placed numerically are included within the
function list, including all currently active UDFs, the Gamma Ray model,
and the Compton Edge model.

To adjust the function numerically, simply change the value of the desired
parameter(s). You may lock a given parameter (its value will not change
during fitting), or you may share a given parameter (its value is shared with
all others at this parameter position also marked as shared). Note that you
must use your own judgment when sharing parameters. For example, it
would normally make no sense to share an amplitude or center value, or to
share width or shape parameters across different peak types.

PeakFit also offers a Common Parameters dialog which allows a given

parameter to be simultaneously set for all peaks. This is the most convenient
way to set a common width or shape parameter across all peaks, or across
any number of selected peaks.

Local Sectioning If a large spectrum or chromatogram has large regions of baseline between
groups of peaks, you may see better fit statistics from fitting the individual
partitions. There may also be instances where you wish to isolate and fit a
single peak to perhaps several models to see which might be most favorable.

Rather than exit the AutoFit Peaks option and section the data from the
main menu, this option allows for a local or temporary sectioning, one that
disables or enables data points only for the duration of the AutoFit
procedure.

AutoFit Peaks I Residuals 6-17

Automated Peak Fitting

Note that sectioning the data results in a complete rebuilding of the

autoscan. All custom adjustments and placements are lost. If you have spent
any time creating a custom placement that you may wish to use with some
future data set, you should explicitly Save it before sectioning.

This sectioning is identical to that offered in the Prepare menu’s Section

option.

Reset The Reset option rescans the data discarding all custom placements. Note
that this is not a full reset, in that control panel settings remain at their
current settings.

If you have spent any time creating a custom placement that you may wish
to use with some future data set, you should explicitly Save it before
sectioning.

This option is used after importing a scan when you wish to discard the
specific peak estimates, and rebuild the scan from the AutoFit control panel
settings in the scan file. This option is also used to discard unwanted
graphical or numerical adjustments.

Toggle Lower Graph In order to better see the constituent peaks, you may wish to remove or
decrease the size of the Residuals and Baseline graph. This toggle works in
two steps. The first decreases the graph to a reduced size and the second
hides it altogether. An additional click restores the equal graph sizing.

List Peak Estimates This option opens a PeakFit text window which displays the scan
characteristics and the peak estimates from the Autoscan When peaks are
cleanly defined, and widths are permitted to vary, PeakFit’s autoscan may
complement other analytical methods. PeakFit’s autoscan uses sophisticated
built-in correlations that map a peaks function’s graphical state to its
parameter estimates. As such, the FWHM and areas often tend to be quite
accurate.

If the Addl Adjust option is selected after fitting, this list will contain
updated peak information based upon the fit. This should not be considered
a replacement for the Numeric Summary in the Review.

Read Scan Setup and This option imports a PeakFit scan file [SCN] that has been saved to disk. A
Parameter Estimates PeakFit scan file contains all AutoFit control panel settings, all peak

6-18 AutoFit Peaks I Residuals

 Automated Peak Fitting

adjustments and customizations, graphical and numerical, and also a UDF

library, if any UDFs have been used in the current placement.

Importing a scan updates the AutoFit control panel’s settings, but does not
trigger an autoscan. Rather are the peaks placed exactly as they were when
the file was saved, including all graphical and numerical adjustments. If
UDFs are used, the internal UDF library is read into the current UDF
positions, replacing any active UDFs.

If you wish to discard custom settings and rebuild the autoscan, simply click
the Reset button.

Save Scan Setup and This option saves the current AutoFit control panel settings, the current
Parameter Estimates peak placement and parameters, and any UDFs to an [SCN] disk file. A scan
file permits any peak placement to be immediately reproduced.

Modify Peak Fit This opens the PeakFit Preferences dialog. These preferences control
Preferences adjustable elements within PeakFit’s non-linear fitting engine. Some of these
options, such as the Robust Minimization and Built-In Peak Fn
Constraints can dramatically affect the fit results. Others such as Fit Extent
and the Curvature Matrix options can significantly affect the time required
for fitting large data sets when many peaks are being fitted.

Fast Peak Fit with This button initiates PeakFit’s Numerical Fitting.
Numeric Update

Full Peak Fit with This button initiates PeakFit’s Graphical Fitting.
Graphical Update

Review Peak Fit This button open’s PeakFit’s Review of the fit. If the current data is not
fitted, either because it was never fitted, or because a fit was invalidated by
subsequent adjustments, the Review is still available although you must
confirm you wish to treat the current state of the peaks as if a fit had
occurred.

AutoFit Peaks I Residuals 6-19

Automated Peak Fitting

AutoFit Peaks II Second Derivative

The AutoFit Peaks II Second Derivative option is one of three AutoFit
Peaks options offered by PeakFit.

In this option, hidden peaks are detected by second derivative minima. A

peak that has no local maximum in the primary data, may very well have a
local minima in a smoothed second derivative.

PeakFit defines a visible peak as one which produces a local maximum in the
input data. A hidden peak is thus defined as one which fails to produce this
local maximum. Peaks which lack this local maximum may range from being
quite apparent in visual inspection to those which are altogether
indiscernible.

This option uses the Savitzky-Golay algorithm to produce a smooth second

derivative. As such, a constant X spacing is required. If data lacks this
uniform X-spacing, or if artifacts or outliers have been excluded, this option
can still be used, provided the data is first processed with PeakFit’s
non-parametric Digital Filter.

This procedure is more demanding of smoothing levels, since second

derivative peaks (local minima) can be “washed out” by excessive
smoothing, and since an ideally smoothed second derivative will require a
greater smoothing power than the raw data. This algorithm locates peaks
exclusively from local minima in the smoothed second derivative data.

Hierarchy of The control panel in the dialog contains the automated options. It is
Processing arranged in a hierarchical order. The change to any option will affect all
options below it. For example, changing the baseline results in baseline
processing, resmoothing, rescanning, and discarding all custom graphical or
numerical adjustments. Changing a scan setting results only in a rescan and
the discarding of any custom adjustments.

Note that changing any of the baseline, smoothing, function, or scanning

options in the control panel, or importing a file-based scan setup, results in a
complete rebuilding of the autoscan. All custom adjustments and placements
are lost. If you spend any time creating a custom placement, you should
explicitly Save it as soon as you are satisfied with it. While all of the AutoFit
control panel’s settings are saved across sessions, custom placements must
be explicitly saved.

6-20 AutoFit Peaks II Second Derivative

 Automated Peak Fitting

As If Processing Even though this option will present smooth baseline-subtracted data, what
you see is only for the purpose of placing peaks. The data that is fitted is
always the PeakFit data table present when this option is invoked.

The only exception is the local sectioning option which permits points or
bands of points to be temporarily enabled or disabled. Even in this case,
data values remain unchanged as only the active state of points is affected.

Baseline Fitting This option does not subtract a baseline prior to fitting. Any baseline selected in this
option is added to the overall fitted model. While this option does pre-fit a
baseline (if one is selected), the parameters derived from this baseline fitting
are used as initial parameter estimates for the baseline in the peak fitting.
These parameters are also used to create a baseline-subtracted temporary
copy of the input data. This exists only to make the task of placing peaks
simpler.

To subtract a baseline from the data prior to fitting, you must use either the
AutoFit and Subtract Baseline for fitting the baseline in the current data,
or the Import and Subtract Baseline for removing a baseline that has been
externally generated.

Smoothing This option does not smooth data prior to fitting. All of the AutoFit options require
smooth curves for the detection of local minima or maxima. As such, the
smoothing observed in this option bears an important role in peak detection.
However, only a copy of the input data is smoothed, and only for the
purpose of detecting and placing peaks. The data to be fitted is not
smoothed in any way.

To smooth data prior to fitting, you must use the Smooth or Fourier
Domain Editing options.

AutoFit Peaks II Second Derivative 6-21

Automated Peak Fitting

Graphical Layout This option will display two PeakFit graphs. Each will have both a Y and Y2
axis.

The upper graph is common to all three AutoFit options. Its Y axis will
contain the component peaks and a copy of the input data,
baseline-subtracted using the current parameter estimates for the baseline. If
no baseline is being fitted, this data will consist of the input data.

The Y2 axis will contain this same data along with a smoothed version of
such. This smoothed data also uses the Savitzky-Golay algorithm at the
current settings, but does not smooth to a derivative. It is furnished for
reference purposes only, to assess the appropriateness of the smoothing
level. The defaults will show the unsmoothed data as points and the
smoothed data as connected lines. The Y2 graph also displays the sum curve
for the constituent peaks. The baseline is not included in the sum curve.

The lower graph is unique to this AutoFit II Second Derivative option.

The Y2 axis will contain the smoothed second derivative. The defaults
display the data as connected lines. The Y axis will consist of the raw input
data, modified to reflect the active states of points used for the baseline
estimation. The Y axis will also display the baseline. If no baseline is fitted,
the Y axis will plot the raw and smoothed data.

6-22 AutoFit Peaks II Second Derivative

 Automated Peak Fitting

In the graphs on the previous page, the four hidden peaks which evidence
no local maxima in the smoothed input data, show quite definite local
minima in the smoothed second derivative. All second derivative minima are
added peaks if the corresponding amplitudes are greater than the Amp %
threshold. The dotted line in the upper graph reflects this amplitude
threshold for peak acceptance.

Zoom-In As with most PeakFit graphs, zoom-in is a simple matter of left clicking the
mouse at one corner of the desired zoom region, holding the mouse button
down, and moving to the opposite corner of the desired region. If you are
zooming in from the constituent peak plot, you must not begin the zoom on
a function anchor.

To restore default scaling, you can click on the Reset Default Scaling button,
or you can right click anywhere within the plot area, except on a function
anchor.

Level 1 of Peak The primary peak placement is fully automatic. This may be the only
Placement - Automatic placement you need concern yourself with if:

• You are fitting a common peak model to all peaks.

• You are fitting one of the built-in baseline models, or no baseline at all.
• You are satisfied that the automatic detection has correctly identified all
visible and hidden peaks present in the data.
• You are willing to allow all peak widths to vary in the fitting or wish to
have all peak widths constant (shared).
• You are willing to allow all peak shape parameters to vary in the fitting or
wish to have all peak shape parameters constant (shared).

To have all peak widths (the a2 parameter of each model) held constant, the
Vary Widths should be unchecked. Frequently in chromatography and
spectroscopy, peak widths will increase with increasing time or wave
number. In such instances, and where peak widths are obviously not
constant, you should check the Vary Widths checkbox.

For the parameters which determine a peak’s shape (parameters a3 and

above), the choice may not be as simple, since it is not always visually
apparent whether or not peaks share a constant shape (asymmetry,
Gaussian-Lorentzian character, etc. ). Ideally, one would like a model and
data where the Vary Shape can remain unchecked. This simplifies the fit

AutoFit Peaks II Second Derivative 6-23

Automated Peak Fitting

considerably, resulting in faster fitting, more stable fitting, and usually better
confidence limits for the fitted parameters. If the F-statistic for a fit with the
Vary Shape on is higher than that with the Vary Shape off, there is good
indication that the peak shape is not constant. For very noisy data with
hidden or overlapping peaks, it is also unlikely that sufficient information
exists within the data to allow these shape parameters to vary.

Steps for Automated 1. Select a baseline to be fitted. If you have subtracted a baseline or wish to
Fitting fit no baseline, choose the No Baseline selection. The baseline options are
identical to those offered in the AutoFit and Subtract Baseline option.
Note that if you select No Baseline and some positive baseline is present,
the scan algorithm may detect a large number of small amplitude peaks
between zero and this baseline.

2. If you have selected a baseline to be fitted, adjust the baseline Tol % so

that the active points in the Y axis of the lower graph represent a meaningful
baseline. Note that this need not be particularly accurate, merely close
enough for peak detection. You may enter the baseline tolerance percent
directly, use the spin buttons, or right click and hold on the field or spin
buttons and select the value desired.

The Include Inactive Scaling button is particularly important for baselines

since this option must be on to see all of the points, and off for a scaling
based on only those points used to determine the baseline.

3. Select a Savitzky-Golay smoothing level for the smoothed second

derivative in the lower graph and the reference data in the upper graph. You
may enter the smoothing level directly, use the spin buttons, or right click
and hold on the field or spin buttons and select the value desired. The
smoothing needs to be sufficient for peak identification and estimation. The
Savitzky-Golay smoothing used here is the same as that offered in PeakFit’s
Smooth option.

You may also click on the AI Expert option to have it seek an optimum
smoothing level for you. Given the narrow range of acceptable smoothing
levels sometimes required with second derivatives, you may need to refine
this level somewhat.

The peak scan algorithm does not use a width rejection criteria. Rather are
all local second derivative minima detected as peaks. It is thus important to
smooth sufficiently to remove all noise perturbations which would appear as
peaks, but not so much as to wash out second derivative data features where

6-24 AutoFit Peaks II Second Derivative

 Automated Peak Fitting

peaks are indicated. A smoothed reference data set using the current
smoothing level, is shown in the Y2 plot of the upper graph (and also in the
Y plot of the lower graph when no baseline is being fitted). This should
assist in determining when undersmoothing or oversmoothing is present.

4. Select the peak function family desired. Spectroscopy, Chromatography,

Statistical, Miscellaneous, and All Peak Functions are options. The last
option will contain all of the peak functions that can be automatically placed.
The functions are covered in detail within Chapter 7.

5. Select the peak type desired. PeakFit offers over 70 built-in peak
functions which can be automatically and graphically placed. The sum of
squared residuals (SSE) may be of some value in peak selection. Note that
this value reflects only PeakFit’s ability to place the function, and not what
the function is capable of achieving in a least-squares fit.

The Peak Labels button offers the means to display center, amplitude, area,
or scan extrema above the individual peaks. Such labels are often very useful
in identifying peaks of small amplitude.

6. Select the amplitude rejection threshold (Amp % ) desired. This should

be high enough to discard the local maxima peaks that appear within and
about the baseline, but low enough to include the smallest of desired peaks.
You may enter the percent directly, use the spin buttons, or right click and
hold on the field or spin buttons and select the value desired.

The amplitude thresholds appear as dotted lines in the graphs. These can be
toggled off if they are not desired.

7. To vary the widths in fitting, check the Vary Widths checkbox. An

additional level of autoscan will seek to carefully refine the widths for each
of the peaks where a FWHM (full width at half-maximum) is clearly defined.
Functions will have anchors at the half-maxima positions for graphical
adjustment of widths.

8. To vary the shape parameter(s) in fitting, if available, check the Vary

Shape checkbox. An additional level of autoscan will seek to carefully refine
the shape for each of the peaks where either a half-maxima asymmetry or
FW25 (full width at 25% maximum) is clearly defined. Asymmetric
functions will now have anchors at the half-maxima positions for graphical
adjustment of asymmetry. Symmetric functions will have additional anchors
at 25% half maxima for graphical shape adjustment.

AutoFit Peaks II Second Derivative 6-25

Automated Peak Fitting

9. If a shape parameter is available within the selected peak mode, and if you
are unsatisfied with the shape of the placed peaks, you may wish to check
the Refine Shape option. This performs a one-dimensional minimization
where a single graphical shape (asymmetry or FW25/FWHM) is set for all
peaks and varied until a minimum chi-square is found. The algorithm uses a
least-absolute deviation minimization.

10. If satisfied with the automatic placement, you can begin the PeakFit
automated fitting:

• Peak Fit with full graphical update

• Peak Fit with fast numerical update

If you are unable to achieve the desired placement automatically, or if the fit
produced unacceptable results, you can then go on to the secondary level of
peak placement.

In the case of an unsatisfactory fit, before resorting to graphical adjustment,

you may wish to explore the Peak Fit Preferences. PeakFit’s defaults use
somewhat rigid constraints on the initial three parameters to insure fitted
peaks remain very close to where they were initially placed. This is especially
critical when fitting a large number of peaks. If you are having difficulty with
a given fit, and constraints are shown as being violated during the fit, you
may wish to widen these constraints, or disable them altogether.

Level 2 of Peak The secondary placement operation is fully graphical. This is done within the
Placement - Graphical components plot of the upper graph. This secondary level of placement
requires no numerical knowledge of functions or function parameters.

Each peak will have a primary anchor defining its amplitude and center.
Additional anchors will be present if widths or shapes are allowed to vary.

Note that Peak Anchors must be selected in order for these anchors to be
displayed.

Graphically Modifying A peak is added simply by left clicking the mouse at the desired center and
Peaks amplitude. This must be done in the components plot of the upper graph.

A peak is deleted by right clicking on the peak’s principal anchor and then
clicking on the Delete Peak button in the Peak Popup dialog.

6-26 AutoFit Peaks II Second Derivative

 Automated Peak Fitting

A peak is toggled on and off by left clicking the mouse on a peak’s principal
anchor. An inactive peak is not used in fitting. Toggling a peak on and off
may be a good way to see its approximate contribution to the overall data.

A peak is graphically adjusted by clicking and holding down the left mouse
button when on an anchor and moving it to a new position. Moving the
primary anchor results in a new center and amplitude. Moving a
half-maxima anchor on a symmetric peak changes the overall peak width.
On an asymmetric peak, moving a half-maxima anchor changes both the
width and asymmetry. On symmetric peaks with shape parameters, the
quarter maxima anchors adjust the FW25 (full width at 25% maximum),
thus changing the peak shape.

An individual peak can be changed to a different function by right clicking

on the principal anchor and selecting a different function from the popup
dialog. here you can also access supplemental functions, such as transition
equations, which can also be graphically placed and adjusted.

Level 3 of Peak Numerical peak placement is for special cases that cannot be graphically
Placement - Numerical placed and adjusted, such as UDFs and unusual models such as the Gamma
Ray and Compton functions. Here placement is by numerically adjusting the
parameter values.

To modify an existing function, simply right click on the primary anchor. If

desired, a different function can be selected from the popup dialog.

To create a new function, add a peak at a desired anchor position by clicking

the mouse at this X,Y position. This position need not bear any relation to
the peak or data feature it represents, but it is where you must go to access
numerical adjustment for the function. The function will initially be the peak
model currently selected in the dialog’s control panel. Right click the new
anchor, and select the desired function from the popup dialog.

Certain functions that can only be placed numerically are included within the
function list, including all currently active UDFs, the Gamma Ray model,
and the Compton Edge model.

To adjust the function numerically, simply change the value of the desired
parameter(s). You may lock a given parameter (its value will not change
during fitting), or you may share a given parameter (its value is shared with
all others at this parameter position also marked as shared). Note that you
must use your own judgment when sharing parameters. For example, it

AutoFit Peaks II Second Derivative 6-27

Automated Peak Fitting

would normally make no sense to share an amplitude or center value, or to

share width or shape parameters across different peak types.

PeakFit also offers a Common Parameters dialog which allows a given

parameter to be simultaneously set for all peaks. This is the most convenient
way to set a common width or shape parameter across all peaks, or across
any number of selected peaks.

Local Sectioning When a large spectrum or chromatogram has large regions of baseline
between groups of peaks, you may see better fit statistics from fitting the
individual partitions. There may also be instances where you wish to isolate
and fit a single peak to perhaps several models to see which might be most
favorable.

Rather than exit the AutoFit Peaks option and section the data from the
main menu, this option allows for a local or temporary sectioning, one that
disables or enables data points only for the duration of the AutoFit
procedure.

Note that sectioning the data results in a complete rebuilding of the

autoscan. All custom adjustments and placements are lost. If you have spent
any time creating a custom placement that you may wish to use with some
future data set, you should explicitly Save it before sectioning.

This sectioning is identical to that offered in PeakFit’s Section option.

The Reset option rescans the data discarding all custom placements. Note
Reset that this is not a full reset, in that control panel settings remain at their
current settings.

If you have spent any time creating a custom placement that you may wish
to use with some future data set, you should explicitly Save it before
sectioning.

This option is used after importing a scan when you wish to discard the
specific peak estimates, and rebuild the scan from the AutoFit control panel
settings in the scan file. This option is also used to discard unwanted
graphical or numerical adjustments.

Toggle Lower Graph In order to better see the constituent peaks, you may wish to remove or
decrease the size of the Second Derivatives and Baseline graph. This toggle
works in two steps. The first decreases the graph to a reduced size and the

6-28 AutoFit Peaks II Second Derivative

 Automated Peak Fitting

second hides it altogether. An additional click restores the equal graph

sizing.

List Peak Estimates This option opens a PeakFit text window which displays the scan
characteristics and the peak estimates from the autoscan. When peaks are
cleanly defined, and widths are permitted to vary, PeakFit’s autoscan may
complement other analytical methods. PeakFit’s autoscan uses sophisticated
built-in correlations that map a peaks function’s graphical state to its
parameter estimates. As such, the FWHM and areas often tend to be quite
accurate.

If the Addl Adjust option is selected after fitting, this list will contain
updated peak information based upon the fit. This should not be considered
a replacement for the Numeric Summary in the Review.

Read Scan Setup and This option imports a PeakFit scan file [SCN] that has been saved to disk. A
Parameter Estimates PeakFit scan file contains all AutoFit control panel settings, all peak
adjustments and customizations, graphical and numerical, and also a UDF
library, if any UDFs have been used in the current placement.

Importing a scan updates the AutoFit control panel’s settings, but does not
trigger an autoscan. Rather are the peaks placed exactly as they were when
the file was saved, including all graphical and numerical adjustments. If
UDFs are used, the internal UDF library is read into the current UDF
positions, replacing any active UDFs.

If you wish to discard custom settings and rebuild the autoscan, simply click
the Reset button.

Save Scan Setup and This option saves the current AutoFit control panel settings, the current
Parameter Estimates peak placement and parameters, and any UDFs to an [SCN] disk file. A scan
file permits any peak placement to be immediately reproduced.

Modify Peak Fit This opens the PeakFit Preferences dialog. These preferences control
Preferences adjustable elements within PeakFit’s non-linear fitting engine. Some of these
options, such as the Robust Minimization and Built-In Peak Fn
Constraints can dramatically affect the fit results. Others such as Fit Extent
and the Curvature Matrix options can significantly affect the time required
for fitting large data sets when many peaks are being fitted.

AutoFit Peaks II Second Derivative 6-29

Automated Peak Fitting

Fast Peak Fit with This button initiates PeakFit’s Numerical Fitting.
Numeric Update

Full Peak Fit with This button initiates PeakFit’s Graphical Fitting.
Graphical Update

Review Peak Fit This button open’s PeakFit’s Review of the fit. If the current data is not
fitted, either because it was never fitted, or because a fit was invalidated by
subsequent adjustments, the Review is still available although you must
confirm you wish to treat the current state of the peaks as if a fit had
occurred.

6-30 AutoFit Peaks II Second Derivative

 Automated Peak Fitting

AutoFit Peaks III Deconvolution

The AutoFit Peaks III Deconvolution option is one of three AutoFit
Peaks options offered by PeakFit.

In this option, hidden peaks are detected by the “sharpening” achieved by

deconvolving a Gaussian instrument response with the raw data. A peak that
initially evidences no local maximum in the initial data, may very do so after
a successful deconvolution.

PeakFit defines a visible peak as one which produces a local maximum in the
input data. A hidden peak is thus defined as one which fails to produce this
local maximum. Peaks which lack this local maximum may range from being
quite apparent in visual inspection to those which are altogether
indiscernible.

This option is the least forgiving of PeakFit’s automated peak placement

procedures. Since the deconvolution and filtration take place in the
frequency domain, this algorithm also requires a uniform X-spacing. Lacking
this constant X-spacing, or if artifacts or outliers have been excluded from
the data, uniform spacing can be restored with PeakFit’s non-parametric
Digital Filter.

This procedure requires a response function width that is sufficient to

produce a significant sharpening of the peaks so as to uncover hidden peaks.
Such widths will also push the deconvolution to the limits of numeric
stability, since deconvolution introduces a great deal of noise into the data.
Fourier domain filtration is critical, and this algorithm determines such
automatically.

This automatic filter level determination is subject to failure when response

function widths start to approach the FWHM of the peaks. This option
locates peaks exclusively from local maxima in the deconvolved data. A
hidden peak in the input data stream must become a peak with a local
maximum in the deconvolved data in order to be automatically detected.

AutoFit Peaks III Deconvolution 6-31

Automated Peak Fitting

Hierarchy of The control panel in the dialog contains the automated options. It is
Processing arranged in a hierarchical order. The change to any option will affect all
options below it. For example, changing the baseline results in baseline
processing, deconvolution, rescanning, and discarding all custom graphical
or numerical adjustments. Changing a scan setting results only in a rescan
and the discarding of any custom adjustments.

Note that changing any of the baseline, deconvolution, function, or scanning

options in the control panel, or importing a file-based scan setup, results in a
complete rebuilding of the autoscan. All custom adjustments and placements
are lost. If you spend any time creating a custom placement, you should
explicitly Save it as soon as you are satisfied with it. While all of the AutoFit
control panel’s settings are saved across sessions, custom placements must
be explicitly saved.

As If Processing Even though this option will present smooth baseline-subtracted data, what
you see is only for the purpose of placing peaks. The data that is fitted is
always the PeakFit data table present when this option is invoked.

The only exception is the local sectioning option which permits points or
bands of points to be temporarily enabled or disabled. Even in this case,
data values remain unchanged as only the active state of points is affected.

Baseline Fitting This option does not subtract a baseline prior to fitting. Any baseline selected in this
option is added to the overall fitted model. While this option does pre-fit a
baseline (if one is selected), the parameters derived from this baseline fitting
are used as initial parameter estimates for the baseline in the peak fitting.
These parameters are also used to create a baseline-subtracted temporary
copy of the input data. This exists only to make the task of placing peaks
simpler.

To subtract a baseline from the data prior to fitting, you must use either the
AutoFit and Subtract Baseline for fitting the baseline in the current data,
or the Import and Subtract Baseline for removing a baseline that has been
externally generated.

Deconvolution This option does not deconvolve data prior to fitting. The deconvolution observed in
this option is strictly for peak detection and placement.

To deconvolve data prior to fitting, you must use the Deconvolve Gaussian
IRF or Deconvolve Exponential IRF options.

6-32 AutoFit Peaks III Deconvolution

 Automated Peak Fitting

Graphical Layout This option will display two PeakFit Graphs. Each will have both a Y and
Y2 axis.

The upper graph is common to all three AutoFit options. Its Y axis will
contain the component peaks and a copy of the input data,
baseline-subtracted using the current parameter estimates for the baseline. If
no baseline is being fitted, this data will consist of the input data.

The Y2 axis will contain this same data along with a smoothed version of
such. This smoothed data uses an automatic FFT filtration. It is furnished
for reference purposes only. The defaults will show the unsmoothed data as
points and the smoothed data as connected lines. The Y2 graph also displays
the sum curve for the constituent peaks. The baseline is not included in the
sum curve.

The lower graph is unique to this AutoFit Peaks III Deconvolution option.
The Y2 axis will contain the deconvolved data. The defaults display the data
as connected lines. The Y axis will consist of the raw input data, modified to
reflect the active states of points used for the baseline estimation. The Y axis
will also display the baseline. If no baseline is fitted, the Y axis will plot the
raw and smoothed data.

AutoFit Peaks III Deconvolution 6-33

Automated Peak Fitting

In the graphs above, the four hidden peaks which evidence no local maxima
in the FFT-filtered input data, show quite definite local maxima in the
deconvolved data. All deconvolved local maxima are added peaks if the
corresponding amplitudes are greater than the Amp % threshold. The
dotted line in the upper graph reflects this amplitude threshold for peak
acceptance.

Zoom-In As with most PeakFit graphs, zoom-in is a simple matter of left clicking the
mouse at one corner of the desired zoom region, holding the mouse button
down, and moving to the opposite corner of the desired region. If you are
zooming in from the constituent peak plot, you must not begin the zoom on
a function anchor.

To restore default scaling, you can click on the Reset Default Scaling button,
or you can right click anywhere within the plot area, except on a function
anchor.

Level 1 of Peak The primary peak placement is fully automatic. This may be the only
Placement - Automatic placement you need concern yourself with if:

• You are fitting a common peak model to all peaks.

• You are fitting one of the built-in baseline models, or no baseline at all.
• You are satisfied that the automatic detection has correctly identified all
visible and hidden peaks present in the data.
• You are willing to allow all peak widths to vary in the fitting or wish to
have all peak widths constant (shared).
• You are willing to allow all peak shape parameters to vary in the fitting or
wish to have all peak shape parameters constant (shared).

To have all peak widths (the a2 parameter of each model) held constant, the
Vary Widths should be unchecked. Frequently in chromatography and
spectroscopy, peak widths will increase with increasing time or wave
number. In such instances, and where peak widths are obviously not
constant, you should check the Vary Widths checkbox.

For the parameters which determine a peak’s shape (parameters a3 and

above), the choice may not be as simple, since it is not always visually
apparent whether or not peaks share a constant shape (asymmetry,
Gaussian-Lorentzian character, etc. ). Ideally, one would like a model and
data where the Vary Shape can remain unchecked. This simplifies the fit

6-34 AutoFit Peaks III Deconvolution

 Automated Peak Fitting

considerably, resulting in faster fitting, more stable fitting, and usually better
confidence limits for the fitted parameters. If the F-statistic for a fit with the
Vary Shape on is higher than that with the Vary Shape off, there is good
indication that the peak shape is not constant. For very noisy data with
hidden or overlapping peaks, it is also unlikely that sufficient information
exists within the data to allow these shape parameters to vary.

Steps for Automated 1. Select a baseline to be fitted. If you have subtracted a baseline or wish to
Fitting fit no baseline, choose the No Baseline selection. The baseline options are
identical to those offered in the AutoFit and Subtract Baseline option.
Note that if you select No Baseline and some positive baseline is present,
the scan algorithm may detect a large number of small amplitude peaks
between zero and this baseline.

2. If you have selected a baseline to be fitted, adjust the baseline Tol % so

that the active points in the Y axis of the lower graph represent a meaningful
baseline. Note that this need not be particularly accurate, merely close
enough for peak detection. You may enter the baseline tolerance percent
directly, use the spin buttons, or right click and hold on the field or spin
buttons and select the value desired.

The Include Inactive Scaling button is particularly important for baselines

since this option must be on to see all of the points, and off for a scaling
based on only those points used to determine the baseline.

3. Specify the Gaussian response Width to be used in the deconvolution.

You may enter the width directly, use the spin buttons, or right click and
hold on the field or spin buttons and select the value desired. Check SD if
this width is a standard deviation, or FWHM is the width is a full width at
half-maxima. The width needs to be sufficient to resolve hidden peaks. In
general, this requires a width somewhere between 50 and 80% of the
FWHM of the peaks. The deconvolution used here is the same as that
offered in PeakFit’s Deconvolve Gaussian IRF option. Note that
deconvolutions using a width close to or exceeding the FWHM of the peaks
will generally produce nonsense.

4. Specify the frequency domain filter level to be used to deal with noise
produced by the deconvolution. Typically, values between and 60 and 85%
will be required. The filtration is identical to the FFT procedure in PeakFit’s
Smooth option.

AutoFit Peaks III Deconvolution 6-35

Automated Peak Fitting

You may also click on the AI Expert option to have it seek an optimum
filtration level for you. This is a highly sensitive setting, one you may need to
refine to achieve the best possible filtration.

The peak scan algorithm does not use a width rejection criteria. Rather are
all local maxima in the deconvolved data detected as peaks. It is thus
important to filter sufficiently to remove all anomalous sinusoidal effects
that could appear as peaks.

5. Select the peak function family desired. Spectroscopy, Chromatography,

Statistical, Miscellaneous, and All Peak Functions are options. The last
option will contain all of the peak functions that can be automatically placed.
PeakFit’s functions are covered in detail in Chapter 7.

6. Select the peak type desired. PeakFit offers over 70 built-in peak
functions which can be automatically and graphically placed. The sum of
squared residuals (SSE) may be of some value in peak selection. Note that
this value reflects only PeakFit’s ability to place the function, and not what
the function is capable of achieving in a least-squares fit.

The Peak Labels button offers the means to display center, amplitude, area,
or scan extrema above the individual peaks. Such labels are often very useful
in identifying peaks of small amplitude.

7. Select the amplitude rejection threshold (Amp % ) desired. This should

be high enough to discard the local maxima peaks and any sinusoidal effects
that appear within and about the baseline, but low enough to include the
smallest of desired peaks. You may enter the percent directly, use the spin
buttons, or right click and hold on the field or spin buttons and select the
value desired.

The amplitude thresholds appear as dotted lines in the graphs. These can be
toggled off if they are not desired.

8. To vary the widths in fitting, check the Vary Widths checkbox. An

additional level of autoscan will seek to carefully refine the widths for each
of the peaks where a FWHM (full width at half-maximum) is clearly defined.
Functions will have anchors at the half-maxima positions for graphical
adjustment of widths.

9. To vary the shape parameter(s) in fitting, if available, check the Vary

Shape checkbox. An additional level of autoscan will seek to carefully refine
the shape for each of the peaks where either a half-maxima asymmetry or

6-36 AutoFit Peaks III Deconvolution

 Automated Peak Fitting

FW25 (full width at 25% maximum) is clearly defined. Asymmetric

functions will now have anchors at the half-maxima positions for graphical
adjustment of asymmetry. Symmetric functions will have additional anchors
at 25% half maxima for graphical shape adjustment.

10. If a shape parameter is available within the selected peak mode, and if
you are unsatisfied with the shape of the placed peaks, you may wish to
check the Refine Shape option. This performs a one-dimensional
minimization where a single graphical shape (asymmetry or FW25/FWHM)
is set for all peaks and varied until a minimum chi-square is found. The
algorithm uses a least-absolute deviation minimization.

11. If satisfied with the automatic placement, you can begin the PeakFit
automated fitting:

• Peak Fit with full graphical update

• Peak Fit with fast numerical update

If you are unable to achieve the desired placement automatically, or if the fit
produced unacceptable results, you can then go on to the secondary level of
peak placement.

In the case of an unsatisfactory fit, before resorting to graphical adjustment,

you may wish to explore the Peak Fit Preferences. PeakFit’s defaults use
somewhat rigid constraints on the initial three parameters to insure fitted
peaks remain very close to where they were initially placed. This is especially
critical when fitting a large number of peaks. If you are having difficulty with
a given fit, and constraints are shown as being violated during the fit, you
may wish to widen these constraints, or disable them altogether.

Level 2 of Peak The secondary placement operation is fully graphical. This is done within the
Placement-Graphical components plot of the upper graph. This secondary level of placement
requires no numerical knowledge of functions or function parameters.

Each peak will have a primary anchor defining its amplitude and center.
Additional anchors will be present if widths or shapes are allowed to vary.

Note that Peak Anchors must be selected in order for these anchors to be
displayed.

Graphically A peak is added simply by left clicking the mouse at the desired center and
Modifying Peaks amplitude. This must be done in the components plot of the upper graph.

AutoFit Peaks III Deconvolution 6-37

Automated Peak Fitting

A peak is deleted by right clicking on the peak’s principal anchor and then
clicking on the Delete Peak button in the Peak Popup dialog.

A peak is toggled on and off by left clicking the mouse on a peak’s principal
anchor. An inactive peak is not used in fitting. Toggling a peak on and off
may be a good way to see its approximate contribution to the overall data.

A peak is graphically adjusted by clicking and holding down the left mouse
button when on an anchor and moving it to a new position. Moving the
primary anchor results in a new center and amplitude. Moving a
half-maxima anchor on a symmetric peak changes the overall peak width.
On an asymmetric peak, moving a half-maxima anchor changes both the
width and asymmetry. On symmetric peaks with shape parameters, the
quarter maxima anchors adjust the FW25 (full width at 25% maximum),
thus changing the peak shape.

An individual peak can be changed to a different function by right clicking

on the principal anchor and selecting a different function from the popup
dialog. here you can also access supplemental functions, such as transition
equations, which can also be graphically placed and adjusted.

Level 3 of Peak Numerical peak placement is for special cases that cannot be graphically
Placement - Numerical placed and adjusted, such as UDFs and unusual models such as the Gamma
Ray and Compton functions. Here placement is by numerically adjusting the
parameter values.

To modify an existing function, simply right click on the primary anchor. If

desired, a different function can be selected from the popup dialog.

To create a new function, add a peak at a desired anchor position by clicking

the mouse at this X,Y position. This position need not bear any relation to
the peak or data feature it represents, but it is where you must go to access
numerical adjustment for the function. The function will initially be the peak
model currently selected in the dialog’s control panel. Right click the new
anchor, and select the desired function from the popup dialog.

Certain functions that can only be placed numerically are included within the
function list, including all currently active UDFs, the Gamma Ray model,
and the Compton Edge model.

To adjust the function numerically, simply change the value of the desired
parameter(s). You may lock a given parameter (its value will not change

6-38 AutoFit Peaks III Deconvolution

 Automated Peak Fitting

during fitting), or you may share a given parameter (its value is shared with
all others at this parameter position also marked as shared).

Note that you must use your own judgment when sharing parameters. For
example, it would normally make no sense to share an amplitude or center
value, or to share width or shape parameters across different peak types.

PeakFit also offers a Common Parameters dialog which allows a given

parameter to be simultaneously set for all peaks. This is the most convenient
way to set a common width or shape parameter across all peaks, or across
any number of selected peaks.

Local Sectioning When a large spectrum or chromatogram has large regions of baseline
between groups of peaks, you may see better fit statistics from fitting the
individual partitions. There may also be instances where you wish to isolate
and fit a single peak to perhaps several models to see which might be most
favorable.

Rather than exit the AutoFit option and section the data from the main
menu, this option allows for a local or temporary sectioning, one that
disables or enables data points only for the duration of the AutoFit
procedure.

Note that sectioning the data results in a complete rebuilding of the

autoscan. All custom adjustments and placements are lost. If you have spent
any time creating a custom placement that you may wish to use with some
future data set, you should explicitly Save it before sectioning. This
sectioning is identical to that offered in PeakFit’s Section option.

Reset The Reset option rescans the data discarding all custom placements. Note
that this is not a full reset, in that control panel settings remain at their
current settings.

If you have spent any time creating a custom placement that you may wish
to use with some future data set, you should explicitly Save it before
sectioning.

This option is used after importing a scan when you wish to discard the
specific peak estimates, and rebuild the scan from the AutoFit control panel
settings in the scan file. This option is also used to discard unwanted
graphical or numerical adjustments.

AutoFit Peaks III Deconvolution 6-39

Automated Peak Fitting

Toggle Lower Graph In order to better see the constituent peaks, you may wish to remove or
decrease the size of the Deconvolution and Baseline graph. This toggle
works in two steps. The first decreases the graph to a reduced size and the
second hides it altogether. An additional click restores the equal graph
sizing.

List Peak Estimates This option opens a PeakFit text window which displays the scan
characteristics and the peak estimates from the Autoscan When peaks are
cleanly defined, and widths are permitted to vary, PeakFit’s autoscan may
complement other analytical methods. PeakFit’s autoscan uses sophisticated
built-in correlations that map a peaks function’s graphical state to its
parameter estimates. As such, the FWHM and areas often tend to be quite
accurate.

If the Addl Adjust option is selected after fitting, this list will contain
updated peak information based upon the fit. This should not be considered
a replacement for the Numeric Summary in the Review.

Read Scan Setup and This option imports a PeakFit scan file [SCN] that has been saved to disk. A
Parameter Estimates PeakFit scan file contains all AutoFit control panel settings, all peak
adjustments and customizations, graphical and numerical, and also a UDF
library, if any UDFs have been used in the current placement.

Importing a scan updates the AutoFit control panel’s settings, but does not
trigger an autoscan. Rather are the peaks placed exactly as they were when
the file was saved, including all graphical and numerical adjustments. If
UDFs are used, the internal UDF library is read into the current UDF
positions, replacing any active UDFs.

If you wish to discard custom settings and rebuild the autoscan, simply click
the Reset button.

Save Scan Setup and This option saves the current AutoFit control panel settings, the current
Parameter Estimates peak placement and parameters, and any UDFs to an [SCN] disk file. A scan
file permits any peak placement to be immediately reproduced.

Modify Peak Fit This opens the PeakFit Preferences dialog. These preferences control
Preferences adjustable elements within PeakFit’s non-linear fitting engine. Some of these
options, such as the Robust Minimization and Built-In Peak Fn
Constraints can dramatically affect the fit results. Others such as Fit Extent

6-40 AutoFit Peaks III Deconvolution

 Automated Peak Fitting

and the Curvature Matrix options can significantly affect the time required
for fitting large data sets when many peaks are being fitted.

Fast Peak Fit This button initiates PeakFit’s Numerical Fitting.

with Numeric Update

Full Peak Fit with This button initiates PeakFit’s Graphical Fitting.
Graphical Update

Review Peak Fit This button open’s PeakFit’s Review of the fit. If the current data is not
fitted, either because it was never fitted, or because a fit was invalidated by
subsequent adjustments, the Review is still available although you must
confirm you wish to treat the current state of the peaks as if a fit had
occurred.

AutoFit Peaks III Deconvolution 6-41

Automated Peak Fitting

Peak Fit Preferences

These preferences control adjustable elements within PeakFit’s non-linear
fitting engine.

Some of these options, such as the Robust Minimization and Built-In

Peak Fn Constraints can dramatically affect the fit results. Others such as
Fit Extent and the Curvature Matrix options can significantly affect the
time required for fitting large data sets when many peaks are being fitted.

Maximum Iterations The default number of iterations is 500. You may enter any value between
10 and 10000. There is usually very little to gain beyond 500 iterations.

Converge to The default non-linear convergence precision is 6. This means that the
Significant Digits in chi-square (goodness of fit) must be unchanging in the sixth decimal place
Chi-Square for five consecutive iterations to signal convergence. You may enter any
value between 3 and 14. Note that often only a few additional iterations are
needed to converge to a high precision. A convergence precision of 12 may
thus require very little additional fitting time than the default of 6.

Robust Minimization In addition to standard least-squares minimization, PeakFit’s

Levenburg-Marquardt non-linear fitting engine is capable of three different
robust estimations. These minimizations are sometimes referred to as
maximum likelihood or m-estimate fitting. Robust fitting is covered in more
detail at the end of this chapter.

6-42 Peak Fit Preferences

 Automated Peak Fitting

If your data spans a large number of orders of magnitude in the Y variable

and the low-valued Y points are not factoring into the fit, a robust fit will
remedy this problem. This may well be a superior solution to seeking to
weight the data so that these low-valued Y points can factor into a
least-squares solution.

The other instance where robust fitting is recommended is when it is known

that there are significant outliers within the data. Robust estimation will
minimize the impact of outliers. This may be important when relatively few
points define an individual peak.

Least-squares corresponds to a Gaussian maximum likelihood distribution

of errors. All of the robust minimizations correspond with maximum
likelihood probability distributions significantly less compact than the
Gaussian. These wider tails mean that the errors associated with outliers are
expected. When outliers are suspected or likely, the Lorentzian minimization
is highly recommended. Although the same non-linear fitting engine is used
for all of these minimizations, you will generally find that a higher number
of iterations will be required for a robust fit as compared to least squares.

Fit Extent This option offers a convenient way to fit a reduced size data set without
having to digitally filter the data. With large data sets, where building the
curvature matrix dwarfs the time required to invert the matrix, fitting times
are roughly proportional to the count of data points being fitted. As such,
fitting every other point will result in close to a halved fitting time. You may
choose to fit Every Point, Every Other Point, Every Third Point, or
Every Fourth Point. These will appear as 1/1, 1/2, 1/3, or 1/4 in the
Extent shown in the Details of Fit subsection of the Numeric Summary.

Curvature Matrix This is another preference useful for minimizing the overall fitting time.
When constructing the curvature matrix, the partial derivatives appearing in
each matrix element need not be summed for all X values in the data.
Rather, a sparse procedure can be used where each element is updated only
when a peak function is close enough to that X value to affect the overall
model.

The Full option makes no use of sparse matrix processing. The curvature
matrix is built by full evaluations at each X value for every element in the
matrix.

The Sparse by Limit Tests is a bi-directional procedure. On even iterations,

the matrix is built by ascending X and checks are made for insignificance at

Peak Fit Preferences 6-43

Automated Peak Fitting

the right decay of peaks. On odd iterations, the matrix is built by descending
X and insignificance checks are made at the left decay of peaks. It is the
fastest of the sparse methods, but suffers the limitation that only one side of
peaks will have updated limits with each iteration.

The Sparse by Root Finding procedure does not suffer this limitation. At
each iteration, each peak’s significance limits are determined by a root
finding procedure. The drawback is that the root finding algorithm tends to
be slower than the bi-directional limit tests.

When fitting only a few peaks, sparse curvature matrix processing offers
little to no benefit. Even with a large number of peaks, the benefit may be
modest with peaks whose partial derivatives are computed very rapidly, such
as Gaussians. On the hand, the processing can yield up to tenfold reduction
in fitting time for fitting large numbers of complex peaks such as the NLC,
whose partial derivative computations are particularly demanding.

Built-In Peak Fn This is one of the most important preferences within PeakFit, especially
Constraints when fitting large numbers of peaks.

The original releases of PeakFit were limited to eight peaks, each of which
had to be manually placed. Under such an environment, you tended to know
when something went wrong. Now that PeakFit places up to 100 peaks, all
automatically, it is essential that the fitting algorithm exert additional
controls to assure peaks remain close to where they are originally and
automatically placed. Non-linear fitting, in and of itself, can hardly assure
such.

Local minima in multidimensional non-linear fitting refers to a limitation in

iterative minimization algorithms. Unlike linear least-squares, where a global
minimum always occurs (and within a single step), achieving a global
minimum (the optimum fit) in iterative non-linear fitting is not assured.
Indeed peaks are amongst the worst of culprits for functions producing local
minima. It is probable that only waveform functions are more
problemsome.

When a peak’s center shifts right to where the left side of the peak function
rests on the right hand decay of the peak data, a local minima condition
develops. It is therefore especially important that center values be
constrained and not be permitted unlimited freedom of movement.

6-44 Peak Fit Preferences

 Automated Peak Fitting

PeakFit’s autoplacement algorithms go to great lengths to place peaks as

accurately as possible. This makes it possible to impose constraints on the
principal parameter values.

The constraints are specified as percents on the parameters. Any active

constraint will apply to all built-in peak functions, regardless of model.
Supplemental functions and UDFs are not included (UDFs have their own
separate constraints when they are built).

PeakFit’s algorithm is designed to deal with minimization of constraint

violations as well as a goodness of fit minimization. In particularly difficult
fits, you may see upwards of several dozen constraints violated on the initial
iteration. In time the algorithm should work itself to a position in
multidimensional parameter space where no constraints are violated.

In conditions where constraints are readily violated, the fit algorithm may
spend more iterations working itself free of constraint violations than in
resolving the optimum fit parameters.

You may disable constraints by unchecking them. In normal practice, it is

recommended that you keep the a1 constraint active, and set at a percent
that prevents small peaks from wandering into the regions of adjacent ones,
and that you preserve the a0 constraint, at least sufficiently to insure that
amplitudes retain their sign.

In fitting large numbers of peaks, we have found repeated instances where a

given iteration shows an overall improvement in the goodness of fit merit
function, but where one small amplitude peak has gone into oblivion,
perhaps into a local minimum position with an adjacent peak, perhaps into a
zero or negative amplitude condition, perhaps decaying to zero width, or
even shifting outside the X range of the data.

PeakFit’s implementation of constraints are a byproduct of our experience

in seeking to introduce a greater stability into the peak fitting algorithm.
Admittedly these constraints may be a nuisance to fine tune, but if you
consider them a form of numerical analysis insurance, you may find yourself
well rewarded as they enable you to put a greater reliance upon the
algorithm than would have otherwise been possible.

Peak Fit Preferences 6-45

Automated Peak Fitting

Saving and Reading Use the Save item to save the current preferences to disk. The default file
Fit Preferences extension is [PRF]. These are binary files that can only be produced within
the program. The current preferences are always saved automatically across
sessions. You will want to save preferences to disk if you plan to regularly
use more than one fit configuration in your work.

Preference files can be recalled at anytime using the Read item.

Reset The Reset button restores PeakFit default preferences.

6-46 Peak Fit Preferences

 Automated Peak Fitting

Peak Adjust
This is a popup dialog that appears when you right click the principal anchor
of any peak within any one of the three AutoFit Peaks options. The dialog
uses a tiny caption to minimize the screen area occupied. You can close the
window with the small system menu button, or with the OK or Cancel
buttons.

The dialog can remain open while different peaks are selected with the right
mouse button. When this dialog is open, the left mouse button can also be
used to select peaks.

The Reset button restores the peak type and parameters to the settings
preset when the dialog was opened.

The Delete Peak button is used to completely remove a peak from the
current scan.

Function Selection To change the current function to a different function, simply select the
function desired. Unlike the functions available in the AutoFit control panel,
this function list includes functions that can only be placed numerically.
These include all currently active UDFs, the Gamma Ray model, and the
Compton Edge model. The function list also includes supplemental
functions, such as transition equations, which can be both graphically and
numerically adjusted.

Parameter Values To adjust the function numerically, simply change the value of the desired
parameter(s). All pertinent graphs will be updated.

You may lock a given parameter (its value will not change during fitting) by
checking the lock option.

You may share a given parameter (its value is shared with all others at this
parameter position also marked as shared) by checking the share option.
Note that you must use your own judgment when sharing parameters. For
example, it would normally make no sense to share an amplitude or center
value, or to share width or shape parameters across different peak types.

Adjusting Multiple The AutoFit Peak options each offer a Common Estimates dialog to adjust
Peaks Simultaneously the parameters for any number of peaks simultaneously.

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Automated Peak Fitting

Common Estimates
The Common Estimates dialog is similar to the Peak Adjust popup dialog,
except that it is used to set the parameter values for any number of peaks
simultaneously. It is particularly useful for setting width or shape parameters
numerically for all peaks.

• This dialog does not allow deletion of peaks.

• This dialog does not allow changing of peak functions.
• The dialog contains a selection list containing all peaks currently placed.
• The Clear All Selections button deselects all peaks in the selection list.
• The Select All Peaks buttons selects all peaks in the list.

Selection List As with multiple selection lists common to Windows, simply hold the Ctl
key to make multiple selections. To select a sequential group of peaks,
simply hold the mouse button down after the initial selection and move the
mouse up or down to highlight the range of selections desired.

Parameter Values Empty values are not processed. You should enter values only for those
parameters you wish to set as common for the current selected group of
peaks. Usually this will be an a2 or higher parameter.

You may also lock or share any parameter entered. Locked parameters are
those whose values do not change during fitting). Shared parameters are
those whose value is shared during fitting.

Update The parameters for the selected peaks are not updated until the dialog is
closed with the OK option. The graphs are then also updated.

6-48 Common Estimates

 Automated Peak Fitting

Fast Peak Fit with Numerical Update

This is the numerical peak fitting algorithm. It is accessed from the various
AutoFit Peaks options. In contrast to the graphical fitting, this option does
not graph a fit as it progresses. Rather, you are shown numerical indices of
the fit, and an r² goodness of fit meter as an indicator of fit progress.

Both the numerical and graphical peak fits are governed by the current fit
Preferences accessible from the AutoFit Peaks options. Some of these
options, such as the Robust Minimization and Built-In Peak Fn
Constraints can dramatically affect the fit results. Others such as Fit Extent
and the Curvature Matrix options can significantly affect the time required
for fitting large data sets when many peaks are being fitted.

If a fit consistently indicates that it is wrestling with one or more constraints,

it may mean that the constraints need to be opened somewhat, or perhaps
cautiously disabled.

When a numeric fit is initiated, a separate thread is used to launch the peak
fitting on Win95 and Win NT systems. Responsiveness to intervention
actions, such as stopping or aborting the fitting, may not be immediate on
Win32S systems.

Informational • Iteration - Indicates the current iteration in fitting.

Elements
• Constraints - Number of constraint violations occurring on the current
iteration. This is an important indicator to watch. If most of a fit is spent
escaping constraint violations, it may suggest the need for more open
constraint percents or better initial placement. Constraints are set in the
Fit Preferences and when UDFs are created.

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Automated Peak Fitting

• Merit Function - The goodness of fit criterion that governs

minimization. For least-squares, it will be the sum of squared residuals.
For robust minimizations, it will be that robust procedure’s merit
function. The value will be the best merit function thus far achieved in
the fit. A robust procedure is set in the Fit Preferences.
• Std Error - The best fit standard error thus far achieved. This will only be
be the standard error for the current iteration’s parameters if such
estimates are adopted.
• F-Stat - The best F statistic thus far achieved in the fit. This will only be
be the F-statistic for the current iteration’s parameters if such estimates
are adopted.
• r² - The r² coefficient of determination thus far achieved. This will only
be be the r² for the current iteration’s parameters if such estimates are
adopted. The r² value is also displayed graphically in the r² progress
meter. An r² of 1.0 is a perfect fit; an r² of 0.0 is a complete lack of fit.
• Time - The total time thus far spent in fitting.
• Time/Iter - The average time per iteration.
• Fn Eval - The total number of function evaluations occurring in the fit. A
single function evaluation consists of the computation of a given peak or
background function and all associated partial derivatives. If one of the
sparse curvature matrix options is set in the Fit Preferences, this value
may vary iteration to iteration.
• FnEval/Iter - The average number of function evaluations per iteration.
If one of the sparse curvature matrix options is set in the Fit Preferences,
this value may vary across iterations.
• Data Points - The total number of data points and the extent of fit.
“500/2" would mean every other point of a 500 point table is being
fitted. The extent of fit is set in the Fit Preferences.
• Parameters - The total number of parameters fitted. This represents the
size of matrix inverted with each iteration. A fit consisting of 35
parameters requires the inversion of a 35x35 matrix with each iteration.

r² Meter The progress meter offers a graphical representation of the r² achieved. The
r² bar is divided into five equal log sections.

Stop Current The Stop Current button will stop the fitting at its current iteration. It can
be used when the fit is changing extremely slowly. Some fits may find

6-50 Fast Peak Fit with Numerical Update

 Automated Peak Fitting

themselves within a very narrow valley in parameter space, and even though
convergence is not signaled, very little is to be gained by continuing the fit.
The Stop Current button is only visible while a fit is underway.

Abort Fit The Abort Fit button is available both during and at the conclusion of a fit.
When the estimates of one or more parameters go widely astray, the fitted
result may be useless. This is especially possible if all constraints have been
disabled. When a fit is aborted you are returned to the AutoFit operation as
if no fit had occurred.

Additional Adjust The Adjust Fit is available at the conclusion of the fit. When using this
numeric fit option, it is useful if you wish to see the graphical results of the
fit before reviewing such. If one or more of the peaks are not as you feel
they should be, you can then graphically adjust an offending peak, or
perhaps numerically adjust or lock a parameter, prior to reinitiating the
fitting. The Review is still accessible from the AutoFit screen should you
choose to make no adjustments.

Review Fit The Review Fit option is available at the conclusion of a fit. It is used when
you wish to proceed directly to PeakFit’s Review.

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Automated Peak Fitting

Full Peak Fit with Graphical Update

This is the graphical peak fitting algorithm. It is accessed from the various
AutoFit Peaks options. In contrast to the numeric fitting, this option graphs
the fit as it progresses. You are also shown numerical indices of the fit.
Updates at each successful iteration are optional.

This operation displays a PeakFit graph containing both a Y and a Y2 plot.

The Y plot will contain the constituent peaks. The Y2 plot will contain the
data and the sum curve for the fitted model.

Both the numerical and graphical peak fits are governed by the current fit
Preferences accessible from the AutoFit options. Some of these options,
such as the Robust Minimization and Built-In Peak Fn Constraints can
dramatically affect the fit results. Others such as Fit Extent and the
Curvature Matrix options can significantly affect the time required for
fitting large data sets when many peaks are being fitted.

If a fit consistently indicates that it is wrestling with one or more constraints,

it may mean that the constraints need to be opened somewhat, or perhaps
cautiously disabled.

When a graphical fit is initiated, a separate thread is used to launch the peak
fitting on Win95 and Win NT systems. Responsiveness to intervention

6-52 Full Peak Fit with Graphical Update

 Automated Peak Fitting

actions, such as toggling the iteration updating, or stopping or aborting the

fitting, may not be immediate on Win32S systems.

Informational • Iter - Indicates the current iteration in fitting.

Elements
• Constraints - Number of constraint violations occurring on the current
iteration. This is an important indicator to watch. If most of a fit is spent
escaping constraint violations, it may suggest the need for more open
constraint percents or better initial placement. Constraints are set in the
Fit Preferences and when UDFs are created.
• Merit - The goodness of fit criterion that governs minimization. For
least-squares, it will be the sum of squared residuals. For robust
minimizations, it will be that robust procedure’s merit function. The value
will be the best merit function thus far achieved in the fit. A robust
procedure is set in the Fit Preferences.
• StErr - The best fit standard error thus far achieved. This will only be be
the standard error for the current iteration’s parameters if such estimates
are adopted.
• Fstat - The best F statistic thus far achieved in the fit. This will only be be
the F-statistic for the current iteration’s parameters if such estimates are
adopted.
• r² - The r² coefficient of determination thus far achieved. This will only
be be the r² for the current iteration’s parameters if such estimates are
adopted. An r² of 1.0 is a perfect fit; an r² of 0.0 is a complete lack of fit.
• Time - The total time thus far spent in fitting.
• Time/Iter - The average time per iteration.
• Evl - The total number of function evaluations occurring in the fit. A
single function evaluation consists of the computation of a given peak or
background function and all associated partial derivatives. If one of the
sparse curvature matrix options is set in the Fit Preferences, this value
may vary iteration to iteration. This count does not include the various
function evaluations necessary for updating the graph.
• Evl/Iter - The average number of function evaluations per iteration. If
one of the sparse curvature matrix options is set in the Fit Preferences,
this value may vary across iterations.

Full Peak Fit with Graphical Update 6-53

Automated Peak Fitting

• Data - The total number of data points and the extent of fit. “500/2"
would mean every other point of a 500 point table is being fitted. The
extent of fit is set in the Fit Preferences.
• Parms - The total number of parameters fitted. This represents the size
of matrix inverted with each iteration. A fit consisting of 35 parameters
requires the inversion of a 35x35 matrix with each iteration.

Iter Update When Iter Update is checked, the peak fit is graphically updated following
each iteration where the parameter estimates are adopted by the fitting
algorithm. This option can be toggled on and off while a fit is underway.

When the iteration update is unchecked, the peaks are graphed only at the
beginning and at the conclusion of the fit.

Graphical Zoom-In You may zoom in the graph while a fit is underway. If some peak becomes
suspect in the course of fitting, simply place the cursor at one corner of the
desired zoom in region, press and hold the left mouse button while moving
to the opposite corner of the desired region. You may zoom-in either plot.

To restore the default scaling, simply right click the mouse in the graph area.

Toggling Outliers Off An outlier is a data point lying outside the general trend of the data being
During Fitting fitted. If you observe one or more outliers adversely impacting the fit, you
may be able to remedy such while the fit is still in motion. Simply place the
cursor on the point you wish to exclude from the fit and click the left mouse
button. From this moment onward, the disabled point will not be used in
the fitting computations.

Stop Current The Stop Current button will stop the fitting at its current iteration. It can
be used when the fit is changing extremely slowly. Some fits may find
themselves within a very narrow valley in parameter space, and even though
convergence is not signaled, very little is to be gained by continuing the fit.
The Stop Current button is only visible while a fit is underway.

Abort Fit The Abort Fit button is available both during and at the conclusion of a fit.
When the estimates of one or more parameters go widely astray, the fitted
result may be useless. This is often very apparent graphically. Such problems
may arise if all parameter constraints have been disabled. When a fit is
aborted you are returned to the AutoFit operation as if no fit had occurred.

6-54 Full Peak Fit with Graphical Update

 Automated Peak Fitting

Additional Adjust The Adjust Fit is available at the conclusion of the fit. When using this
graphical fit option, you may observe a peak which does not fit as you may
have wished it to. This option offers the means to update the scan with the
current results of the fit, enabling you to graphically adjust the offending
peak, or perhaps to numerically adjust or lock a parameter, prior to
reinitiating the fitting. The Review is still accessible from the AutoFit screen
should you choose to make no adjustments.

Review Fit The Review Fit option is available at the conclusion of a fit. It is used when
you wish to proceed directly to PeakFit’s Review.

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Automated Peak Fitting

Peak Fit Review

The PeakFit Review is available at the conclusion of numeric fitting, at the
conclusion of graphical fitting, and from the three AutoFit Peaks options. It
is available whether or not the data has been fitted, although cautions are in
order when reviewing unfitted data.

The Review consists of a PeakFit graph containing both a Y and Y2 plot.

The component peak functions are shown in the Y-axis plot. The Y2 plot
contains the fitted curve, the data that was fitted, and the baseline, if one
was included in the fit. By default, the data points are colored by relation to
the fit standard error.

Intervals
The Review graph offers the option of displaying confidence and prediction
intervals about the fitted curve.

The Set Confidence/Prediction Intervals, % Confidence button is used to

set confidence interval type and percentage. The option opens a simple

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 Automated Peak Fitting

dialog where you can individually check Confidence Intervals and

Prediction Intervals, and where you can select from 90% Confidence,
95% Confidence, and 99% Confidence.

The Show Confidence/Prediction Intervals button toggles the currently

set intervals on and off.

The selected confidence level applies not only to the confidence and
prediction intervals in the graph, but also to confidence limits on the
parameters in the Numeric Summary, and to the confidence and prediction
ranges for the data in the Data Summary.

Confidence Intervals Confidence intervals are useful for replicated data where, for each X, you
have either multiple Y observations or a single Y value which represents an
average from multiple Y observations (when each Y value in the data set
consists of an average from multiple observations at the same X, the point
should be weighted with the inverse square of the standard deviation). With
many Y observations at a given X, the average Y can be said to approach the
true Y value, since the errors will sum to zero.

A 95% confidence interval is the Y-range for a given X that has a 95%
probability for containing the true Y value.

A confidence interval cannot be used to predict, for a given X, where the

next individual observation of Y will occur.

Prediction Intervals Prediction intervals are useful for predicting, for a given X, the Y value of
the next experiment. It is often used when a fit represents a single
experiment, where each Y value is a single observation rather than an
average. In this case, the weight for each Y value isn’t based upon a standard
deviation from multiple observations, but rather is inversely related to the
experimental uncertainty for the individual measurement, if such is known.
If the uncertainty of the Y measurement is unknown or thought to be equal
for all X, all points can use equal 1.0 weights.

A 95% prediction interval is the Y range for a given X where there is a 95%
probability that the next experiment’s Y value will occur, based upon the fit
of the present experiment’s data.

Local Measure of Confidence and prediction intervals measure the confidence only at a
Error specific X, not for the entire X data region. These must be computed for
each X value in the curve. For fits containing many peaks or peaks that are

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Automated Peak Fitting

computationally intensive, it may take PeakFit some time to compute these

intervals.

Relation to Standard There is often a strong similarity between points which lay outside 2
Error standard errors and the 95% prediction interval. These are not identical,
however.

The standard error of fit is based upon the overall fit. It matters not if a
point is near a portion of the curve well characterized by the fit, or one only
poorly so. A certain standard error in Y exists for the overall fit, and a point
with a residual whose magnitude lay beyond a given multiple of this value is
drawn in a certain color.

The prediction interval, on the other hand, is a more localized measure of

error. Points near a region of the curve strongly determined by the fit with
have a slightly tighter confidence interval than those points near a region of
the curve only weakly determined in the fit. Generally, strongly determined
regions will have a good number of accurate data points whereas poorly
determined regions will have few points with considerable variability. With
peak-type data, confidence and prediction intervals are often narrowest near
the centers of the various peaks and wider furthest from these centers.

Export
All PeakFit graphs have the means to export the graph image to the
clipboard or file as various forms of bitmaps and metafiles. Also included is
an option that copies to the clipboard in spreadsheet format all of the
numeric information used to produce the graph.

The Export button is used specifically to create ASCII, Excel, Lotus 123,
Quattro Pro, or SigmaPlot disk files. These files use a generated X data
stream and contain columns for the individual peaks and baseline as well as
the overall sum curve. The five formats are:

• ASCII (Full Precision)

• Excel XLS v4
• Lotus 123 WK3
• Quattro Pro Windows WB1
• SigmaPlot SP5

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 Automated Peak Fitting

Exported Information The first two columns will contain the X and Y data used in the peak fit.

The third column will contain the X data generated from the X Start, X
Incr, and X End values in the dialog. By default, these values will reproduce
the current count of X data values. If the data consists of uniformly-spaced
X values, the default generated X will match the X data values. You can
modify the increment to produce any density of generated information you
like, subject to a 16384 maximum count. You can also modify the start and
end values to generate only a portion of the fitted spectrum.

The fourth column will contain the predicted Y values from the peak fit for
the generated X in column 3.

Subsequent columns will contain the predicted Y values for each of the
component peaks within the fit. There will be one column for each peak
fitted.

A final column will contain the predicted baseline, if one was fitted.

ASCII The values will be separated by a space delimiter and will be written to 15
digits numeric precision.

Excel The exported file will be in the non-OLE XLS v4 format. This format can
be imported not only by Excel, but also by most applications which accept
Excel XLS worksheet files.

Lotus 123 The exported file will be in the ubiquitous WK3 format. Most applications
can read WK3 files.

Quattro Pro The exported file will be in the WB1 Quattro Pro Windows format.

SigmaPlot The exported file will use the SigmaPlot SP5 worksheet format. This file can
be imported by both DOS and Windows versions of SigmaPlot.

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Automated Peak Fitting

Numeric
The Numeric button is used to display a full numeric peak analysis.

This analytical summary can consist of the following items, all selected from
the Options menu:

• Fitted Parameters - goodness of fit and fitted parameter values

• Measured Values - measured peak characteristics; also includes analytic
areas if available
• Parameter Statistics - peak by peak parameter statistics
• Chromatographic Analysis - theoretical plates and resolutions
• Overlap Areas - matrix of overlap areas for all peaks
• ANOVA - Analysis of Variance
• Details of Fit - Convergence State, Iterations, Fit Settings

The Options menu also includes a probabilities option:

• Add Probabilities - adds probability values to the parameter statistics and
analysis of variance table

All elements of this Numeric Summary are computed when the Review is
opened with the exception of Overlap Areas. Because this is a matrix of
integrations, it can require a considerable time with a large number of peaks
or with those that are particularly demanding to compute. Overlap Areas
should not be routinely checked unless you really need this information.

The Numeric Summary is toggled on and off by the Numeric button in the
Review. The window size and position you choose for the Numeric
Summary is automatically saved across sessions. The Numeric Summary
uses the PeakFit text viewer to display, modify, and print the information.
This is covered in Chapter 3.

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 Automated Peak Fitting

Fitted Parameters The Fitted Parameters section begins with PeakFit’s four sum of squares
based goodness of fit criteria. These are the r² coefficient of determination, a
degree of freedom adjusted r², the standard error for the fit and the
F-statistic for the fit.

The summary then presents each function and its fitted parameter values. If
a background was fitted, it is last in the function list.

Measured Values The initial Measured Values section uses minimization and root-finding
algorithms to report amplitude, center, full width at half-maxima (FWHM),
asymmetry at half-maxima (Asym50), full width at base (FW Base), and

asymmetry at 10% of maxima (Asym10) measured values.

The final Measured Values section first reports an analytic area and its
percentages if an analytic area is available for the functions. An integrated
area and its percentages follows. The final two elements are the first and
second moments. The area and moment computations use an automated

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Automated Peak Fitting

integration routine which uses a Gaussian Quadrature, Romberg, and

Adaptive Quadrature methods. Target precision is 1E-8. The baseline is not
included in this option.

Parameter Statistics

The Parameter Statistics section is a peak by peak display of parameter

statistics, which include each parameter’s value, standard error, t-value
(parameter value/std error), and confidence limits.

Note that these standard errors and confidence limits assume normal
(Gaussian) errors. If you plan to use these statistics, you should confirm
normal errors using a Residuals Graph, inspecting both a distribution and
SNP plot.

The confidence level must be set in the Review’s Confidence Intervals

option.

Chromatographic The Chromatographic Analysis section furnishes two computations of the

Analysis theoretical plate count. The first is the rigorous procedure using moments.
The second uses a Gaussian approximation.

The next columns are the full width at the base (extrapolated from a straight
line between 90% and 10% of peak maximum) and the asymmetry at 10% of
peak maximum. The final column contains the resolution between adjacent
peaks.

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 Automated Peak Fitting

Overlap Areas The Overlap Areas section consists of a square matrix whose size is the
peak count. This option is computed only as requested since it can be quite
time consuming with a large number of peaks or with peaks whose
functions are computationally demanding.

To find the overlap area between any two peaks simply find the desired
position in the matrix. The computations use an automated integration
routine which uses a Gaussian Quadrature, Romberg, and Adaptive
Quadrature methods. Target precision is 1E-8.

Analysis of Variance The Analysis of Variance section includes a standard ANOVA table.

Also reported is the r² coefficient of determination, the degree of freedom

adjusted r², and the standard error for the fit (the square root of the Mean
Square Error).

Details of Fit The Details of Fit section is helpful for subsequently referencing how a fit
was made, and its success or failure relative to convergence.

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Automated Peak Fitting

Also reported are the number of parameter constraints violated on the final
iteration. Many of the items summarize the current Peak Fit Preferences
since these can significantly affect the results of any given fit.

Data
The Data Summary is used to display a point by point data summary for the
peak fit with the X and Y value of each data point, the predicted value, the
residual and %residual, the confidence limits, and the prediction limits.

The Data Summary is toggled on and off by the Data button in the Review
window. The window size and position you choose for the Data Summary is
automatically saved across sessions. The Data Summary uses the PeakFit
text viewer to display, modify, and print the information.

Type Menu The Type menu is used to select the display of residuals, confidence limits,
prediction limits, or all three types of information. You will need to select a
small font size or landscape mode in order to fully print all three types of
information on a standard size page.

Eval
The Eval button is used to perform a full-featured numeric evaluation of the
peak fit. While the Evaluation procedure is active, all other Review processes
are suspended. The Evaluation feature is covered in Chapter 3.

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 Automated Peak Fitting

Residuals
The Residuals Graph is a separate Peak Fit Graph activated in the Review to
window graphically displaying the residuals for the current peak fit.

The residuals can be displayed in five different formats:

• Basic Residuals - the simple difference between the Y data value and the
Y predicted from the peak fit
• Percent Residuals - the residuals as a % of the Y data value
• Standardized Residuals - the residuals as fraction of fit standard error
• Distribution - the residuals in a binned histogram
• Delta SNP - the residuals as a delta stabilized normal probability

By default, the Basic Residuals graph also doubles as a standardized residuals

graph since the Point format specifies the coloring of residuals by fit
standard error.

The Residuals Graph is toggled on and off by the Residuals button in the
Peak-Fit Review window. You may also close the Residuals Graph directly.
The window size and position you choose for the Residuals Graph is
automatically saved across sessions.

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Automated Peak Fitting

Residuals Distribution The least-squares coefficient standard errors and confidence ranges as well
Graph as the curve’s confidence and prediction intervals reported by PeakFit
contain an implicit assumption that the residuals are normally distributed.
These uncertainty statistics cannot be assumed correct unless this condition
of normality is verified.

The Distribution Graph option displays a histogram of the residuals.

Distributions with obvious asymmetry or wide tails would readily disqualify
this assumption of Gaussian errors.

The above example illustrates a histogram where errors are easily judged
Gaussian.

PeakFit requires at least 16 active points in order to produce a residuals

distribution. Note that any histogram is of dubious merit when data table
sizes are small because of the large bin spacings. The greater the number of
data points, the more accurate the distribution will be.

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 Automated Peak Fitting

Delta Stabilized PeakFit introduces this approach as the best way to assure errors are normal.
Normal Probability A stabilized normal probability (SNP) plot uses an arctangent
Plot transformation on both X and Y to produce a normal probability plot that
uses a linear scale for both the X and Y axes. On such a plot, perfectly
normal errors plot as a 45 degree line. Critical limits also have a 45 degree
slope, and lay equally above and below this line.

PeakFit modifies the SNP slightly and uses a delta SNP, where the X value is
subtracted from the Y. This produces a horizontal y=0 for pure normal data,
and horizontal critical limit lines.

PeakFit plots 90, 95, 99, and 99.9% critical limit lines on the SNP plot. A
99% critical limit means that in only 1 out of 100 data sets should even a
single point violate this limit. You may find the 99% critical limit the most
useful. If even a single data point in the SNP violates this 99% limit, it is
reasonable to assume that the errors fail this normality test. The above graph
is from a peak fit which yielded normal errors.

By default, the 90% lines will be blue, the 95% green, the 99% yellow, and
the 99.9% red.

If you go directly to the Review without fitting the data, you should inspect
the SNP before attempting to use the parameter confidence statistics in any

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Automated Peak Fitting

way. It is altogether likely that the errors present in such a case will be
non-normal.

Maximum Likelihood When the normal assumption is invalidated due to appreciable tails in the
residuals distribution, equally invalidated is the assumption that least-squares
is furnishing the maximum likelihood fit. In such a case, one of PeakFit’s
robust minimizations may represent a better maximum likelihood model.

If you choose to fit a robust model, please remember that PeakFit’s

goodness of fit statistics are all based on a least-squares common frame of
reference. As such, the goodness of fit values will fail to reflect the
improvement derived from switching to a robust method. Also, you should
not assume simply because a distribution of errors is Gaussian that a robust
procedure is if no value. Two key reasons for using a robust minimization
are to deal effectively with outliers and a wide dynamic range on the
y-variable.

Robust fitting is covered in greater detail at the end of this chapter.

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User-Defined Functions
You may install up to fifteen non-linear User-Defined Functions (UDFs)
into PeakFit’s equation set at any given time. This User-Defined Functions
option is used as the control center for entering, editing,and saving UDFs.

From this UDF dialog, you can work with individual functions or with UDF
libraries.

PeakFit UDFs Active UDFs are available for placement in PeakFit’s three AutoFit Peaks
options. Note that PeakFit’s UDFs cannot be used to fit data directly.
Rather are they assumed to describe components, usually individual peaks,
that are placed during the AutoFit placement process.

Within the AutoFit procedures, a UDF is placed by right clicking on the

primary anchor of an existing peak (usually the default peak type). The
popup dialog will contain a selection list with all PeakFit functions and all
active UDFs. After changing the peak to a UDF, it is then refined by
numeric adjustment of the UDF parameters. The initial estimates within a
PeakFit UDF are thus not used for final placement or fitting, but only for
initially placing the UDF in a visible manner, somewhere within the X-Y
range of the data.

User-Defined Functions 6-69

Automated Peak Fitting

If you create a special peak function UDF and you wish to fit this model to
data containing 10 peaks, note that you normally create a single UDF whose
estimates position the peak somewhere near the midrange of the current
data. You then place 10 of these functions in the AutoFit procedure,
adjusting the parameters of each to fit the individual peaks.

If your model contains a center and amplitude parameter, you may wish to
use the data constants XMEAN and YRANGE as estimates. This assures
that the UDF will visually appear with any data set, regardless of its X or Y
ranges.

After the UDFs are all placed, the AutoFit Peak’s Save Scan Setup and
Parameter Estimates option will store the UDF formula and all instances of
its placement. Similar data sets can then be processed by simply importing
this setup.

Note also that UDFs need not be peak functions. They can be unusual
baselines, transitions, or any other function desired to describe any usual
feature in the data.

UDF Content A PeakFit UDF contains all information necessary to describe and initially
place the function, including the function name, the parameter count, the
function’s formula, and starting estimates and constraints for each
parameter. A special Adjust item allows you to graphically adjust the starting
estimates to better assure a successful (visual) initial placement. You may
also inspect the partial derivatives for the UDF to find instances of multiple
constants, insignificant parameters, and to expose conditions where the
function would be likely to fail.

UDF Selection The keypad with buttons labeled 1 through 15 is used to select the current
UDF. UDFs active at any given time need not be sequential and they will
remain in the position where you install them, even if lower number UDFs
are empty.

Function Name This name will be used to represent the function in the function selection
list and in the numeric summary.

Coefficient Count This is the number of adjustable parameters in the UDF model. The UDF
must contain the number of parameters entered. If, for example, you enter 4
as the coefficient count, the UDF must contain parameters A0,A1,A2,A3 (or
#A,#B,#C,#D). The maximum coefficient count is 10.

6-70 User-Defined Functions

 Automated Peak Fitting

Expression Entry You are furnished an ASCII multi-line editor for entering the user function.
You can use the Cut, Copy, and Paste items to move text about or to paste
in the UDF formula if you placed it into the clipboard via another program.
You may have as many lines for constant expressions as you like, up to 9
lines with auxiliary functions referencing X or the adjustable parameters, and
a line for constructing the overall function.

Y= Expression : The last entry must construct the fitting function and it
must begin Y=. This main function expression is compiled and evaluated
once for each data point’s X value in every iteration using that iteration’s
particular parameters.

Adjustable Parameters : #A (or A0) must be used for the first parameter, #B
(or A1) for the second, #C (or A2) for the third, and so on. The # symbol
makes it easier to spot the adjustable parameters which can easily become
lost within complex expressions. The adjustable parameters #A-#J or
A0-A9 should be used only in main and function expressions.

Functions : These must be referenced by F1 through F9 (or #F1 through

#F9) in expressions and assignments. Functions include X and/or the
adjustable parameters #A-#J or A0-A9. Functions should be used only in
the Y= main expression and in other function expressions. Functions are
compiled and are evaluated once for each data point’s X value in every
iteration using that iteration’s parameters.

Constants: Constant expressions, which are evaluated only once when the
UDF is compiled, can be assigned any unused variable name. You may
define as many constants as you wish. As an example, the expression
SQRT2PI=SQRT(2*PI) would be defined on an initial line and referenced
in subsequent lines. Any assignment to a variable other than F1-F9 (or
#F1-#F9) and Y is assumed to be a constant.

Here is a simple UDF example, first with the #A-#J nomenclature and then
with the A0-A9 format:

S2=SQRT(2)
F1=ERFC(-#D/S2+LN(X/#C)/(#D*S2))
Y=#A+0.5*#B*F1

S2=SQRT(2)
F1=ERFC(-A3/S2+LN(X/A2)/(A3*S2))
Y=A0+0.5*A1*F1

User-Defined Functions 6-71

Automated Peak Fitting

The S2 expression is a constant that is evaluated numerically and stored for

use in subsequent expressions. The F1 expression calls the complementary
error function with an argument expression containing the third (#C or A2)
and fourth (#D or A3) adjustable parameters. The last line is the required
Y= function expression. It contains the first (#A or A0) and second (#B or
A1) adjustable parameters. The F1 and Y expressions are compiled. In the
fit, these expressions are evaluated at the X of every data point for each
iteration’s particular adjustable parameters.

Directly Accessing UDFs based upon PeakFit equations will execute faster if they use the
PeakFit’s Built-In built-in functions available via the Peak Fns or Supplemental function
Functions insert help in the UDF entry dialog. The built-in functions also contain the
conditional code insuring that the functions will return a valid value for all
X.

UDFs with Derivative The first derivative function is DX(n) where n is from 1 to 9 and references
Functions an F1 to F9 expression. The second derivative function is DX2(n). The
following example is for the first derivative of a Voigt function:

F1=VOIGT(A0,A1,A2,A3)
Y=DX(1)

UDFs with Integration The primary integration function is AIP(n,start,end,prec). It first seeks to
Functions achieve the target precision using a successive step Gaussian Quadrature. If
this is unsuccessful, a Romberg procedure follows. If the Romberg fails to
achieve the desired precision, an adaptive Quadrature procedure is used. The
following example uses the AIP() function to fit the cumulative of the
Log-Normal distribution:

LOWER=1E-8
F1=LN($/A2)/A3
F2=EXP(-0.5*F1*F1)
Y=A0+A1*AIP(2,LOWER,X,1E-6)

Note that the $ symbol is used as the variable of integration. In this example,
the second function expression is integrated with $ ranging from a lower
limit to X, to a precision of 6 significant figures.

PeakFit also offers an easy way to integrate built in functions. By adding an

underscore (_) to the beginning of the function name, you can access a
modified form of the built-in functions where the first argument becomes
the variable X in the function (normally X is implicit, and is assumed to be

6-72 User-Defined Functions

 Automated Peak Fitting

the X in the PeakFit data table). The following function creates a cumulative
for the Voigt function:

LOWER=0
F1=_VOIGT($,A0,A1,A2,A3)
Y=AIP(1,LOWER,X,1E-5)

Validation A UDF is extensively validated before it is compiled. If there is a math or

parser error, you will be given a clear indication of the error and the cursor
will be placed at the location where the validation failed. If you are having
trouble with a UDF, you may save the failed UDF to disk. You can then
recall it at some future time in an effort to fix what is wrong.

Starting Estimates and You must enter starting estimates for the parameters in the model. Since
Constraints PeakFit UDF’s will normally be placed and adjusted within one of the
AutoFit procedures, it is important the estimates be sufficient to have the
function appear within the X and Y ranges of the data being fitted.
Minimum and maximum constraints are optional. If a parameter violates a
constraint that you set, a penalty function is added to the chi-square to
attempt to bring the parameter back into a valid range.

Entering Formulas for You may enter formulas for the estimates and lower and upper limits in
Estimates and UDFs. These enable a UDF to be constructed so as to accommodate widely
Constraints varying X and Y data ranges. Since these formulas are evaluated sequentially,
any given formula can reference a previous estimate. For example, A3 can
reference A0, A1, or A2, but not A4 and higher. More commonly, a UDF
estimate formula will reference an X-Y data table constant which PeakFit
computes for each data set. These can consist of the basic X-Y data table
constants, such as XMEAN, XSTD, YRANGE, etc. as well additional
constants used specifically for UDFs.

Since UDFs are often used for fitting single peaks and transitions, PeakFit
determines the following data constants:

For peak functions:

• XCTR, X at Peak Center

• XL50, X at Half-Maxima Left
• XR50, X at Half-Maxima Right
• XW50, X Width at Half-Maxima (FWHM)

User-Defined Functions 6-73

Automated Peak Fitting

For transition functions:

• X50, X at Ymin+Yrange/2
• X25, X at Ymin+Yrange/4
• X75, X at Ymin+3*Yrange/4
• XWTR, X transition width X75-X25

For waveform functions, there are:

• XWL, wavelength
• XPH, phase for sine
• XPH2, phase for sine-squared wave
Note that these constants will not likely have any meaning in data sets
containing multiple peaks or transitions.

When a UDF contains one or more formulas for the adjustable parameter
estimates and limits, it is saved in a binary rather than ASCII form. These
UDFs can only be edited inside of PeakFit. The [UDF] extension is used for
both the binary and ASCII formats.

Reading a UDF Use the Read button to read a UDF from disk. The UDF will be read into
the current UDF position, replacing any UDF that might currently be in this
same position. You may read any UDF into any of the 15 available
positions, even if lower numbered positions are empty. This option will read
both the ASCII format UDFs and the binary format UDFs containing one
or more formulas for estimates or constraints. It is recommended that
UDFs be created only within the program.

Clearing a UDF Use the Clear Current UDF button to immediately free the currently
selected UDF from memory and to reset the UDF entry screen.

Saving a UDF Use the Save button to save a UDF to disk. If the estimates and constraints
are numeric, the UDF is saved in an ASCII format with a [UDF] extension.

If formulas are present, the UDF is saved in a binary format that requires
approximately 4K of disk space. A UDF is always validated before a Save is
made. If the validation fails, you are given an option to save the UDF even
though it contains an error. You can then recall it at some future time in an
effort to repair what is wrong.

6-74 User-Defined Functions

 Automated Peak Fitting

Saving a UDF Library To make it easy to work with up to 15 user functions, any set of installed
UDFs can be saved as a user function library. UDF positions are preserved
as you enter them and installation of empty UDF positions are permitted.

UDF libraries are saved as binary [UDL] files, whether or not estimates
contain formulas. In a UDF library, each UDF consumes about 4K, so a full
15 UDF set will require about 60K disk space. To save the current set of
UDFs to a library, use the Save UDF Library button.

For maximum flexibility, you may wish to save individual UDFs as separate
files in addition to having them in libraries. If you choose to keep UDFs
only in libraries, you can save out individual UDF components simply by
reading the UDL library, selecting the UDF of interest, and then using the
individual Save option.

Reading a UDF Library To read the user functions within a PeakFit UDF library, use the Read UDF
Library button. This will install and validate all of the UDFs in the UDL
library. If a validation fails, you are notified of such and that specific
function is not installed. To extract a UDF from a library, use this option,
select the UDF desired, and then use the individual Save option to create the
UDF file.

Clearing All UDFs To clear all of the currently installed UDFs from memory and to clear the
current entry screen, use the Clear All UDFs button. You must confirm this
option before this the UDFs are freed. Note that it is not necessary to clear
the existing UDFs when reading a UDF library since all current UDFs are
cleared prior to this read operation.

Parameter Even with a successful UDF compilation, you may still get the warning
Contribution Warning Parameter n Makes Less Than .1% Fractional Contribution to Equation
in X-Range of Data. Adjustment is Recommended.

PeakFit determines the minimum and maximum partial derivative for each
parameter. This range of partial derivative is multiplied by the current
estimate for that parameter and then compared to the overall Y-data range.
If a parameter does not make at least a .1% contribution relative to the
Y-data range, this warning is given.

This warning is often a good indicator that the starting estimates are
inadequate for placing the UDF within the range of the current data. It may
also indicate a poorly designed model that contains a parameter that
minimally impacts the equation in the X-range of the data. When this

User-Defined Functions 6-75

Automated Peak Fitting

message occurs, it is recommended that you use the Adjust procedure to

refine the estimates, and if necessary, the Adjust procedure’s Derivatives
option to see if an insignificant parameter is present.

If You are Uncertain of When working with a new or exploratory model, you may not know what
Starting Estimates values constitute good starting estimates. In such a case, for each parameter
you are uncertain of, simply enter 1.0 for the starting estimate (or some
other value for which the UDF is defined). If you achieve a successful
compile but get the parameter contribution warning, use the Adjust item to
graphically set the estimates.

If you do not get the parameter contribution warning, there is a good chance
that the UDF will be placed within the current range of the data.

If there is a revision of a parameter initially entered as a formula in the main

UDF dialog, you must confirm that the original formula for the estimate is
to be replaced with the new numeric value. Note that any save operations
occurring after this replacement will use this numeric value.

Graphical Adjustment Use the Adjust item to open a graph of the current PeakFit data and the
of Starting Estimates UDF.

Do not be surprised if you do not see the UDF at first. If the estimates are
too far off, it is possible that no part of the UDF graph will be anywhere
within the field of the X-Y data.

6-76 User-Defined Functions

 Automated Peak Fitting

For PeakFit, the goal of graphical adjustment is to simply set the parameters
so that the UDF appears somewhere within the field of the data. Final
adjustment of the UDF will occur when it is placed during the AutoFit
procedure.

You may adjust each parameter with the scrollbar or you may enter the
actual value. If you enter an actual value, there will be a brief delay before
the screen is updated, allowing you to complete your entry.

For each parameter that has user-specified minimum and maximum

constraints, as opposed to the defaults, the scrollbar adjustment will range
from this minimum to maximum in 100 steps.

If such constraints are absent, the scrollbar’s sensitivity will depend on

computed partial derivatives. The partial derivatives are updated with each
value entered. If you enter a value for which the partial derivative range is
very nearly zero, or if such is true when you begin, the scroll bar adjustment
may produce very large jumps. In this case you must enter a value that
brings the UDF back into range.

UDF Failures The two most common causes of UDF failures are having the UDF produce
essentially a constant value across the X-range of the data, or to have a
parameter set at such a value, that its impact totally masks the impact of X
varying across the range of the data. You have to be especially cautious
when using a UDF with a very narrow range of X-data.

As you adjust the starting estimates graphically, the UDF is being updated.
When you return to the UDF screen, the estimates will reflect those set in
the graphical procedure. By careful attention in the Adjust feature, you
should have no trouble placing UDFs in the AutoFit procedure.

Inspecting Partial From the graphical UDF Adjustment screen, you can choose the
Derivatives Derivatives option. This displays a similar set of adjustment controls except
instead of seeing a graph, you see a table of partial derivative information.

The % of Y column consists of the % of the Y data range represented by

the range of the partial derivative multiplied by the estimate. The last
column reports a relative percent based on all non-constant parameters, and
these sum to 100%.

The fitting algorithm relies on changes in partial derivatives to guide the

revisions in the parameters that occur with each iteration. If a partial

User-Defined Functions 6-77

Automated Peak Fitting

derivative is constant, then it should have a value of 1.0, meaning that it is a

true constant in the expression. If you see two constants, then at least one of
these parameters should be removed. If you find that across wide
adjustments of a given parameter there is no significant contribution to the

Y-range of the data, whereas the others are making such a contribution, you
should seriously question whether this parameter belongs in your model.

Note that you can set the estimate of a given parameter so far out of range
that no legitimate partial derivatives are found for any parameter in the X
range of the data.

Last Session User Functions

The last session UDFs are automatically saved as a UDF library.

The configuration file PF2.CFG, is actually this UDF library. This file is
automatically written upon exit, but an existing file is only overwritten when
a UDF has been installed in the current session.

When this Last Session User Functions item is selected, these UDFs are
loaded, validated, and compiled.

6-78 Last Session User Functions

 Automated Peak Fitting

Robust Fitting
By selecting a robust minimization procedure in the Peak Fit Preferences
option, robust fitting can be done for any peak fit, including those
containing UDFs.

Limitations of Least-squares fitting involves the minimization of the sum of the squared
Least-Squares residuals. There are two instances where this minimization produces a less
than satisfactory fit. The first is where significant outliers are present. In this
case, the square of the residuals of these outlier points may, within a given
region, significantly shift the fitted curve away from the bulk of the data.

The other instance is when the Y-data spans more than several orders of
magnitude. The squared residuals of the largest valued Y-points can
overwhelm the influence of the squared residuals of the smallest Y-valued
points, causing the smallest Y-value points to either be poorly fitted or not
fitted at all. Data that requires a logarithmic Y-scale to see all of the points
may be a good candidate for robust fitting, especially if four or more major
log divisions are present.

Least Absolute The essence of robust fitting is to use a minimization that is less influenced
Deviation by outliers and the dynamic range of the Y-variable. Instead of minimizing
the sum of the squares of the residuals, an obvious alternative is the
minimization of the sum of the absolute value of the residuals. This is
probably the best known robust method, though not necessarily the best,
and is usually designated as least absolute deviation. Of PeakFit’s three
robust methods, least absolute deviation is the least powerful in terms of
managing outliers.

Lorentzian The intermediate method in terms of power of robustness is a Lorentzian

Minimization minimization. Here what is actually being minimized is the sum of
LN(1+(ABS(residual))^2). If you are uncertain of which robust method to
use, we strongly recommend this Lorentzian minimization. It is very
effective when you have noisy data with outliers or if your peak data spans
many orders of magnitude in Y. These fits also tend to converge as rapidly
as the least absolute deviation minimization.

Pearson VII Limit This is the most robust of the three methods. Here the minimization is of
Minimization the sum of LN(SQRT(1+(ABS(residual))^2)). With this method, outliers
tend to have almost no impact on the fit. This minimization represents an
extreme one where wild and random errors are expected as a natural course.

Robust Fitting 6-79

Automated Peak Fitting

Gaussian Error Each of the various minimization formulas corresponds with a maximum
Distribution likelihood probability distribution.

For least squares, this corresponding error distribution is Gaussian or

normal. Relative to fit standard error (SE), a least-squares goodness of fit,
95.4% of the data points should have residuals with a magnitude less than 2
SE (1 in 21.98 points should lay outside 2 SE). The value is 99.73% for
residuals within 3 SE (1 in 370.4 points should lay outside 3 SE).

The Gaussian distribution decays very rapidly. It suggests that only 1 of

15787 points should lay outside 4 SE, only 1 of 1.74 million points should
lay outside of 5 SE, and those outside 6 SE should be only 1 in a half billion.
Where data is subject to human error, it is generally agreed that major errors
are more likely or probable than the Gaussian distribution suggests.

Double-Sided The least absolute deviation fitting corresponds with a double-sided

Exponential Error exponential probability distribution. While such a distribution produces
Distribution reasonably wide tails, it also has a discontinuous first derivative at the peak
center. As such, the actual error profile isn’t likely to match this
double-sided exponential in this center region. The tails of the double-sided
exponential may however, be very practical.

This distribution suggests that 75.7% of the points should be within 2 SE

and 88.0% within 3 SE. This distribution expects 1 of 16.9 points to be
outside 4 SE, 1 of 34.3 to be outside 5 SE, and 1 of 69.6 to be outside of 6
SE. It is important here to recognize that such SE-relative values are based
upon a least-squares goodness of fit standard deviation.

Lorentzian Error The Lorentzian minimization is strongly recommended for data with
Distribution significant outliers because the Lorentzian distribution is a very natural one
both at the center and the very wide tails. Such broad tails in effect state that
significant errors are expected or likely and that points with such errors
should minimally influence the fit.

The Lorentzian distribution suggests that 68.5% of the points should be

within 2 SE and 83.3% within 3 SE. In terms of outliers, the Lorentzian
expects 1 of 7.9 points to be outside of 4 SE, 1 of 9.8 to be outside of 5 SE,
and 1 of 11.4 to be outside of 6 SE. Even out to 10 SE, the Lorentzian
contains only 94.9% of the points, less than the Gaussian contains within 2
SE.

6-80 Robust Fitting

 Automated Peak Fitting

Pearson VII Limit PeakFit also offers a distribution with the same naturalness about the peak
Distribution center as the Lorentzian but with extremely wide tails. This is based upon

the Pearson VII function with a power term of 0.5. This is the smallest
power term that can be used in the Pearson VII function and still have the
peak converge to a finite area.

It is difficult to estimate areas for this distribution since area convergence

does not occur until near infinity. We can, however assume a functional
range of +/- 50 SE. In this case, 38.3% of the points are expected to be
within 2 SE and 45.9% within SE. Here 1 of every 2.25 points is likely to be
outside 5 SE and 1 of every 3.22 is likely to be outside 10 SE. If you know
your model is very appropriate to your data, but that there is a high
likelihood that a significant number of the data points are very seriously in
error or perhaps even fabricated, this highly robust method may be a good
choice for removing the impact of such points.

Analysis of Residuals For least-squares fits, the parameter confidence limits, parameter standard
errors, and fitted curve confidence and prediction bands can be considered
valid only if the hypothesis of normally distributed errors is confirmed.
PeakFit offers two procedures for confirming this normality of residuals.

Robust Fitting 6-81

Automated Peak Fitting

The Distribution Graph option in the Review’s Residuals option displays a

histogram of the residuals. Although subjective, this may assist you in
making the judgment as to whether or not errors are normally distributed.

The principal approach PeakFit offers for determination of normality is a

Delta Stabilized Normal Probability plot. This plot will contain critical limits
that can confirm or reject the assumption of normality for residuals.

Evenness of Residuals All minimization models are symmetric ones, which assume a symmetric
Distribution distribution of residuals. Such is often not the case, as the errors at one end
of the X-values may have one sign and those at the other may have the
opposite sign. In such cases, a more robust method may be desirable to
lessen the impact of this uneven distribution of residuals.

Using a Robust Small peaks will influence an overall fit’s merit function far less in a
Procedure least-squares fit than in a robust minimization. If you fit with very open
constraints or with no constraints at all, and small peaks are not sufficiently
defined by the data so as to remain in place during the fitting, a robust
procedure may be of value. The phenomena of peaks diminishing to zero or
negative amplitudes, of narrowing to near zero width, and of shifting outside
the x-range of the data occur less frequently when using a robust method.

Using Least-Squares If the Y-range spans no more than a few orders of magnitude, small peaks
are reasonably defined by data, and there are no obvious outliers, you should
generally use least-squares minimization. Most distributions in nature tend
toward the Gaussian, and least-squares is perfectly sufficient most of the
time. You should bear in mind that the robust methods do not have the
same dynamic for convergence, and additional iterations and lengthier fits
may be the price paid for utilizing one of the robust methods.

Curve-Fit Statistics To maintain a true basis for comparing fits in PeakFit, all goodness of fit
statistics are based upon sum-of-squares, even when a fit is made using a
robust method. This raises several important points:

The computation of standard errors and confidence ranges, as well as

prediction and confidence intervals, are based upon sum-of-squares
computations and can be assumed accurate only when the residuals are
normally distributed. If the residuals distribution has narrower or wider tails
than the Gaussian, or has appreciable asymmetry, then this assumption of
normality fails, and these statistics should not be used as absolute measures
of uncertainty. This is true for both least-squares and robust minimizations.

6-82 Robust Fitting

 Automated Peak Fitting

It is quite possible to use a robust method to deal with rare outliers or a wide
dynamic range on Y, and still see a residuals distribution that is essentially
normal. In this case, even though the maximum likelihood estimation was a
robust minimization rather than least squares, the reported confidence
statistics can still be considered valid. If this seems confusing, consider what
happens when a least-squares non-linear fit is stopped somewhat
prematurely. Although the true least-squares minimum was not quite
realized, the reported confidence statistics are not invalid because of
this—they simply fail to reflect the least-squares fit at full minimization. A
robust fit is similar in that it will likewise not reflect this least-squares
minimum.

It should be borne in mind that the various goodness of fit statistics are
sum-of-squares relative, and as such the the results from different
minimizations cannot be compared, even for the same peak setup. Even
though a robust fit may offer a clearly superior estimation by visual
inspection, the goodness of fit statistics for robust fits may approach but will
never match or exceed those for a least-squares fit. This is simply one more
reason to rely on graphical inspection as to the value of a robust fit. The
instances where a robust fit is less influenced by outliers or where it fully
manages a wide dynamic range in Y will not be at all apparent from the
statistical indices of fit.

Robust Fitting 6-83

 PeakFit Functions

7 PeakFit Functions

Gaussian (Amplitude)
 1x − a  

y = a  exp −   
 2  a  
a0 = amplitude
a1 = center
a2 = width (>0)

Fit Time Index = 1.0

(reference for all functions)

Gaussian (Area)
a
 1x − a  

y = exp −   
2π a  2  a  

a0 = area
a1 = center
a2 = width (>0)

Fit Time Index= 0.9

Normal Distribution The Gaussian is also known as the normal distribution function. It is
encountered in virtually every field of science. It is a symmetric function
whose mean µ is equal to a1, the center parameter. Its standard deviation σ
is equal to a2, the width parameter. PeakFit’s area version of the Gaussian is
the standard statistical form.

Gaussian (Amplitude) 7-1

PeakFit Functions

Instrumental In spectroscopy, line broadening from instrument optical and electrical

Broadening effects, usually referred to as the spread function or instrument response
function, is generally Gaussian. The removal or undoing of this instrumental
smearing is the objective of most deconvolution procedures.

Count Statistics In high energy spectra where events are counted, a binomial distribution
would reflect the variance due to count statistics. In the limit of n
approaching infinity, the binomial distribution converges to the Gaussian.

Doppler Broadening Gaussian peak shapes do not occur exclusively from instrumental
broadening and count statistics. In the optical spectra of gases, an inherent
Gaussian line broadening is found.

Molecular velocities also follow a Gaussian form (Maxwell distribution). The

Doppler effect states that light from molecules in motion toward the
observer will be shifted upwards in frequency at the point of observation,
whereas the light from molecules in motion away from the observer is
shifted downward. The probability per unit frequency of observing light at a
frequency ν is given by the following Gaussian form (Jansson reference):

1 ln 2

exp −
(ν − ν )
ln 2 

J
D =  
∆ν D π  ∆ν D 

2kT ln 2
where the Doppler half-width ∆ν D = ν  ,
mc

and m is the molecular mass, k is Boltzmann’s constant, T is the absolute

temperature of the gas, and c is the velocity of light.

Spectral Fitting Note that the x in the Gaussian above is in frequency units. When fitting
Considerations spectral peaks, the x variable must be proportional to frequency, wave
number, or energy. The y variable must be a quantitative measure. As such,
you must convert wavelengths to wave number and transmission to
absorption prior to fitting.

When spectra consist of an energy emission due to a transition between

energy states, the natural line shape is Lorentzian. Molecular collisions in
gases add additional Lorentzian broadening. This means that most spectral
lines will have some measure of Lorentzian character. Unless the Gaussian
instrumental response broadening completely dominates the line shape, the
Voigt function is the theoretical line shape for most spectral peaks.

7-2 Gaussian (Area)

 PeakFit Functions

Central-Limit Theorem The central-limit theorem states that when a function f(x) is convolved with
itself n times, in the limit n->∞, the convolution product is Gaussian with
variance n times the variance of f(x), provided that the area, mean, and
variance of f(x) are finite.

In practice, only a small number of self-convolution sequences produce a

near-Gaussian shape. This is true even of the rectangular distribution. Any
process that could thus be viewed as a series of successive self-convolutions
would thus be expected to produce a Gaussian shape.

While the transport phenomena within a chromatographic column can be

viewed as producing a series of mathematical self-convolutions, the
non-idealities producing band broadening (diffusion, dispersion, mass
transfer resistances, kinetic adsorption and desorption resistances, and
extracolumn effects) generally produce a net spread function that is
asymmetric.

Suggested References Further information on spectral line broadening, and the measuring and
deconvolution of Gaussian instrument response functions can be found in
Peter A. Jansson, Deconvolution with Applications in Spectroscopy, Academic
Press, 1984, ISBN 0-12-380220-2.

Mathematical properties of the normal statistical distribution function can be

found in Merran Evans, Nicholas Hastings, and Brian Peacock, Statistical
Distributions, p.114-118, John Wiley and Sons, 1993, ISBN 0-471-55951-2.

Further information on the normal distribution function is available in

Norman L. Johnson and Samuel Kotz, Continuous Univariate Distributions, Vol
1, p.40-111, Houghton Mifflin, 1970.

Gaussian (Area) 7-3

PeakFit Functions

Lorentzian (Amplitude)
a

y =
 x − a 
1+  
 a 

a0 = amplitude
a1 = center
a2 = width (>0)

Fit Time Index= 0.8

Lorentzian (Area)
a

y =
 x − a  

π a 1 +   
  a  
a0 = area
a1 = center
a2 = width (>0)

Fit Time Index= 1.4

Cauchy Distribution The Lorentzian peak function is also known as the Cauchy distribution
function. It is a symmetric function whose mode is a1, the center parameter.
The tails of the Lorentzian are much wider than that of a Gaussian.
Moments do not exist.

The Lorentzian is encountered primarily in spectroscopy and is sometimes

referred to as the “natural” shape of a spectral line.

Natural Line This broadening is associated with the lifetimes of energy states and the
Broadening Heisenberg principle where the energy uncertainty is inversely proportional
to the uncertainty in time for the occupation of a particular energy state.
There are upper and lower energy states associated with a simple transition
where a photon is absorbed or emitted.

7-4 Lorentzian (Amplitude)

 PeakFit Functions

The probability per unit frequency of a transition yielding a frequency v is

given by the following Lorentzian form (Jansson reference.):
γn
Jn =
γn
4π ( ν − ν  ) +
4

where γn is the sum of the reciprocals of the natural upper and lower state
lifetimes, and ν0 is the center frequency of the emission.

Any spectral emission due to a transition between energy states would thus
be expected to have some degree of Lorentzian broadening.

Collision Broadening In the optical spectra of gases, one must also account for molecular
interactions. Additional line broadening occurs from molecular collisions.
When natural and collision broadening effects are combined, the resulting
line shape is also a Lorentzian, but of greater width than that which would
occur absent collisions (Jansson reference):
γn + γc
J n+ c =
(γ n + γ c )
4π ( ν − ν  ) +
4

where γc is equal to twice the collision frequency. At low pressures, natural

line broadening will dominate the peak width. At high pressures, the peak
width will be primarily due to collision broadening. A shift in line frequency
may also occur with high pressures.

Spectral Fitting Note that the x in the Lorentzian above is in frequency units. When fitting
Considerations spectral peaks, the x variable must be proportional to frequency, wave
number, or energy. The y variable must be a quantitative measure. As such,
you must convert wavelengths to wave number and transmission to
absorption prior to fitting.

Most instrument response functions are Gaussian. This means that most
spectral lines will have some measure of Gaussian character. Unless the
Gaussian instrumental response broadening is nearly absent, the Voigt
function is the theoretical line shape for most spectral peaks.

Central Limit Theorem The central-limit theorem states that when a function f(x) is convolved with
itself n times, in the limit n->∞, the convolution product is Gaussian with
variance n times the variance of f(x), provided that the area, mean, and
variance of f(x) are finite.

Lorentzian (Area) 7-5

PeakFit Functions

As the equation for both natural and collision broadening suggests, this
theorem does not hold for Lorentzians. When two Lorentzian distributions
are convolved with one another, the result is also Lorentzian whose width is
equal to the sum of the widths of the components.

When two Gaussians with equal half-maxima widths are convolved, the
result is a Gaussian with 2 times the width. Convolving two Lorentzians
with equal half maxima widths produces a Lorentzian with twice the width.

Lorentzian Areas It is not particularly easy to envision a peak function without a variance or
standard deviation, but this is true of the Lorentzian. Random samples of a
Lorentzian distribution do not converge to a single mean and standard
deviation as the size of the sample set increases.

Although the area of a Lorentzian is analytically defined and finite, it is not

uncommon for some portion of a fitted Lorentzian’s area to lay outside the
range of the spectrum. As such, analytical peak areas will be greater than the
measured areas which report a numeric integration using the lower and
upper x limits of the data.

Suggested References Further information on spectral line broadening, and the measuring and
deconvolution of Gaussian instrument response functions can be found in
Peter A. Jansson, Deconvolution with Applications in Spectroscopy, Academic
Press, 1984, ISBN 0-12-380220-2.

Mathematical properties of the Cauchy statistical distribution function can

be found in Merran Evans, Nicholas Hastings, and Brian Peacock, Statistical
Distributions, p.42-44, John Wiley and Sons, 1993, ISBN 0-471-55951-2.

Further information on the Cauchy distribution function is available in

Norman L. Johnson and Samuel Kotz, Continuous Univariate Distributions, Vol
1, p.154-165, Houghton Mifflin, 1970.

7-6 Lorentzian (Area)

 PeakFit Functions

Voigt (Amplitude) ∞ exp (− ) t

a
 ∫
−∞  x − a 
dt

a
! + − t
 a 
=
(− )
y
∞ exp t

∫
−∞ a ! +t
dt

a0 = amplitude
a1 = center
a2 = width (>0)
a3 = shape (≥0)

Time Index = 13.2

Voigt (Area)
a a
∞ exp (− ) t

∫
 !
y = dt
π πa −∞  x − a 
a
! + − t
 a 
a0 = amplitude
a1 = center
a2 = width (>0)
a3 = shape (≥0)

Fit Time Index = 11.0

The effects which give rise to a Gaussian line shape, such as instrumental
and Doppler broadening tend to be independent of those which give rise to
a Lorentzian shape, as in the natural broadening from energy state
transitions and collision broadening. As such, the convolution of these two
types of functions results in the theoretical model for a spectral line when
both types of broadening are present. This is the Voigt function.

PeakFit offers amplitude and area forms for two different parametrizations
of the Voigt function. The traditional parametrization is above where a2 is
the Gaussian width and a3 is proportional to the ratio of Lorentzian and
Gaussian widths.

Voigt (Amplitude) 7-7

PeakFit Functions

Voigt (Amplitude, Gaussian/Lorentzian Widths)

∞ exp (− ) t
a
 ∫
−∞ a  x − a 
dt

!
+ − t
2a  2a 
=
(− )
y
∞ exp t

∫
−∞ a !
dt

+t
2a

a0 = amplitude
a1 = center
a2 = width1, Gaussian (>0)
a3 = width2, Lorentzian (≥0)

Fit Time Index = 15.0

Voigt (Area, Gaussian/Lorentzian Widths)

exp(− ) a a
∞ t

∫
 !
= y dt
2π πa −∞ a  x − a 
!
+ − t
2a  2a 
a0 = area
a1 = center
a2 = width1,Gaussian (>0)
a3 = width2, Lorentzian (≥0)

Fit Time Index = 12.6

PeakFit also offers amplitude and area forms for a Voigt parametrization
that directly computes the Gaussian and Lorentzian widths. This enables
you to get a standard error and confidence limits for the computation of
each of the widths.

In PeakFit, the option to vary widths affects the a2 parameter and the option
to vary shape affects the a3 and higher parameters. When fitting these forms
of the Voigt, unless you can justify holding only one of these widths as
constant, you should either vary both width and shape (a2 and a3 are

7-8 Voigt (Amplitude, Gaussian/Lorentzian Widths)

 PeakFit Functions

computed for each peak), or have niether vary (a single a2 and a3 is shared
by all peaks).

Computation of Voigt The Voigt functions are shown containing integrals simply because the
Function convolution integrals lack real closed form solutions. There are, however,
closed form complex solutions. PeakFit implements these complex analytical
closed-form solutions, and as a result, computes exact Voigt functions to
|ε|<1E-14 (to at least 14 significant figures). The Voigt function is now
computed as accurately as a Gaussian, to full precision.

Voigt Approximation Prior versions of PeakFit used an approximation (Puerta and Martin, 1981,
Appl. Opt. 20, 3923-3928) which computed the Voigt to 3-5 digits precision.
This approximation is included in PeakFit for backward compatibility only.
Its fit time index is 5.3.

Spectral Fitting When fitting spectral peaks, the x variable must be proportional to
Considerations frequency, wave number, or energy. The y variable must be a quantitative
measure. As such, you must convert wavelengths to wave number and
transmission to absorption prior to fitting.

Voigt Areas Depending on the extent of a Voigt’s Lorentzian character, it is not

uncommon for some portion of the fitted Voigts’s area to lay outside the
range of the spectrum being fitted. When this occurs, analytical peak areas
will be greater than the measured areas which report a numeric integration
using the lower and upper x limits of the data being fitted.

True Non-Linear Although “deconvolution” seems to be inappropriately applied to all peak

Deconvolution fitting, with the Voigt a true deconvolution does occur, and in a manner that
no noise is introduced into the analysis. Another way to state the Voigt is
Lorentz(a0,a1,a3) ⊗ GaussArea(1,0,a2). As such, the fitted parameters in the
G/L forms directly produce the deconvolved Lorentzian and instrument
response function width. Because convolution of an instrument response
function is area invariant, one assumes that Lorentz(a0,a1,a3) represents the
true peak (as measured by a perfect instrument).

Suggested References Further information on convolution, the Voigt function, and spectral line
broadening can be found in Peter A. Jansson, Deconvolution with Applications in
Spectroscopy, Academic Press, 1984, ISBN 0-12-380220-2.

Voigt (Area, Gaussian/Lorentzian Widths) 7-9

PeakFit Functions

Pearson VII (Amplitude)

a

y = a!
  x − a   a 
1 + 4   2 ! − 1 
 a   
   
a0 = amplitude
a1 = center
a2 = width (>0)
a3 = shape (>0) Fit Time Index = 2.4

Pearson VII (Area)


a!
−1
Γ (a ! )
2
a a
 !
a
!
y = a!
 1
  x − a   a !  


a π a ! Γ  a ! −  1 + 4  2 − 1 

 2   a    

a0 = area
a1 = center
a2 = width (>0)
a3 = shape(>0.5) Fit Time Index = 20.1

One hundred years ago, Karl Pearson pioneered a system of frequency

curves which is still used widely today. His Type VII model has been used as
an approximation for the Voigt function. The reasons for such are readily
apparent. The parameter a2 is the FWHM (full-width at half-maxima). When
a3 is 1.0, the function is an exact Lorentzian. As this a3 power term
increases, the function tends toward the Gaussian. At an a3 of 50, the
function is essentially Gaussian.

The Pearson VII function is a different parametrization of the Student-t

distribution function, also included in PeakFit. Pearson’s work is found in
many statistical references. One that may be of interest is J. K. Ord, Families
of Frequency Distributions, Charles Griffin and Co., London, 1972. ISBN 0
85264 137 0.

7-10 Pearson VII (Amplitude)

 PeakFit Functions

Gaussian-Lorentzian Sum (Amplitude)

 a ln 2   x − a   1 − a!

 ! exp −4 ln 2  + 
  a   
 a π   x − a   
 πa 1 + 4  
   a  

y = a  
! ln 2 1 − a!
 a
+ 
 a π πa 
 
 
 a0 = amplitude 
a1 = center
a2 = width (>0)
a3 = shape (≥0, ≤1)

Fit Time Index = 2.9

Gaussian-Lorentzian Sum (Area)

 
 
 a ! ln 2   x − a   1 − a! 
y = 2a  exp −4 ln 2  + 
   
 a π   a    x − a   
 πa 1 + 4  
   a  

a0 = area
a1 = center
a2 = width (>0)
a3 = shape (≥0, ≤1)

Fit Time Index = 2.3

Another approximation for the Voigt, this model simply sums equal FWHM
Lorentzians and Gaussians. The parameter a2 directly computes the
full-width at half-maximum (FWHM). The parameter a3 varies from 0 to 1,
with 0 being a pure Lorentzian and 1 being a pure Gaussian.

Gaussian-Lorentzian Sum (Amplitude) 7-11

PeakFit Functions

Gaussian-Lorentzian Cross Product (Amplitude)

a

y =
 x − a    x − a  
1+ a !   (
exp 1 − a ! ) 12   
 a    a  

a0 = amplitude
a1 = center
a2 = width (>0)
a3 = shape (≥0, ≤1)

Fit Time Index = 1.8

Yet another Voigt approximation, this model has been used for fitting XPS
spectra. It combines the Gaussian and Lorentzian in a multiplicative format.

As with the Gaussian-Lorentzian sum function, the a3 parameter varies from

0 to 1. Here though, the pure Lorentzian occurs with a3=1 and the pure
Gaussian with an a3 of 0. Another difference is that the degree of Lorentzian
character is not a linear function of a3.

Constrained Gaussian (Amplitude)

 1 x − a  

y = a  exp −   
 2  a a + a  
 !

a0 = amplitude
a1 = center
a2 = width1, fixed (>0)
a3 = width2, freq dependent (>0)

Fit Time Index = 1.2

7-12 Gaussian-Lorentzian Cross Product (Amplitude)

 PeakFit Functions

Constrained Gaussian (Area)

a
 1 x − a  
 
y = exp −   
2π ( a a
 ! +a )  2  a a ! + a  

a0 = area
a1 = center
a2 = width1, fixed (>0)
a3 = width2, freq dependent (>0)

Fit Time Index= 1.7

Event-related data that depend upon counting statistics, such as high energy
spectra, will often have peak widths which increase with energy. This model
allows a simplification of Gaussian fitting when fitting multiple peaks.

Multiple Peaks Only Note that this model has no validity for fitting a single peak. In such a case
the denominator is clearly overspecified, where a1a3+a2 is but a single
parameter. What gives this model validity is sharing a2 and a3 across all
peaks. The concept of this model is to fit many peaks with only two widths.

Single a2, a3 The a2 width represents a constant line spread function, the width of each
peak due to effects which have no frequency or energy dependence. The a3
term simply creates a scaled width which is linearly proportional to energy. It
is not a width per se, but is used to produce a unique frequency-dependent
width component for each peak. When fitting constrained Gaussians, a
single a2 and a3 is always fit. Widths and shapes cannot be varied.

7-13
PeakFit Functions

Gamma Ray Peak (Gaussian + Compton Edge)

 1x − a  

y = a  exp −   +
 
2 a  
  me
 +
a a
 ! −t
−
(
2me a a !− t)  

 t

  a a ! a
 a a t
 ! 
me
2 + 
+
a
m A  ( a a
 ! − t) 
aa ! 
  1  a! x − t 
a a
 " ∫ a
!
2π a
exp − 
 2  a a
  dt
!  
−∞  a! 

a0 = amplitude (photopeak)
a1 = center (energy photopeak and edge)
a2 = width (photopeak and edge smearing)
a3 = calibration (channels/MeV)
a4 = edge magnitude (as fraction of a0)
me = mass electron (0.511004116)

Fit Time Index = 216

Gaussian Photopeak, The Gamma Ray model combines an amplitude Gaussian with a
Shared Response Fn Gaussian-smeared Compton edge function. This is a five parameter model
which assumes the photopeak is Gaussian, and also that the Compton edge
is smeared by the same Gaussian response width as the photopeak.

Energy Scale The model requires that channel 0 represent an energy of 0. If this is not so,
you must first use a calculation to adjust the data, compensating for any
non-zero offset.

Evaluation of Integral We currently lack a closed form for the convolution integral representing the
Gaussian-smeared edge, and as such this function uses numeric integration,
making it PeakFit’s slowest function. If you find this function useful, please
let us know so that we can explore ways to speed up its fitting.

7-14 Gamma Ray Peak (Gaussian + Compton Edge)

 PeakFit Functions

Manual Placement This function cannot be automatically placed. Right click the main anchor of
the peak automatically positioned at the photopeak and change to the
Gamma Ray model. You will then need to adjust the five parameters
directly. All values must be positive. The initial estimates assume a 1 MeV
midpoint in the graph. You must set the calibration parameter a3 so that the
correct edge matches the photopeak. You can lock a3 if you know the
calibration accurately. If the edge itself was automatically detected as a
separate peak, you will need to either toggle this peak off or delete it.

Fit Options You must set the Curvature Matrix in Fit Preferences to Full. In the gap
between the photopeak and Compton, it is possible that both functions will
have decayed to where one of the sparse matrix procedures will detect a
significance limit for the function. These procedures should never be used
with fitting a Gamma Ray model.

We thank Dr. Larry Levit for his expertise and assistance in making this
function possible in PeakFit. At present, this model should be considered
experimental. Nothing has yet been published outside of the PeakFit
materials.

Gamma Ray Peak (Gaussian + Compton Edge) 7-15

PeakFit Functions

Compton Edge
  me
 +
a a
 ! −t
−
(
2me a a !− t)  

 t

  a a ! a
 a a t
 ! 
me
2 + 
+
a
m A  ( a a
 ! − t) 
aa ! 
  1  a! x − t 
y = a ∫ a
!
2π a
exp − 
 2  a a
  dt
!  
−∞  a! 

a0 = amplitude (edge magnitude)

a1 = center (energy edge)
a2 = width (edge smearing)
a3 = calibration (channels/MeV)
me = mass electron (0.511004116)

Fit Time Index = 213

Custom Gamma Ray If the photopeak is non-Gaussian, or if its width does not match the
Models smearing of the edge, you can create your own custom Gamma Ray function
by fitting a Gaussian-smeared Compton Edge alongside any peak model you
like. First place the desired photopeak, such as a Lorentzian or Voigt.

Then, if there is no peak placed at the edge position, click the left mouse
button at the edge site to place one there. The initial parameter estimates
assume a 1 MeV midpoint in the graph. Right click on the peak anchor and
select the Compton Edge function. The parameters here match those in the
Gamma Ray model except that a0 is now an edge magnitude rather than a
fraction of photopeak amplitude. All parameters must likewise be positive.

As with the Gamma Ray model, be sure to match the correct photopeak and
edge, to set the Curvature matrix option in Fit Preferences to Full before
fitting, and be sure that channel 0 corresponds with zero energy.

7-16 Compton Edge

 PeakFit Functions

Notes on Chromatography Functions

Basic Chromatography Dynamics

Retention The position or retention time of a chromatographic peak is governed
largely by the fundamental thermodynamics of solute partitioning between
mobile and stationary phases. This retention time is called the capacity
factor, k′,and it is measured experimentally as

′ = r −1
t
k
t

where tr is the retention time of the peak (usually measured at the peak
maximum, but most rigorously measured as the first statistical moment or
centroid), and t0 is the dead time of the column (the time required for an
unretained solute to traverse the length of the column).

Broadening In chromatographic peaks, band broadening arises from a wide array of

real-world phenomena that are generally categorized as “non-idealities”.
These band broadening processes include:

• Axial diffusion
• Dispersion (varying flow paths through a packed column)
• Resistances to mass transfer in both mobile and stationary phases
• Kinetic resistances to adsorption and desorption
• Extracolumn instrument response effects from detector, tubing, and
electronics

All of these non-idealities are present to widely varying degrees with the
different forms of chromatography. The width and distortion parameters in
a chromatographic model may account for only a portion of these effects, or
they may all be dealt with in a combined way. The physical meaning
attributed to the broadening parameter is highly dependent upon the
chromatographic model from which it was derived, and the assumptions
which underlie that model.

Asymmetry Unlike spectroscopy where bands generally broaden symmetrically,

chromatographic band broadening processes are often asymmetric in nature
because of the directional constraints of column dynamics.

Notes on Chromatography Functions 7-17

PeakFit Functions

Tailing Instrumentally, a detector cannot sense a component prior to its arrival, but
if the detector has a slow response, it may record that component as having
arrived further along in time. Many chromatographic effects can also be
viewed as having directional constraints. In affinity chromatography, for
instance, components tend to desorb very slowly, thereby “drizzling” off the
column with a long extended tail. Column overload, which is a non-linear
effect, causes peaks to elute earlier than they otherwise would because the
column adsorption capacity has been effectively decreased by the overload.
Peaks come out with a sharp right-shifted skew which is commonly called
“tailing”, and in extreme cases, they can even take on a right triangular
profile.

Fronting A more unusual case occurs in gas-liquid chromatography, where column

overload, rather than causing peaks to elute earlier, causes them to elute later
because the solute has effectively increased the column adsorption capacity.
This phenomenon is called “fronting”. In strictly thermodynamic terms,
tailing results from a Langmuir adsorption isotherm, while fronting results
from an anti-Langmuir isotherm.

The essential point is that chromatography exhibits a very wide variety of

asymmetric processes. This explains both the diversity and complexity of the
various mathematical models used to describe the process.

Parameter Inferences The PeakFit chromatographic models each offer an a1 parameter which
relates to the retention time of the peak. More advanced users can relate this
directly to the thermodynamic capacity factor (k’) by applying an X=X/t0-1
calculation to the time axis of the data prior to the data fit, where t0 is the
dead time value.

Band broadening is modeled by the a2 parameter in PeakFit’s

chromatographic models. Peak asymmetry is modeled by a fourth parameter,
a3, and possibly by a fifth parameter, a4. These parameters relate to different
physical broadening and peak skewing processes depending upon the peak
model which is chosen. These are covered in the separate function
descriptions.

If you are only trying to find the best fit for a peak and have no use for the
parameter values, you should fit whichever model works best.

7-18 Notes on Chromatography Functions

 PeakFit Functions

Column Efficiency Chromatographers often characterize band width by the common

half-maxima and extrapolated baseline widths, as well as the second
statistical moment of the peaks. Also used is the concept of column
efficiency as a function of the number of “theoretical plates”.
µ
N moment = where µ1=moment1(centroid), µ2=moment2.
µ
 Ctr
For a Gaussian, this translates to: N
Gauss = 5545178
. 

 .

FWHM

While PeakFit does report the historical Gaussian form of this computation
in its numeric analysis, its use in column efficiency determinations is not
recommended, except for nearly symmetrical peaks.

Reduced Plate Height When comparing columns, a more useful figure of merit is the “reduced
plate height”:
L
h = where L=column length and dp=sorbent particle size.
Nd p

This calculation will allow you to directly compare the efficiency of columns
which have different lengths, and different sorbent particle sizes—factors
which can dramatically affect the number of theoretical plates in the column.
A value of 2 is the theoretical minimum of the reduced plate height (the
maximum possible efficiency).

Resolution Resolution is defined between two adjacent peaks. This is why PeakFit’s
chromatographic analysis reports no resolution for the first peak.
Resolutions are computed by:
tr − tr

Res =
2(W + W )

where tr2 and tr1 are the retention times of the second and first peaks, and
W2 and W1 are their full widths at peak base.

Peak Skew Peaks that are right skewed are commonly referred to as “tailed” and those
left-skewed as “fronted”. Traditionally, chromatographers measure the peak
asymmetry at 10% of the peak maximum while statisticians do so at 50% of
maximum. An asymmetry is a ratio of the width to the right of the mode to
that left of the mode (the mode is the position of the peak apex). As such
tailed peaks have asymmetries > 1, and fronted peaks have asymmetries < 1.

Notes on Chromatography Functions 7-19

PeakFit Functions

Capillary In electrophoretic separations, the migration that occurs under the influence
Electrophoresis of the applied field represents a highly directional constraint, the kind that
may well be suited to the various convolution models. If you find a
particular PeakFit model effective in capillary electrophoretic separations, we
would very much welcome a copy of your published work.

Acknowledgment The chromatographic capabilities within PeakFit owe much to the kind
contributions of Dr. James L. Wade at the Hercules Research Center. Dr.
Wade built upon the earlier work of H. C. Thomas, and derived the NLC
function you find in PeakFit’s chromatographic function set. With the
enhancements in this current version, PeakFit offers an implementation of
Dr. Wade’s NLC function that is fast enough to be used for routine analysis.
If your work involves affinity chromatography, we hope you will seriously
explore Dr. Wade’s NLC model. We also wish to thank Dr. Wade for
assisting us in determining the analytical needs of chromatographers, and for
those portions of these chromatography and function notes he furnished to
us.

Suggested References J. C. Giddings, Dynamics of Chromatography, Part I, Marcel Decker, New York,
1965
J. R. Conder and C. L. Young, Physicochemical Measurement by Gas
Chromatography, John Wiley and Sons, 1989
P. R. Brown and R. A. Hartwick (eds.), High Performance Liquid
Chromatography, John Wiley and Sons, 1989
J. C Sternberg, in Advances in Chromatography, Vol. 2, Eds. J. C. Giddings and
R. A. Keller, Marcel Decker, New York, 1966

7-20 Notes on Chromatography Functions

 PeakFit Functions

HVL (Haarhoff-Van der Linde)

a a
 − a  
1x

exp −  
 ! 2π
a a  2  a  
y =
1 1  x − a  
+ 1 + erf  
 a a !  2   2 a  
exp − 1
 a 
a0 = area
a1 = center (>0)
a2 = width (>0)
a3 = distortion (≠0)

Fit Time Index = 4.4

Gas Chromatography This function is based upon the work of P. H. Haarhoff and H. J. Van der
Linde (Analytical Chemistry 38, 573, 1966). This function was derived
mainly for describing gas chromatographic systems where axial diffusion and
axial dispersion are the primary sources of band broadening, and equilibrium
conditions apply. The HVL function will also describe band profiles under
certain conditions in reversed-phase chromatography.

Tailed and Fronted All physical and chemical resistances to interphase mass transfer are
Peaks neglected by the assumption of an equilibrium adsorption isotherm. A
polynomial expansion is used to approximate the adsorption isotherm, and
by adjusting the sign of the second order term, column overloading under
both Langmuir conditions (tailed peaks) and anti-Langmuir conditions
(fronted peaks) can be modeled. The HVL is the only non-empirical model
capable of modeling fronted peaks.

a1 Assuming the data has been properly transformed, the a1 “center” parameter
is the true thermodynamic capacity factor k’. The time transformation can
be carried out as follows: X=X/t0-1, where t0 is the dead time of the
column. In the case of tailed peaks, k’ will be some time after the peak
maximum. For fronted peaks, k’ is at a time before the peak maximum.

HVL (Haarhoff-Van der Linde) 7-21

PeakFit Functions

a2 The a2 width parameter represents a “lumped” dispersion parameter that

represents all non-ideal band broadening processes.

a3 The a3 distortion parameter represents the curvature of the adsorption

isotherm at the origin and has the following physical significance:

a
! = −Y ′′C 

where Y" is the second derivative of the adsorption isotherm at the origin,
and C0 is the amount of solute injected into the column. For Langmuir
isotherms (downward curvature, tailed peaks), the distortion parameter is
positive. For anti-Langmuir isotherms (upward curvature, fronted peaks),
the distortion parameter is negative.

Using the HVL As an empirical model, the HVL can model a wide variety of tailed and
fronted peaks, including some with extreme asymmetry. It will give very
accurate fitted areas and moments if applied to the appropriate peaks. No
physical significance can be ascribed to the statistical moments of the HVL
distribution, however. The HVL equation is best suited to fitting distorted
peaks in gas chromatography, particularly those whose distortion is caused
by overly large sample concentrations. The fitted parameters can yield the
thermodynamic k’ and isotherm curvature if the chromatographic data was
obtained under conditions consistent with the model’s assumptions.

Automated Fitting of PeakFit’s graphical mapping of this function operates over a half-width
the HVL asymmetry range of 0.1667-6.0. If you are fitting extreme asymmetries, and
the fitting algorithm fails to resolve them when starting at one of these
limits, you may need to adjust the a3 parameter directly prior to fitting.

7-22 HVL (Haarhoff-Van der Linde)

 PeakFit Functions

Giddings
a a 2 a x

  − x − a 
I   exp
 
y =   
a x  a   a 

a0 = area
a1 = center (>0)
a2 = width (>0)

Fit Time Index = 4.7

The Giddings equation was derived by J. C. Giddings (Dynamics of

Chromatography, Part I, Marcel Decker, New York, 1965). The equation
provides a theoretical description for chromatographic peaks obtained under
linear conditions (no column overload), where the kinetic rates of
adsorption and desorption are the primary source of band broadening.
Diffusion, dispersion, and extracolumn effects are assumed negligible.
Where interphase mass transfer can be modeled with first order “rate
constants” of mass transfer, the equation is also valid. The equation models
only kinetic effects, and can fit only slightly tailed peaks.

a1 Assuming the data has been properly transformed, the a1 “center” parameter
is the true thermodynamic capacity factor k’. The time transformation can
be carried out as follows: X=X/t0-1, where t0 is the dead time of the
column. The a1 parameter is also the first statistical moment, or centroid.

Split Peak Effect Under certain unusual conditions corresponding with large a2, a peak
appears to lose mass. This is the “split peak” condition where a fraction
elutes at the column void volume (t/t0=1), and a fraction is retained. It arises
when the kinetics of adsorption and desorption are so slow that a solute
molecule has a finite probability of traversing the column without adsorbing
even once. The split peak effect is occasionally observed in affinity
chromatography.

a2 The width parameter has physical significance in that its inverse is actually a
dimensionless rate constant:
1
a =
k
d t

Giddings 7-23
PeakFit Functions

where kd is the solute desorption constant, and t0 is the dead time of the
column.

As a practical matter, the rate constants derived by fitting this model will
reflect lumped contributions from effects other than chemical desorption
such as slow interphase mass transfer, axial dispersion, and extracolumn
effects. As such, it is generally unwise to regard rate constants derived in this
way as representing only one chemical process.

Relation to NLC The Giddings equation represents the limiting case for the NLC function for
the infinitely dilute case of zero overload (the NLC a3=0).

Using the Giddings As an empirical model, the Giddings function is likely to be most useful only
Functions in cases where the peak shape deviates slightly from Gaussian, and where
detector non-idealities have been greatly minimized. Some types of gas
chromatography and normal phase liquid chromatography are most likely to
give rise to this peak shape. As with all three-parameter models, the shape of
the function is determined by the mathematical constructs within the model
rather than by an adjustable shape parameter.

7-24 Giddings
 PeakFit Functions

NLC (Non-Linear Chromatography)

 a  2 a x   − x − a  
  I   exp
 
a

  a !   x  a   a 
y = 1 − exp −    
a
!   a    a x    a!   
1 − T ,  1 − exp −  
 a a   a 
 
u
(
= exp( − v )∫ exp( − t ) I  2 vt dt
T( u , v ) )

I0() and I1() are Modified Bessel functions

a0 = area
a1 = center (>0)
a2 = width (>0)
a3 = distortion (>0)

Fit Time Index (avg.) = 33.4

An analog of the NLC function was originally derived by H. C. Thomas (J.

Am. Chem. Soc., 66:1664, 1944) for modeling chromatographic
breakthrough curves.

J. L. Wade later extended this approach to chromatographic peaks resulting

from a delta function solute input (James L. Wade, Alan F. Bergold, and
Peter W. Carr, Anal. Chem., 59, 1286-1295, 1987).

The NLC function was derived under an identical set of assumptions as the
Giddings equation, except that no assumption is made about the amount of
solute injected. Kinetic rates of adsorption and desorption are assumed to be
the primary primary source of band broadening and peak skew. Dispersive
and extracolumn effects are assumed negligible. Chemical and diffusive
resistances to interphase mass transfer are “lumped” together in a single,
dimensionless rate parameter. The equation can model extremely tailed, even
right triangular, peaks. In the limiting case of an infinitely dilute solute, as
the a3 distortion parameter goes to zero, the NLC model reduces to the
Giddings equation.

NLC (Non-Linear Chromatography) 7-25

PeakFit Functions

Split Peak Effect Under certain unusual conditions corresponding with large a2, the split peak
effect, described in the section on the Giddings function, can also be
observed.

a1 Assuming the time scale has been properly transformed, the a1 “center”
parameter is the true thermodynamic capacity factor k’, the retention time at
infinite dilution. With highly tailed peaks, k’ will occur well after the peak
maximum. The time transformation can be carried out as follows:
X=X/t0-1, where t0 is the dead time of the column.

a2 The a2 width parameter is exactly analogous the Giddings equation. It has

physical significance in that its inverse is actually a dimensionless rate
constant:
1
a =
k
d t

where kd is the solute desorption constant, and t0 is the dead time of the
column.

a3 The a3 distortion parameter can be used to derive the equilibrium constant,

and thus the adsorption rate constant as well (since kd is known from a2):

a
! = KC 

where K is the equilibrium constant for adsorption (ka/kd), and C0 is the

amount of solute injected. Using the fitted value k’ (a1) and K, it is then
possible to derive the density of adsorption sites on the column which is
simply the ratio k’/K.

Using the NLC Because of its computationally demanding nature, the NLC is not generally
recommended as an empirical fitting function. If your peaks are extremely
tailed or distorted, however, the NLC may offer the best, or perhaps only,
solution.

The NLC model is most valid, in terms of the physical meaning of its
parameters, in liquid affinity chromatography, which is usually characterized
by the slow kinetics of solute adsorption and desorption.

It may also be used in reversed phase and normal phase HPLC to derive
physically meaningful parameters, although this tends to be more difficult in
the case of the desorption rate constant.

7-26 NLC (Non-Linear Chromatography)

 PeakFit Functions

In liquid chromatography, the NLC offers advantages over the HVL in that
it is more likely to give physically meaningful parameters, and it generally
produces a better goodness of fit with moderately or highly distorted HPLC
peaks.

Automated Fitting of PeakFit’s graphical mapping of this function operates over a half-width
the NLC asymmetry range of 1.0-6.0. If you are fitting extreme asymmetries, and the
fitting algorithm fails to resolve them when starting at one of these limits,
you may need to adjust the a3 parameter directly prior to fitting.

Fitting time for a single NLC function averages approximately 33 times that
of a simple Gaussian. When fitting multiple peaks to the NLC model, very
appreciable gains in fitting time are realized (up to tenfold improvements
have been observed) by setting one of sparse curvature matrix processing
options in the fitting engine preferences. Still, this function is not
recommended for machines lacking built-in or coprocessor hardware
floating point.

NLC (Non-Linear Chromatography) 7-27

PeakFit Functions

EMG (Exponentially Modified Gaussian)

a  a a − x    x − a a  a 
y = 
exp + 
 erf  − +

!

2a !  2a ! a
!    2 a 2 a!  a
! 

(Bi-directional form)

a0 = area
a1 = center
a2 = width (>0)
a3 = distortion (≠0)

Fit Time Index = 4.4

The EMG model is currently the most widely used chromatographic model
capable of modeling tailing. It has been quite extensively studied. One
review paper (M. S. Jeansonne and J. P. Foley, J. Chrom. Sci., 29, p258-266,
1991) documents 127 separate EMG-related references.

The EMG is the mathematical convolution of a Gaussian with a first order

decay or exponential response function. There are only two components to
this model, a primary Gaussian, and a one-sided exponential response
function which convolves or smears the Gaussian as in the profiles above.
As this exponential element increases, peaks become more right asymmetric
or tailed. Since this model’s components consist of two of the simplest
functions in nature, it is not difficult to see why various theoretical
underpinnings have been proposed for such.

Gaussian Component: In the EMG, the intracolumn band broadening processes such as axial
Intracolumn diffusion, dispersive effects, mass transfer resistances, and slow kinetics of
Convolution adsorption of desorption are not assumed negligible, but are simply assumed
in the aggregate to distribute or broaden the solute as a Gaussian. This is the
most rudimentary description of intracolumn dynamics.

7-28 EMG (Exponentially Modified Gaussian)

 PeakFit Functions

Exponential Response The exponential response function is easily attributed to extracolumn

Function Component: effects. The EMG assumes a simple first order response from a detector and
Extracolumn Effects its associated electronics. This response function has a non-zero time
constant and is one sided or directionally constrained—a detector does not
have an instantaneous response nor can it sense material prior to its arrival
at the detector.

Note that no intracolumn effects are modeled by the a3 parameter. Indeed,

to assume that any form of intracolumn-originated asymmetry can be
independently attributed to something as rudimentary as a first order decay
model seems a vast oversimplification (note the complexity of the kinetic
chromatographic models which directly deal with only a portion of
intracolumn dynamics).

Tailed and Fronted When treating the exponential component as a detector response function,
Peaks only tailed peaks can be modeled. To enable the EMG to be used as an
empirical function for fronted peaks, PeakFit offers the EMG in a
bi-directional form. The a3/|a3| term simply modifies the model so that it
represents the convolution product of a Gaussian with an exponential that
decays backward in time when a3<0.

True Non-Linear In a convolution model, the parameters actually represent a true

Deconvolution deconvolution of different physical effects. Fitting an analytical form for a
convolution product is also the finest form of deconvolution in that, unlike
discrete Fourier procedures, no noise is introduced into the data.

Another way to state the EMG is Gauss(a0,a1,a2) ⊗ Exp(a3). As such, the

fitted parameters directly produce the deconvolved Gaussian and
deconvolved detector response function time constant. Because convolution
of an instrument response function is area invariant, one simply assumes
that Gauss(a0,a1,a2) represents the true peak (the peak with a perfect
detector and electronics).

Parameter The higher moments of the convolved product cannot be assumed to have
Significance any significance, but the moments of the deconvolved Gaussian are exactly
significant, where a1 represents the centroid or first moment, and a2
represents the standard deviation or square root of the second moment.
Assuming the time scale of the data has been pre-transformed with the
X=X/t0-1 calculation, a1 also represents the true thermodynamic capacity
factor k’. As with all models, the degree to which parameters can be judged
significant must be in proportion to the quality of the fit.

EMG (Exponentially Modified Gaussian) 7-29

PeakFit Functions

Fitting a Constant a3 The nature of this model suggests that a single a3 should be shared across all
peaks since it should represent an invariable time constant for the
instrument’s detector. When a3 is varied in order to account for peak shape
differences across the chromatographic data, you are in effect fitting an
empirical model, and you should no longer assume any significance for the
parameters.

Using the EMG as an As an empirical model, the EMG can model a wide variety of tailed and
Empirical Model fronted peaks, including those with moderate asymmetry. It can be applied
to any form of chromatography where a significant fraction of peak
asymmetry can be ascribed to extra-column effects. Its best use is to derive
accurate peak areas and chromatographic performance measures from
moderately asymmetric peaks.

Automated Fitting of PeakFit’s graphical mapping of this function operates over a half-width
the EMG asymmetry range of 0.45-2.225. If you are fitting greater asymmetries, and
the fitting algorithm fails to resolve them when starting at one of these
limits, you may need to adjust the a3 parameter directly prior to fitting.

7-30 EMG (Exponentially Modified Gaussian)

 PeakFit Functions

GMG (Half-Gaussian Modified Gaussian)

 1 ( − a )  
x
 a ( x − a ) 
  1 + erf  ! 
a
 exp − 
 2
! + a   +a 
a
 2a a
! 
y =
2π a
! +a

a0 = area
a1 = center
a2 = width (>0)
a3 = distortion (≠0)

Fit Time Index = 4.2

The GMG is the mathematical convolution of a Gaussian with a

half-Gaussian response function. As with the EMG, there are only two
components to this model, a primary Gaussian, and a response function
which convolves or smears the Gaussian as in the profiles above. As the
width of the half-Gaussian response increases, peaks become more
asymmetric or tailed.

This empirical model was developed for and is unique to PeakFit. It is

intended to model intracolumn effects in a way the EMG does not. It is not,
however, capable of modeling the extracolumn impact of a slow detector.
For this model to be valid, the detector’s time constant must be negligible
relative to the width of the half-Gaussian accounting for the
intracolumn-originated asymmetry.

In convolution science, anything which assumes a Gaussian in any form,

even a directionally-constrained Gaussian, tends to be on very solid ground.
Clearly there are intracolumn elements which have a directional basis
(adsorption necessarily preceding desorption, diffusion into a dead zone
necessarily preceding diffusion out of such, etc.).

We present the GMG as simple extension to the EMG for describing what
goes on inside the column. Instead of assuming net intracolumn dynamics
result in a singular convolution that produces a perfect Gaussian, the GMG

GMG (Half-Gaussian Modified Gaussian) 7-31

PeakFit Functions

assumes that no chromatographic process can be perfectly symmetrical

because of the directional constraints of intracolumn band broadening
processes.

A testimony to the naturalness of this function is that, unlike the EMG, the
convolution of either half of a response Gaussian produces the same GMG
function. Another interesting element is that the area of an EMG function
can be expressed solely as a constant*amplitude*FW25 (any impact due
asymmetry drops out when using FW25 for the width). For the GMG, this
occurs exactly at FWHM.

In the GMG, the intracolumn band broadening processes such as axial

diffusion, dispersive effects, mass transfer resistances, and slow kinetics of
adsorption of desorption are similarly assumed to exist, but now they are
assumed in the aggregate to distribute or broaden the solute as either a full
Gaussian or as a directionally constrained half-Gaussian. One component of
the process reflects a two dimensional freedom for convolution and the
other reflects a single degree of freedom.

Tailed and Fronted The GMG can directly fit both tailed and fronted peaks. The transition from
Peaks tailed to smooth is continuous and occurs at a3=0.

Parameter The higher moments of the convolved product cannot be assumed to have
Significance any significance, but the moments of the deconvolved Gaussian are exactly
significant, where a1 represents the centroid or first moment, and a2
represents the standard deviation or square root of the second moment.
Assuming the data has been pre-transformed with the X=X/t0-1 calculation,
a1 also represents the true thermodynamic capacity factor k’, the retention
time at infinite dilution.

As with all models, the degree to which parameters can be judged significant
must be in proportion to the quality of the fit. When your data’s later peaks
are well fitted by the GMG but narrower peaks earlier in time are better
fitted by the EMG, there are clearly extracolumn elements which cannot be
ignored.

Fitting a Variable a3 The nature of this model suggests that a single a3 should not be shared
across all peaks unless there is some evidence to suggest a form of steady
state exists where peaks elute at constant asymmetries regardless of retention
time.

7-32 GMG (Half-Gaussian Modified Gaussian)

 PeakFit Functions

Using the GMG as an As an empirical model, the GMG can model a wide variety of tailed and
Empirical Model fronted peaks, including those with moderate asymmetry. Because this is a
new function which has not previously been described in the
chromatographic literature, we would very much welcome your sharing with
us your experiences with this model.

Automated Fitting of PeakFit’s graphical mapping of this function operates over a half-width
the GMG asymmetry range of 0.45-2.225. If you are fitting greater asymmetries, and
the fitting algorithm fails to resolve them when starting at one of these
limits, you may need to adjust the a3 parameter directly prior to fitting.

GMG (Half-Gaussian Modified Gaussian) 7-33

PeakFit Functions

GEMG4 (4 Parameter EMG-GMG Hybrid)

 1 ( − a )  
x
 a ( x − a ) 
  1 + erf  ! 
a
 exp − 
 2
! + a   +a 
a
 2a a
! 
y =
2π a
! +a

a0 = area
a1 = center
a2 = width (>0)
a3 = distortion (>0)

Fit Time Index = 7.2

GEMG5 (5 Parameter EMG-GMG Hybrid)


 1
a exp −
( a x
" − a a " + a ! ) 

1 + erf
 a! −a
 ( )
+ a " x − a a "  

  2

a
" ( a
! +a )  


 2a a
" a! + a


y =
 2a ! 
2π a
! +a erf  − 1
 2a " 

a0 = area
a1 = center
a2 = width (>0)
a3 = distortion1 (>0)
a4 = distortion2 (>0)

Fit Time Index = 9.3

7-34 GEMG4 (4 Parameter EMG-GMG Hybrid)

 PeakFit Functions

EMG+GMG
a

 2a a ! − 2a ! x + a   a a − a! x + a 
y = exp  erfc   ! +
4a !  a
!   2a a ! 

a


exp −
1 ( a
 − x)  
 erfc 
a
" ( a
 − x) 

 + a "   
2 2π a + a"  2 a
 2a a + a" 

a0 = area
a1 = center
a2 = width (>0)
a3 = distortion1 (>0)
a4 = distortion2 (>0)

Fit Time Index = 8.6

The GEMG4, GEMG5, and EMG+GMG function combine the GMG and
EMG in various ways. These are empirical functions furnished by PeakFit
for attempting to model peaks that have an intracolumn-originated
asymmetry suited to the GMG, and an extracolumn detector response
described by the EMG.

Ideally, one would seek a three element convolution:

Gaussian ⊗ Half-Gaussian Response Fn ⊗ Exponential Response Fn, or

GMG ⊗ Exponential Response Fn, or
EMG ⊗ Half-Gaussian Response Fn.

Unfortunately, we do not have a closed form analytical solution for such a

convolution product. The GEMG4, GEMG5, and EMG+GMG models are
offered only as approximations to such a three element convolution. Such
models can no doubt describe peak shapes between those of the EMG and
GMG. They should be regarded entirely as empirical functions, possibly of
some value when the tailing profile is between that of an EMG and GMG.

EMG+GMG 7-35
PeakFit Functions

GEMG5 The GEMG5 is produced by convolving a Gaussian with a hybrid response

function consisting of a half-Gaussian multiplied by an exponential decay. In
the GEMG5, the standard deviation of the half-Gaussian and the time
constant of the exponential are separate parameters, a3 and a4 respectively.

GEMG4 The GEMG4 is similar to the GEMG5, except that a single a3 parameter is
used for both the half-Gaussian SD and the exponential’s τ. Note that there
isn’t a single reason why the two responses would share anything at all in
common. The GEMG4 is simply provided as a 4-parameter empirical
alternative whose shape rests between that of the EMG and GMG.

EMG+GMG As the name implies, the EMG+GMG is not a separate convolution

product, but simply an addition of the two models in the form
½*EMG(a0,a1,a2,a3)+½*GMG(a0,a1,a2,a4). This model represents pure
empiricism.

Using the GEMG4, These are purely empirical models which can describe a wide variety of
GEMG5, and the tailed peaks with peak shapes between the EMG and GMG, including those
EMG+GMG with moderate asymmetry. Because these are new functions which have not
previously been described in the chromatographic literature, we would very
much welcome your sharing with us your experiences with these models.

Automated Fitting PeakFit’s graphical mapping of the GEMG5 and GEMG4 functions
operates over a half-width asymmetry range of 1.01-3.857. For the
EMG+GMG the range is 1.01-8.5. These models are limited to fitting tailed
peaks.

If you are fitting greater asymmetries, and the fitting algorithm fails to
resolve them when starting at one of these limits, you may need to adjust the
a3 and a4 parameters directly prior to fitting.

7-36 EMG+GMG
 PeakFit Functions

Log Normal (Amplitude)

  x  
  ln  
 1  a  
y = a  exp − 
2 a 
   
   
a0 = amplitude
a1 = center (≠0)
a2 = width (>0)

Fit Time Index = 1.6

Log Normal (Area)

  x  
  ln  
a
  1  a  
y = exp −
 2 a  
2π a a ( )
exp a
 




 
a0 = area
a1 = center (≠0)
a2 = width (>0)

Fit Time Index = 3.0

The log-normal appears as a normal distribution when plotted on a log-x

scale. The mode is a1. For x ≤ 0, this function returns 0.

Log Normal (Amplitude) 7-37

PeakFit Functions

Log Normal 4-Parameter (Amplitude)


 ln(2) ln  ( (
 x − a ) a ! − 1  
+ 1 
)
  a a  
  !  
y = a  exp −
 ln( a ! ) 
 
 
 
a0 = amplitude
a1 = center
a2 = width (>0)
a3 = shape (>0,≠1)

Fit Time Index = 2.7

Log Normal 4-Parameter (Area)


 ln(2) ln  ( ( )
 x − a ) a ! − 1  
+ 1 
y =
a
 (
ln 2 a ! )
−1 

exp −


a a
!
 
 

( ) π exp
( )
 ln a ! 

ln( a ! ) 

a a
! ln a !
 4 ln 2   
   

a0 = area
a1 = center
a2 = width (>0)
a3 = shape (>0,≠1)

Fit Time Index = 4.2

This function is included in both the chromatographic and statistical

function sets. The mode is a1, the full-width at half-maximum (FWHM) is
a2, and the asymmetry at half maxima is a3. The function returns 0 for those
x where it is undefined.

7-38 Log Normal 4-Parameter (Amplitude)

 PeakFit Functions

Extreme Value (Amplitude)

  x − a  x − a 
y = a  exp − exp − − + 1
  a  a


a0 = amplitude
a1 = center
a2 = width (>0)

Fit Time Index = 1.0

Extreme Value (Area)

a

  x − a  x − a 
y = exp − exp − − 
a
  a  a


a0 = area
a1 = center
a2 = width (>0)

Fit Time Index = 1.9

The area version consists of the standard statistical form. The mode is a1.

Extreme Value (Amplitude) 7-39

PeakFit Functions

Extreme Value 4 Parameter Tailed (Amplitude)



 x +a ( )−
ln a ! a



 − x + a + a − a a
! exp −
 
= a  exp
a
y
 a a 
!
 
 

a0 = amplitude
a1 = center
a2 = width (>0)
a3 = shape (>0)

Fit Time Index = 2.6

Extreme Value 4 Parameter Tailed (Area)

  1     
1 
 −x + a ln   + a    −x + a ln   + a  
a
   a!      a!  
y = exp
  exp − exp
 
 1 a a
!
 a
a Γ       
 a!       

a0 = area
a1 = center
a2 = width (>0)
a3 = shape (>0)

Fit Time Index = 11.1

This 4-parameter Extreme Value function was created for PeakFit’s

chromatographic function set. This model fits only tailed peaks. The mode is
a1. Note that the amplitude form is much faster.

7-40 Extreme Value 4 Parameter Tailed (Amplitude)

 PeakFit Functions

Extreme Value 4 Parameter Fronted (Amplitude)



 x −a ( )−
ln a ! a



 x + a − a − a a
! exp −
 
= a  exp
a
y
 a a 
!
 
 

a0 = amplitude
a1 = center
a2 = width (>0)
a3 = shape (>0)

Fit Time Index = 2.7

Extreme Value 4 Parameter Fronted (Area)

   1     1 
x+ a ln   − a   x+ a ln   − a  
a
   a!      a!  
y = exp
  exp − exp
 
 1 a a
!
 a
a Γ       
 a!       

a0 = area
a1 = center
a2 = width (>0)
a3 = shape (>0)

Fit Time Index = 11.0

This modified Extreme Value function was created for PeakFit’s

chromatographic function set. This model fits only fronted peaks. The mode
is a1. Note that the amplitude form is much faster.

Extreme Value 4 Parameter Fronted (Amplitude) 7-41

PeakFit Functions

Logistic (Amplitude)
 x − a 
4a  exp − 
 a 
y =
  x − a  
1 + exp − 
  a 

a0 = amplitude
a1 = center
a2 = width (>0)

Fit Time Index = 0.9

Logistic (Area)
 x − a 
a −
 exp 
 a 
y =
  x − a  
1 + exp −
a 
  a 
a0 = area
a1 = center
a2 = width (>0)

Fit Time Index = 1.7

The Logistic peak function is symmetric. The mode is a1. The area version
consists of the standard statistical form.

7-42 Logistic (Amplitude)

 PeakFit Functions

Laplace or Double Exponential (Amplitude)

 2x − a 
y = a  exp − 

 a 

a0 = amplitude
a1 = center
a2 = width (>0)

Fit Time Index = 1.2

Laplace or Double Exponential (Area)

a  2x − a 
exp − 

y =  
2a  a 

a0 = area
a1 = center
a2 = width (>0)

Fit Time Index = 1.3

The Laplace or double exponential peak is symmetric with a discontinuous

first derivative at the mode, a1. The area version represents the standard
statistical form with a reparametrization so that a2 directly yields σ, the
standard deviation.

Laplace or Double Exponential (Amplitude) 7-43

PeakFit Functions

Error (Amplitude)
   
 
 1 x − a  a !  
y = a  exp − 
 2 a

 
a0 = amplitude
a1 = center
a2 = width (>0)
a3 = shape (>0)

Fit Time Index = 2.6

Error (Area)
   
 
a

 1 x − a  a !  
y = a ! +  exp − 
a
 ! 

  a   2 a

a  2 Γ ! + 1  
 2 
a0 = area
a1 = center
a2 = width (>0)
a3 = shape (>0)

Fit Time Index = 12.4

The Error model is symmetric. The mode is a1. The area version consists of
the standard statistical form. For a3=1, the function reduces to the normal
distribution; for a3=2, to the Laplace or double exponential distribution.
Note that the amplitude form is much faster.

7-44 Error (Amplitude)

 PeakFit Functions

Student t (Amplitude)
a

y = a ! +  


 (x − a )  
1 +  
 
 a a
! 
a0 = amplitude
a1 = center
a2 = width (>0)
a3 = shape (>0)

Fit Time Index = 2.2

Student t (Area)
a 1
a
 Γ ! + 
 2 2
y = a ! +  


  (x − a )  
a
!  
a π a! Γ   1 +
 2  
 a a
! 
a0 = area
a1 = center
a2 = width (>0)
a3 = shape (>0)

Fit Time Index = 17.8

The Student t distribution is symmetric. The mode is a1. The area version
consists of the standard statistical form with the exception that the
parameter a1 has been added to enable variable x positioning, and a2 to
enable scaling. The Pearson VII function, included in PeakFit’s
spectroscopic function set, is an alternate parametrization of the Student t
model. Note that the amplitude form is much faster.

Student t (Amplitude) 7-45

PeakFit Functions

Gamma (Amplitude)
( a ! − )
 x − a 
 + a ! − 1
 x − a   a 
y = a  exp − 
 a   a
! − 1 
 
 
a0 = amplitude
a1 = center
a2 = width (>0)
a3 = shape (>1.01,<167.92)

Fit Time Index = 2.7

Gamma (Area)
( a ! − )
a
 x+ a a
!− a − a   x +a a
!− a − a 
y =   exp − 
a Γ (a ! )  a   a 

a0 = area
a1 = center
a2 = width (>0)
a3 = shape (>1.01,<167.92)

Fit Time Index = 11.2

The area version represents a reparametrization of the standard statistical

form. The parameter a1 has been added to enable variable x positioning,
whereas the +a2a3-a2 adjusts a1 so that it represents the mode. The function
returns 0 for those x where it is undefined. Note that the amplitude form is
much faster.

7-46 Gamma (Amplitude)

 PeakFit Functions

Weibull (Amplitude)
− a ! a ! − a!
 
   
 
a − 1 a !  x − a a − 1 a !  a
  x − a  a ! − 1 !  a − 1
= a  !  

+ !   exp −
  a +  + !

 a !  
y
 a!   a  a!    a
     !

 
a0 = amplitude
a1 = center
a2 = width (>0)
a3 = shape (>1.01)

Fit Time Index = 8.5

Weibull (Area)
a!
  
 
  a
 a ! − 1 !  
a ! −
 
 x + a   − a  
 a ! − 1 a !    a! 
a a  
exp −   
 !
y = a!  x + a   − a 
a   a!     a  
     
   
 

a0 = area
a1 = center
a2 = width (>0)
a3 = shape (>1.01)

Fit Time Index = 8.3

The area version represents a reparametrization of the standard statistical

form. The parameter a1 has been added to enable variable x positioning. An
additional adjustment term has been added so that a1 represents the mode.
The function returns 0 for those x where it is undefined.

Weibull (Amplitude) 7-47

PeakFit Functions

Beta (Amplitude)
a ! − a " −

 x − a +
a ( − 1) 
a
!


 x − a +
a ( − 1) 
a
!

 a
! + a" − 2 1 − a
! + a" − 2
a
   
a a
   
   
y = a ! − a " −
 a! − 1   a" − 1 
   
 a ! + a " − 2  a ! + a " − 2
a0 = amplitude
a1 = center
a2 = width (>0)
a3 = shape1 (>1.01)
a4 = shape2 (>1.01) Fit Time Index = 7.6

Beta (Area)
a ! − a " − − a!− a"+
y = a  Γ(a ! + a " )(a ! − 1) ( a
" − 1) ( a
! + a " − 2) ⋅
a ! − a " −

 x − a +
a ( − 1) 
a
!


 x − a +
a ( − 1) 
a
!

 a
! + a" − 2 1 − a
! + a" − 2
 a   a 
   
   
a ! − a " −
 a! − 1   a" − 1 
   
 a ! + a " − 2  a ! + a " − 2
a0 = area
a1 = center
a2 = width (>0)
a3 = shape1 (>1.01)
a4 = shape2 (>1.01) Fit Time Index = 40.8

The area version represents a reparametrization of the standard statistical

form. The parameter a1 has been added to enable variable x positioning, and
a2 to enable scaling. The mode is a1. The function returns 0 for those x
where it is undefined. Note that the amplitude form is much faster.

7-48 Beta (Amplitude)

 PeakFit Functions

Inverted Gamma (Amplitude)

 ( x − a )(a + 1)   a x + x + a − a a − a  − a!
exp  !
 !  ! 
a a 
  a ! x + x + a − a a ! − a   
  a
y =
a x + x + a − a a − a
!  ! 
a
0 = amplitude a
1 = center a
2 = width (>0) a
3 = shape (>0, <167.92)

Fit Time Index = 3.5

Inverted Gamma (Area)

a!
 a (a + 1)  a (a + 1) 
a
 ( a
! + 1) exp − !
 !

 a ! x + x + a − a a ! − a   a ! x + x + a − a a ! − a 
y =
(a ! x + x + a − a a ! − a )Γ(a ! )
a0 = area
a1 = center
a2 = width (>0)
a3 = shape (>0, <167.92)

Fit Time Index = 12.2

The area version represents a reparametrization of the standard statistical

form. The parameter a1 has been added to enable variable x positioning.
Adjustment terms have been added so that a1 is the mode. The function
returns 0 for those x where it is undefined. Note that the amplitude form is
much faster.

Inverted Gamma (Amplitude) 7-49

PeakFit Functions

F-Variance (Amplitude)
a ! − a!+ a"

a 
 x − a
+
a
"( a
! − 2)

 a
! − 2
1 + 
!( + 2) 

 a a a
"  a
" + 2
y = a!+ a"
  x − a
 +
a
" (
a
! − 2) 
 a ! −
 a
!( + 2)    a " (a ! − 2)
!
1 +  a a a
"
 
 a
"
  a ! (a " + 2) 
 
 
a0 = amplitude
a1 = center
a2 = width (>0)
a3 = shape1 (>2, <167.92)
a4 = shape2 (>2, <167.92) Fit Time Index = 10.4

F-Variance (Area)
a! a ! −

Γ
a
! + a"   a!   x − a
 +
a
"( a
! − 2)

a  
!( + 2) 

 2   a"   a a a
"
y = a!+ a"
"( − 2) 
  x − a a a
!
 a  + 
!( + 2)  
!
 a!   a"    a a a
"
a Γ   Γ   1+
 2   2  a
"

 
 
a0 = area
a1 = center
a2 = width (>0)
a3 = shape1 (>2, <167.92)
a4 = shape2 (>2, <167.92) Fit Time Index = 28.2

The area version represents a reparametrization of the standard statistical

form. The parameter a1 has been added to enable variable x positioning, and
a2 to enable scaling. The mode is a1. The function returns 0 for those x
where it is undefined. Note that the amplitude form is much faster.

7-50 F-Variance (Amplitude)

 PeakFit Functions

Chi-Squared (Amplitude)
a ! −

a 
 x − a + a ( a
! − 2)


exp −
1 x − a + a ( a
! − 2)


 a   2 a 
y = a ! −  
( − 2)
a
!
a
! exp − + 1
 2 

a0 = amplitude
a1 = center
a2 = width (>0)
a3 = shape (>2.01, <2*167.92)

Fit Time Index = 5.8

Chi-Squared (Area)
a ! −

a 
 x − a + a ( a
! − 2)


exp −
1 x − a + a ( a
! − 2)


 a   2 a 
y = a!
a 
a 2 Γ ! 
 2

a0 = area
a1 = center
a2 = width (>0)
a3 = shape (>2.01, <2*167.92)

Fit Time Index = 12.7

The area version represents a reparametrization of the standard statistical

form. The parameter a1 has been added to enable variable x positioning, and
a2 to enable scaling. The mode is a1. The function returns 0 for those x
where it is undefined. Note that the amplitude form is faster.

Chi-Squared (Amplitude) 7-51

PeakFit Functions

Pearson IV (Amplitude)
− a!
  a a      a a  
"
 x − − a     x −
"
− a  
     + tan −  a "   
2a ! 2a !
  − 
a
 1 +  exp − a " tan
a    a   2a !   
      
      
y = − a!
 a
"

1 + 
 4a ! 

a0 = amplitude
a1 = center
a2 = width (>0)
a3 = shape1 (>0)
a4 = shape2

Fit Time Index = 6.3

The parameter a1 has been added to enable variable x positioning, and an

additional term has been added to x so that the mode is a1. The fronted
peaks have a positive a4, the tailed peaks a negative a4.

7-52 Pearson IV (Amplitude)

 PeakFit Functions

a3=1 Pearson IV (Area)

  a a
! 
 x+ − a  
− 2
a a exp a
 ! ! tan  
  a 
   
y =
  a a
!  
 x + − a  
  a! π   a ! π    2  
a
exp 2  − exp − 2   1 + 
  a

a0 = amplitude  
 
a1 = center
a2 = width (>0)
a3 = shape

Fit Time Index = 4.5

This function is offered for chromatographic peaks. It is an area form of the

Pearson IV where its a3 is set to 1.0. The a3 in this modified form is the a4 in
the original Pearson IV.

a3=1 Pearson IV (Area) 7-53

PeakFit Functions

a3=2 Pearson IV (Area)

  a a
! 
 a! π   x+ − a  
a 
 exp
 2  !
a (4 + a! ) exp a !
−
tan 

4


 a

   
y =
  a a  
 x + !
− a  
  
(exp(a ! π) − 1) 1+
4
2a
a

 
 

a0 = amplitude
a1 = center
a2 = width
a3 = shape

Fit Time Index = 5.0

This function is offered for chromatographic peaks. It is an area form of the

Pearson IV where its a3 is set to 2.0. The a3 in this modified form is the a4 in
the original Pearson IV.

7-54 a3=2 Pearson IV (Area)

 PeakFit Functions

Erfc Peak (Amplitude)

 x − a  

y = a  erfc    
  a  

a0 = amplitude
a1 = center
a2 = width (>0)

Fit Time Index = 2.3

Pulse Peak (Amplitude)

 x − a    x − a  
y = 4a  exp −  1 − exp − 
 a   a 

a0 = amplitude
a1 = center (pulse initiation)
a2 = width (>0)

Fit Time Index = 1.5

This function returns 0 for all x prior to a1, the pulse initiation time.

Erfc Peak (Amplitude) 7-55

PeakFit Functions

Logistic Dose Response Peak (Amplitude)

− a − a +
4a  x a
 a
y =
( a − 1+ a x
−a
a

a
+ x −a a

a
)
a0 = amplitude
a1 = center (>0)
a2 = width (>1.1)

Fit Time Index = 8.2

This function returns 0 for all x ≤ 0.

Asymmetric Logistic (Amplitude)

− a ! −
  x+a ln a ! − a   a ! +  +a ln a ! − a 
− a!
( + 1)
x
y = a  1 + exp −  a
! a
! exp − 
  a   a 

a0 = amplitude
a1 = center
a2 = width (>0)
a3 = shape (>0)

Fit Time Index = 7.4

7-56 Logistic Dose Response Peak (Amplitude)

 PeakFit Functions

Logistic Power Peak (Amplitude)

− a ! −
 x+ a ln a ! − a   a ! x+ a ln a ! − a  a ! +
 (a ! + 1) a !
a

y = 1 + exp  exp
a
!   a   a 

a0 = amplitude
a1 = center
a2 = width (≠0)
a3 = shape (≥1)

Fit Time Index = 3.8

Pulse Peak Modified with Power Term (Amplitude)

a!
  x − a    x − a 
a 1 − exp −  exp − 

  a   a 
y = − a ! −
a
!
a!
( a
! + 1)

a0 = amplitude
a1 = center (pulse initiation)
a2 = width (>0)
a3 = shape (>0)

Fit Time Index = 6.7

This function returns 0 for all x prior to a1, the pulse initiation time.

Logistic Power Peak (Amplitude) 7-57

PeakFit Functions

Pulse Peak with Second Width Term (Amplitude)

a!
  x − a    x − a 
a 1 − exp −  exp − 

  a   a 
y = − a ! −
a
!
a!
( a
! + 1)

a0 = amplitude
a1 = center (pulse initiation)
a2 = width (>0)
a3 = shape (>0)

Fit Time Index = 4.9

This function returns 0 for all x prior to a1, the pulse initiation time.

Intermediate Peak

y =
a a
  ( (− (
exp a
 x − a ! )) − exp( − a ( x )
− a ! ))
a − a

a0 = initial concentration
a1 = rate 1 (>0, ≠a2)
a2 = rate 2 (>0, ≠a1)
a3 = time reaction begins

Fit Time Index = 2.7

A→a1→B→a2→C, Conc of B

This is the function for a first order intermediate with fitted initial reaction
time. This function returns 0 for all x prior to a3, the time the reaction starts.

7-58 Pulse Peak with Second Width Term (Amplitude)

 PeakFit Functions

Symmetric Double Sigmoidal

  a    a   x − a 
a1 + exp −   1 + exp    exp − 
  2a !     2a !    a 
y =
  a    a 
  x − a +     x − a −  
2 2
1 + exp −   1 + exp − 
  a
!    a
! 
       
a0 = amplitude
a1 = center
a2 = width
a3 = shape (>0)

Fit Time Index = 5.4

Asymmetric Double Sigmoidal

 
 
 
 
a
  1 
y = 1 − 
  a     a  
  x − a +    x − a − 
1 + exp −
2
  1 + exp −
2
 
  a
!     a
"  
         
a0 = maximum amplitude
a1 = center
a2 = width
a3 = shape1 (>0)
a4 = shape2 (>0)

Fit Time Index = 3.4

Symmetric Double Sigmoidal 7-59

PeakFit Functions

Symmetric Double Gaussian Cumulative

  a    a 
  x − a +    x − a − 
2 2
a 1 + erf    1 − erf  

  2 a!    2 a! 
       
y =
  a 
1 + erf  
 2 2 a!  


a0 = amplitude
a1 = center
a2 = width
a3 = shape (>0)

Fit Time Index = 8.9

Asymmetric Double Gaussian Cumulative

  a    a 
a   x − a +    1 1  x − a − 
 2 2
y = 1 + erf     − erf  
2
  2 a
!   2 2  2 a
" 
       

a0 = maximum amplitude
a1 = center
a2 = width
a3 = shape1 (>0)
a4 = shape2 (>0)

Fit Time Index = 6.5

7-60 Symmetric Double Gaussian Cumulative

 PeakFit Functions

Sigmoid
a

y =
 x − a 
1 + exp − 
 a 
a0 = amplitude
a1 = center
a2 = width (≠0)

Gaussian Cumulative
a   x − a  
y = 
1 + erf   

2   2a  

a0 = amplitude
a1 = center
a2 = width (≠0)

Lorentzian Cumulative
a  −  x − a  π 

y =  tan  + 
π   a  2

a0 = amplitude
a1 = center
a2 = width (≠0)

Logistic Dose Response

a

y = a
 x
1+  
 a 
a0 = amplitude
a1 = center (>0)
a2 = width (≠0)

Sigmoid 7-61
PeakFit Functions

Log-Normal Cumulative
  x 
 ln   
a
   a  
y = erfc −
 
2 2a
 
 
a0 = amplitude
a1 = center (>0)
a2 = width (≠0)

Extreme Value Cumulative

  x − ln(ln 2) − a  
y = a  exp − exp − 
  a 

a0 = amplitude
a1 = center
a2 = width (≠0)

Pulse Cumulative
   2 
  x−a ln 1 −  − a  
   2  
y = a  1 − exp − 
 a
  
 
   
a0 = amplitude
a1 = center
a2 = width (≠0)

7-62 Log-Normal Cumulative

 Appendix A: Functions

Appendix A: Functions

PeakFit offers extensive mathematical function support in its Calculation,

Inspect Function(X), and User Defined Function options.

Function Insert Help A special program help exists for accessing the various functions used
within user entered numeric expressions. You simply click on the type of
function you are seeking and then on the specific function of interest. The
function is automatically inserted into the calculation, view function, or
UDF at the current cursor position. You may need to modify the symbols
used as arguments in the function to match the variable intended for the
expression.

General Functions
^ Power

% Modulus Division

! Factorial (of Integer)

ABS(X) Absolute Value

CEIL(X) Integer Above X

EXP(X) Exponential

FLOOR(X) Integer Below X

FRACTION(X) Fractional Part of X

INTEGER(X) Integer Part of X

LN(X) Natural Logarithm

LOG(X) Base 10 Logarithm

General Functions A-1

Appendix A: Functions

PI 3.14159265358979323846

RAND(N) Random Integer Between 0 and N Inclusive

RANDOM Random Number Between 0 and 1 Inclusive

ROUND(X) Round X to Nearest Integer

SQRT(X) Square Root of X

Conditional Expressions
IF(expr,n1,n2) Evaluates to n1 if expr true, n2 if expr false

> .GT. Greater

< .LT. Less

>= .GE. Greater or Equal @MINOR HEAD = <= .LE.

Less or Equal

== .EQ. Equal

<> .NE. Not Equal

Trigonometric Functions
ACOS(X) ArcCosine of X (radians)

ACOSH(X) Hyperbolic ArcCosine

ASIN(X) ArcSine of X (radians)

ASINH(X) Hyperbolic ArcSine

ATAN(X) ArcTangent of X (radians)

A-2 Conditional Expressions

 Appendix A: Functions

ATAN2(X,Y) ArcTangent of X/Y (radians)

ATANH(X) Hyperbolic ArcTangent

COS(X) Cosine of X radians

COSH(X) Hyperbolic Cosine

COT(X) Cotangent of X radians

COTH(X) Hyperbolic Cotangent

CSC(X) Cosecant of X radians

CSCH(X) Hyperbolic Cosecant

DTOR(Deg) Degrees to Radians

RTOD(Rad) Radians to Degrees

SEC(X) Secant of X radians

SECH(X) Hyperbolic Secant

SIN(X) Sine of X radians

SINH(X) Hyperbolic Sine

TAN(X) Tangent of X radians

TANH(X) Hyperbolic Tangent

Trigonometric Functions A-3

Appendix A: Functions

Statistical Functions
! Factorial (of Integer)

B0(N) Bernoulli Number B0(N)

BET(A,B) Beta Function

ERF(X) Error Function

ERFC(X) Error Function, Complement

ERFP1(X) Error Function + 1.0

ERFM1(X) Error Function - 1.0

GAM(X) Gamma Function

HYPM(A,B,X) Hypergeometric M

HYPU(A,B,X) Hypergeometric U

HYPF(A,B,C,X) Hypergeometric F

IBETA(A,B,X) Incomplete Beta Function

IGAM(A,X) Incomplete Gamma

IGAMC(A,X) Incomplete Gamma, Complement

LGNDR(L,M,X) Legendre Polynomial PLM(X)

LNFAC(X) Natural Logarithm Factorial

LNGAM(X) Natural Logarithm Gamma

PFBI(K,N,P) Binomial Probability (Probability >=K Occurrences in N trials,

P=probability/trial)

A-4 Statistical Functions

 Appendix A: Functions

PFCP(K,N) Cumulative Poisson Probability (Probability 0 to K-1 Occurrences for

Mean=N)

PFF(V1,V2,F) F Probability (Probability Variance 1 >= Variance 2 for F=f-statistic,

V1=DOF1,V2=DOF2)

PFST(T,V) Student-t Probability (Probability t < Observed t for v=DOF)

PFX2(X2,V) Chi-Squared Probability (Probability Observed Χ2 < Χ2 for v DOF)

PFX2C(X2,V) Chi-Squared Complement Probability (Probability Observed Χ2 > Χ2 for v

DOF)

PSI(X) Psi (Digamma) Function

Bessel Functions
J0(X) J1(X) JN(N,X) Integer Order J Bessel, orders 0, 1, N

Y0(X) Y1(X) Integer Order Y Bessel, orders 0, 1, N

YN(N,X)
I0(X) I1(X) IN(N,X) Integer Order I Modified Bessel, orders 0, 1, N

I0EIX(X) I1EIX(X) EXP(-X) * Integer Order I Modified Bessel, orders 0, 1

K0(X) K1(X) Integer Order K Modified Bessel, orders 0, 1, N

KN(N,X)
JNU(NU,X) Fractional Order J Bessel

YNU(NU,X) Fractional Order Y Bessel

INU(NU,X) Fractional Order I Modified Bessel

KNU(NU,X) Fractional Order K Modified Bessel

DJNU(NU,X) Fractional Order J Bessel 1st Derivative

DYNU(NU,X) Fractional Order Y Bessel 1st Derivative

Bessel Functions A-5

Appendix A: Functions

DINU(NU,X) Fractional Order I Modified Bessel 1st Derivative

DKNU(NU,X) Fractional Order Modified K Bessel 1st Derivative

SPHJN(N,X) Spherical Bessel j

SPHYN(N,X) Spherical Bessel y

SPHIN(N,X) Spherical Modified Bessel i

SPHKN(N,X) Spherical Modified Bessel k

DSPHJN(N,X) Spherical Bessel j 1st Derivative

DSPHYN(N,X) Spherical Bessel y 1st Derivative

DSPHIN(N,X) Spherical Modified Bessel i 1st Derivative

DSPHKN(N,X) Spherical Modified Bessel k 1st Derivative

AIRYA(X) Airy A Function

AIRYB(X) Airy B Function

DAIRYA(X) Airy A 1st Derivative

DAIRYB(X) Airy B 1st Derivative

A-6 Bessel Functions

 Appendix A: Functions

Integral, Derivative, And Summation Functions

The Calculus functions are only available in UDFs since these require
multi-line expressions.

DX(n) 1st Derivative of function #Fn, d(#Fn)/dX. Derivative is always with

respect to X.

UDF Example (1st Derivative of Gaussian):

F1=A0*EXP(-((X-A1)/A2)^2)
Y=DX(1)

DX2(n) 2nd Derivative of function #Fn, d2(#Fn)/dX2. Derivative is always with

respect to X.

UDF Example (2nd Derivative of Gaussian):

F1=A0*EXP(-((X-A1)/A2)^2)
Y=DX2(1)

AI(n,st,end) Automated Integration of #Fn from st to end with $ as variable of

AIP(n,st,end,prec) integration; supports infinite limits and undefined bounds; AI() seeks 1E-8
precision, AIP() seeks prec fractional convergence.

The AI() and AIP() functions first attempt to achieve the target precision
with a successive step Gaussian Quadrature procedure. If this is
unsuccessful, a Romberg procedure follows. If convergence is still not
achieved, an adaptive quadrature integration is done. For compatibility with
previous versions, less automated integration procedures are also
supported:
• RI(n,st,end)—Romberg Integration
• QI(n,st,end)—Gaussian Quadrature Integration, 24 step
• QIP(n,st,end,prec)—Gaussian Quadrature Integration, prec fractional
convergence
UDF Example (Cumulative of Log-Normal):
LOWER=1E-5
F1=LN($/A2)/A3
F2=EXP(-0.5*F1*F1)
F3=AIP(2,LOWER,X,1E-4)
Y=A0+A1*F3

Integral, Derivative, And Summation Functions A-7

Appendix A: Functions

INF Infinite Limits for Integration functions. You should always use the INF
-INF constant rather than some arbitrarily large number.

SUM(n,st,end,inc) Sums Fn with index $ going from st to end with increment inc

SER(n,st,inc,lim) Sums Fn with index $ beginning at st, incrementing with inc, until
iteration’s fractional contribution lim

PROD(n,st,end,inc) Multiplies Fn with index $ going from st to end with increment inc

XY Data Table Constants And Functions

XMIN Minimum X

XMAX Maximum X

XRANGE Max X - Min X

XMEAN Mean X

XSTD Std Deviation X

XMED Median X

XATYMIN X at Min Y

XATYMAX X at Max Y

YMIN Minimum Y

YMAX Maximum Y

YRANGE Max Y - Min Y

YMEAN Mean Y

YSTD Std Deviation Y

YMED Median Y

A-8 XY Data Table Constants And Functions

 Appendix A: Functions

YATXMIN Y at Min X

YATXMAX Y at Max X

XCTR X at Peak Center (single peak only)

XL50 X at Half-Maxima Left (single peak only)

XR50 X at Half-Maxima Right (single peak only)

XW50 X Width at Half-Maxima (single peak only)

X50 X at Ymin+Yrange/2 (single transition only)

X25 X at Ymin+Yrange/4 (single transition only)

X75 X at Ymin+3*Yrange/4 (single transition only)

XWTR X transition width X75-X25 (single transition only)

XWL Wavelength (single waveform only)

XPH Phase for Sine Wave (single waveform only)

XPH2 Phase for Sine2 Wave (single waveform only)

XYCNT Active Data Point Count

XYINDEX Position within XY Data Table

NOISE(P) Random Uniform P% Y Noise. A calculation Y=Y+NOISE(10) would be

used to add 10% uniform random noise to a data set.

GNOISE(P) Random Gaussian P% Y Noise. A calculation Y=Y+GNOISE(10) would

be used to add 10% Gaussian random noise to a data set.

XY Data Table Constants And Functions A-9

Appendix A: Functions

Built-In Peak Functions

GAUSS(A0,A1,A2) - Gaussian, amplitude

GAUSSA(A0,A1,A2) - Gaussian, area

GAUSSD1(A0,A1,A2) - Gaussian 1st derivative, amplitude

GAUSSD2(A0,A1,A2) - Gaussian 2nd derivative, amplitude

GAUSSC(A0,A1,A2,A3) - Constrained Gaussian, amplitude

GAUSSCA(A0,A1,A2,A3) - Constrained Gaussian, area

LORENTZ(A0,A1,A2) - Lorentzian, amplitude

LORENTZA(A0,A1,A2) - Lorentzian, area

LORENTZD1(A0,A1,A2) - Lorentzian 1st derivative, amplitude

LORENTZD2(A0,A1,A2) - Lorentzian 2nd derivative, amplitude

VOIGT(A0,A1,A2,A3) - Voigt, amplitude

VOIGTA(A0,A1,A2,A3) - Voigt, area

VOIGTAPX(A0,A1,A2,A3) - Voigt, amplitude, approximation

VOIGTGL(A0,A1,A2,A3) - Voigt, Gaussian/Lorentzian widths, amplitude

VOIGTGLA(A0,A1,A2,A3) - Voigt, Gaussian/Lorentzian widths, area

PEARSON7(A0,A1,A2,A3) - Pearson VII, amplitude

PEARSON7A(A0,A1,A2,A3) - Pearson VII, area

GLORXP(A0,A1,A2,A3) - Gaussian-Lorentzian Cross Product, amplitude

GLORSUM(A0,A1,A2,A3) - Gaussian-Lorentzian Sum, amplitude

GLORSUMA(A0,A1,A2,A3) - Gaussian-Lorentzian Sum, area

GAMMARAY(A0,A1,A2,A3,A4) - Gamma Ray(Gaussian+Compton Edge)

COMPTON(A0,A1,A2,A3) - Compton Edge

A-10 Built-In Peak Functions

 Appendix A: Functions

GIDDING(A0,A1,A2) - Gidding, area

EMG(A0,A1,A2,A3) - Exponentially modified Gaussian, area

HVL(A0,A1,A2,A3) - Haarhoff van der Linde, area

NLC(A0,A1,A2,A3) - Non-linear chromatography, area

GMG(A0,A1,A2,A3) - Half-Gaussian modified Gaussian, area

GEMGSUM(A0,A1,A2,A3,A4) - EMG+GMG Sum, area

GEMG4(A0,A1,A2,A3) - 4 parameter EMG-GMG hybrid, area

GEMG5(A0,A1,A2,A3,A4) - 5 parameter EMG-GMG hybrid, area

LOGISTIC(A0,A1,A2) - Logistic, amplitude

LOGISTICA(A0,A1,A2) - Logistic, area

LAPLACE(A0,A1,A2) - LaPlace or Double-Exponential, amplitude

LAPLACEA(A0,A1,A2) - LaPlace or Double-Exponential, area

ERROR(A0,A1,A2,A3) - Error, amplitude

ERRORA(A0,A1,A2,A3) - Error, area

STUDENT(A0,A1,A2,A3) - Student t, amplitude

STUDENTA(A0,A1,A2,A3) - Student t, area

LOGNORM(A0,A1,A2) - Log Normal, amplitude

LOGNORMA(A0,A1,A2) - Log Normal, area

LOGNORM4(A0,A1,A2,A3) - Log Normal, 4-parameter, amplitude

LOGNORM4A(A0,A1,A2,A3) - Log Normal, 4-parameter, area

EXTRVAL(A0,A1,A2) - Extreme Value, amplitude

EXTRVALA(A0,A1,A2) - Extreme Value, area

EXTRVAL4(A0,A1,A2,A3) - Extreme Value, 4-parm tailed, amplitude

Built-In Peak Functions A-11

Appendix A: Functions

EXTRVAL4A(A0,A1,A2,A3) - Extreme Value, 4-parm tailed, area

EXTRVAL4F(A0,A1,A2,A3) - Extreme Value, 4-parm fronted, amplitude

EXTRVAL4FA(A0,A1,A2,A3) - Extreme Value, 4-parm fronted, area

GAMMA(A0,A1,A2,A3) - Gamma, amplitude

GAMMAA(A0,A1,A2,A3) - Gamma, area

WEIBULL(A0,A1,A2,A3) - Weibull, amplitude

WEIBULLA(A0,A1,A2,A3) - Weibull, area

CHISQ(A0,A1,A2,A3) - Chi-Square, amplitude

CHISQA(A0,A1,A2,A3) - Chi-Square, area

BETA(A0,A1,A2,A3,A4) - Beta, amplitude

BETAA(A0,A1,A2,A3,A4) - Beta, area

IGAMMA(A0,A1,A2,A3) - Inverted Gamma, amplitude

IGAMMAA(A0,A1,A2,A3) - Inverted Gamma, area

FVAR(A0,A1,A2,A3,A4) - F-variance, amplitude

FVARA(A0,A1,A2,A3,A4) - F-variance, area

PEARSON4(A0,A1,A2,A3,A4) - Pearson IV, Amplitude

ERFCPK(A0,A1,A2) - Complementary Error Function Peak, amplitude

PULSE(A0,A1,A2) - Pulse, amplitude

LDRPK(A0,A1,A2) - Logistic Dose Response Derivative, amplitude

ASYMLGSTC(A0,A1,A2,A3) - Asymmetric Logistic, amplitude

LGSTCPOWPK(A0,A1,A2,A3) - Logistic w/Power Term, amplitude

PULSEPOW(A0,A1,A2,A3) - Pulse w/Power Term, amplitude

PULSEWID2(A0,A1,A2,A3) - Pulse w/Second Width, amplitude

A-12 Built-In Peak Functions

 Appendix A: Functions

INTMEDPK(A0,A1,A2,A3) - Kinetic Intermediate w/lag

SDS(A0,A1,A2,A3) - Symmetric Double Sigmoidal

SDC(A0,A1,A2,A3) - Symmetric Double Gaussian Cumulative

ADS(A0,A1,A2,A3,A4) - Asymmetric Double Sigmoidal

ADC(A0,A1,A2,A3,A4) - Asymmetric Double Gaussian Cumulative

Built-In Transition and Supplementary Functions

SIGMOID(A0,A1,A2) - Sigmoid, ascending, a2 +

REVSIG(A0,A1,A2) - Sigmoid, descending, a2 -

GAUSSCUM(A0,A1,A2) - Gaussian cumulative, ascending, a2 +

REVCUM(A0,A1,A2) - Gaussian cumulative, descending, a2 -

LDR(A0,A1,A2,A3) - Logistic Dose Response

LORENTZCUM(A0,A1,A2,A3) - Lorentzian cumulative

LOGNORMCUM(A0,A1,A2) - Log-Normal cumulative

EXTRVALCUM(A0,A1,A2) - Extreme value cumulative

PULSECUM(A0,A1,A2) - Pulse cumulative

PULSEPWCUM(A0,A1,A2) - Pulse w/power term cumulative

WEIBULLCUM(A0,A1,A2,A3) - Weibull cumulative

ASYMSIG(A0,A1,A2,A3) - Asymmetric sigmoid

ASYMSIGREV(A0,A1,A2,A3) - Asymmetric sigmoid w/reverse asymmetry

SDSCUM(A0,A1,A2,A3) - Symmetric Double Sigmoidal cumulative

BGCNST(A0) - Constant background

BGLIN(A0,A1) - Linear background

Built-In Transition and Supplementary Functions A-13

Appendix A: Functions

BGQUAD(A0,A1,A2) - Quadratic background

BGCUB(A0,A1,A2,A3) - Cubic background

BGEXP(A0,A1,A2) - Exponential background

BGPOW(A0,A1,A2) - Power background

BGLOG(A0,A1) - Logarithmic background

BGHYP(A0,A1,A2) - Hyperbolic background

SINE(A0,A1,A2) - Sine wave

SINESQ(A0,A1,A2) - Sine-squared wave

SINEDAMP(A0,A1,A2,A3) - Dampened sine wave

SGLEXP(A0,A1) - Single Exponential

DBLEXP(A0,A1,A2,A3) - Sum of Two Exponentials

DECAY1_(A0,A1) - First Order Decay

FORM1_(A0,A1) - First Order Formation

DECAY2_(A0,A1) - Second Order Decay

FORM2_(A0,A1) - Second Order Formation

DECAY2HYP_(A0,A1) - Second Order Decay, Hyperbolic form

FORM2HYP_(A0,A1) - Second Order Formation, Hyperbolic form

DECAY3_(A0,A1) - Third Order Decay

FORM3_(A0,A1) - Third Order Formation

DECAYPT5_(A0,A1) - Half Order Decay

FORMPT5_(A0,A1) - Half Order Formation

DECAYN_(A0,A1,A2) - Variable Order Decay

FORMN_(A0,A1,A2) - Variable Order Formation

A-14 Built-In Transition and Supplementary Functions

 Appendix B: Statistics of Fit

Appendix B: Statistics of Fit

n
Sum of Squares due SSE = ∑ w i ( y$ i − y i )
to Error i =
The SSE, the sum of squares due to error, is the sum of squared residuals.
It is the merit function in least-squares fitting. It is a weighted sum of
squared residuals. The y data value is y i and the estimated y value is y$ i . The
weight value is w i .

n
Sum of Squares SSM = ∑ wi ( y i − y )
about Mean i =
The SSM, the sum of squares about the mean, defines a complete lack of
fit. The mean of the y data values is y .

SSE
Coefficient of r =1−
SSM
Determination

Degree of Freedom DOF =n−m

The total number of data points is n and the total number of parameters
fitted is m.

SSE n (
− 1)
DOF Adjusted r2 DOF r =1−
SSM ( DOF − 1)

SSE
Mean Square Error MSE =
DOF

Appendix B: Statistics of Fit B-1

Appendix B: Statistics of Fit

Fit Standard Error S

E = MSE

The standard error of fit is sometimes known as the root MSE.

SSM − SSE
Mean Square MSR =
m−1
Regression

F-statistic SSM− SSE

MSR m−1
F = =
MSE SSE

DOF

′( X ′X )
−
Confidence Interval CI = y$ i ± t MSE l l

Prediction Interval PI = y$ i ± t MSE (

1 + l ′ X ′X ) − l

In the intervals computations, t is the t-distribution value for the specified

confidence level and degree of freedom, ( X ′X ) is the covariance matrix,
−

and l is the parameter partial derivative vector evaluated at x i .

Suggested Reference: For further information on fit statistics, you may wish to refer to Lyman
Ott, An Introduction to Statistical Methods and Data Analysis, 3rd Edition, 1988,
PWS-Kent.

B-2 Appendix B: Statistics of Fit

 Index

Vary Shape
Index 6-13,6-15,6-23,6-25,6-34,6-36
Vary Widths
6-13,6-15,6-23,6-25,6-34,6-36
Zoom-In 6-12,6-23,6-34
AutoFit Peaks I Residuals 1-2,2-2 - 2-5,6-8,6-10 -
! 6-19
2D View 3-2,3-8 Add Residuals 6-15
Automated Fitting Steps 6-13
A Graphical Layout 6-11
Active Points 1-12,4-4,4-16,5-13 AutoFit Peaks II Second Derivative 1-3,2-6 -
Zero Values 4-4 6-9,6-20 - 6-30
Add Residuals 2-3,6-15 Automated Fitting Steps 6-24
AI Expert Graphical Layout 6-22
2-2,2-6,2-9,5-17,5-27,5-31,5-34,5-36,6-5,6-24,6-36 AutoFit Peaks III Deconvolution 1-4,2-9 -
AIA Import 4-7 2-12,6-9,6-31 - 6-41
Amplitude Labels 3-19 Automated Fitting Steps 6-35
Amplitude Rejection Threshold Graphical Layout 6-33
2-3,2-6,2-9,3-2,6-14,6-25,6-36 Automated Placement 6-7,6-23,6-34
Anchors, Peak 2-9,3-2 AutoScan 1-14
Appending Data 4-2 - 4-3 AutoScan Reset 2-10
Area Labels 3-19
Area Normalize 5-9 B
Area, Cumulative 5-9 Bar Graphs 3-13
ASCII Editor 3-21 - 3-23,4-21 Baseline
Guidelines 4-21 2nd Deriv Zero 6-2 - 6-3
ASCII Export 2-11,6-58 - 6-59 Fit with Peaks 1-13,6-1
ASCII Files Graph 6-4
Appending 4-14 Import and Subtract 5-32
Importing Data 4-8 - 4-9 Non-Parametric 6-3
Listing 4-14 Numeric Option 6-4
Multi-Column 4-10 Parametric Models 6-3
Save Format 4-13 Point Selection 6-5
Single Column 4-10 Pre-Fit and Subtract 1-13,6-1
X-Y Format 4-8 Progressive Linear 6-2
ASCII List 4-14 Subtracting 6-6
Asymmetric Double Gaussian Cumulative 7-60 Tolerance 6-5
Asymmetric Double Sigmoidal 7-59 Two Point 6-2
Asymmetric Logistic Function 7-56 Zeroing Negative Points 6-6
AutoFit Baseline 6-1 - 6-6 Baseline Processing 1-13
AutoFit Peaks Bessel Functions A-5 - A-6
Baseline Fitting 6-10,6-21,6-32 Beta Peak Function 7-48
Overview 6-7 - 6-9 Bitmaps 3-6
Processing Hierarchy 6-10,6-20,6-32 Bold, Titles 3-15
Recommendations 6-8
Refine Shape 6-15,6-26,6-37 C
Smoothing 6-11,6-21 Calculation
Applying 5-7
Cancel 5-7

I-1
Index

Common 5-5 Conditional Expressions A-2

Entering 5-5 - 5-6 Confidence Intervals 2-5,3-26,6-57,B-2
Validating 5-6 Constants, Data Table A-8 - A-9
Cauchy Function 7-4 Constrained Gaussian Function 7-12
Center Labels 3-19 Constrained Gaussian, Constrained Function 7-13
Central-Limit Theorem 7-3,7-5 Constraints 1-15,6-44 - 6-45
Chi-Squared Function 7-51 Convergence, Fit 6-42
Chromatographic Peak Skew 5-29 Convolution 1-5,5-25,5-29,7-28,7-31
Chromatography Copy
Asymmetry 7-17 Data Table to Clipboard 4-22
Broadening 7-17 Copy Data Table to Clipboard 4-22
Column Efficiency 7-19 Copying Data as WK3 4-22
Deconvolving Instrument Response 1-7 Copying Graphs 2-10,3-1,3-5
EMG Function 7-28 - 7-30 Copying Text as WK1 3-21,3-23
EMG+GMG Function 7-35 - 7-36 CPG Files 3-4
Extreme Value 4-Fronted 7-41 Cumulative Area 5-9
Extreme Value 4-Tailed 7-40 Cumulative Areas 3-26
Fronting 7-18 Curvature Matrix, Sparse 1-16,6-43
Gas 7-21,7-24 Custom Colors, Graph 3-17
Gaussian Function 7-1
GEMG4 Function 7-34 D
GEMG5 Function 7-34 Data Enhancement 1-10
Giddings Function 7-23 - 7-24 Data Filtering 1-11
GMG Function 7-31 - 7-33 Data Option 2-11
HVL Function 7-21 - 7-22 Data Smoothing 1-11,5-16 - 5-20
Liquid 7-24 - 7-27 Data Summary 6-64
Log Normal 4-Parameter 7-38 Data Table
NLC Function 7-25 - 7-27 Active Points 4-4
Peak Skew 7-19 Constants A-8 - A-9
Reduced Plate Height 7-19 Marking Data Points Inactive 4-4
Resolution 7-19 Range of Values 4-9
Retention 7-17 Saving 4-13
Tailing 7-18 Saving in Lotus Format 4-13
CLC Files 5-6 Data Transforms 5-5 - 5-6
Clear XY Data 5-12 dBASE
Clipboard Importing Data 4-7
Copy Data Table 4-22 Dead-Time Transformation 4-7,5-5
Importing Data 4-12 Deconvolution 1-4 - 1-5,5-25,5-29
Zero Values 4-12 AutoFit Peaks 6-31 - 6-41
CLR Files 3-17 - 3-18 EMG Function 7-29
Collision Broadening 7-5 Exponential IRF 1-7
Color by Std Error 2-4,3-14 Filtering 5-27,5-30
Colors, Graph 3-2,3-16 Gaussian IRF 1-6,5-24 - 5-27
Column Selection 4-4 GMG Function 7-31 - 7-33
Comma Delimited 3-24 Intrinsic Skew 1-8 - 1-9
Comma to Decimal 3-23 Pitfalls 5-25,5-29
Common Scaling 3-2,3-12 Symmetry 5-26
Compare with Reference 5-3 Voigt Function 7-9
Compton Function 1-14,6-17,6-27,7-16 vs. Peak Separation 1-7

I-2
 Index

Delete Peak 2-10 Fourier Domain Editing 1-11,5-21 - 5-23

Derivative Functions A-7 Fourth Derivative 5-35 - 5-36
DIF Files Inspect 5-35 - 5-36
Multi-Column 4-11 F-statistic B-2
Single Column 4-11 Function Insert Help 3-24,A-1
X-Y Format 4-11 Function Label 2-5
Digital Filter 4-3 Function List A-1 - A-15
Digital Filter, Import 4-2 - 4-3,4-12 Function Popup 2-10
Digital Filter, Non-Parametric 1-10,1-12,4-12,5-10 Function(X)
- 5-11 Cumulative Area 5-38
Doppler Broadening 7-2 First Derivative 5-37
Double Exponential Function 7-43 Inspect 5-37 - 5-38
Drag and Drop Files 4-15 Second Derivative 5-37
Functions
E Bessel A-5 - A-6
EMF Files 3-7 Calculus A-7
EMG Function 1-5,1-8,5-28,7-28 - 7-30 Derivatives A-7
EMG+GMG Function 7-35 - 7-36 General A-1
Enhancement, Data 1-10 Integral A-7
Equivalent Noise 5-20,5-23,5-27,5-31,5-34,5-36 Statistical A-4
Erfc Peak Function 7-55 Trigonometric A-2 - A-3
Error Function 7-44 F-Variance Function 7-50
Evaluation Option 2-11,3-25 - 3-28,5-38,6-64
Excel Export 2-11,6-58 - 6-59 G
Excel Import 2-2,4-4 - 4-6 Gamma Peak Function 7-46
Excluded Points 1-12,4-16 Gamma Ray Function 1-14,6-17,6-27,7-13
Excluded Points 4-4 Gaussian
Explicit Peak Functions 3-24 Instrument Response 1-6,5-24,6-35
Exponential Gaussian Convolution Smoothing 1-11,5-18
Instrument Response 1-7,5-28 - 5-31 Gaussian Cumulative Function 7-61
Export Option 2-11,6-58 Gaussian Function 7-1
Extent, Fit 1-16,6-43 Gaussian-Lorentzian Sum Function 7-11
Extreme Value 4-Fronted 7-41 Gaussian-Lorentzian XP Function 7-12
Extreme Value 4-Tailed 7-40 GEMG4 Function 7-34
Extreme Value Cumulative Function 7-62 GEMG5 Function 7-34
Extreme Value Function 7-39 Generating Data 3-25 - 3-26
Giddings Function 7-23 - 7-24
F GMG Function 1-5,1-8 - 1-9,7-31 - 7-33
FFT Data 5-23 Graphical Adjustment 1-14
FFT Editing 5-21 - 5-23 Graphical Fitting 2-4,6-52 - 6-55
FFT Filtering 1-11,5-17 Graphical Placement 6-7,6-16,6-26,6-37
File Selection 4-3 Graphs, PeakFit 3-1 - 3-19
Fit Preferences 6-42 - 6-46 Grids, State 3-9
Fit Statistics B-1 - B-2
Fit, Aborting 6-51,6-54 H
Fit, Additional Adjust 6-51,6-55 Help
Fitting, Numeric 2-7 Function Insert 3-24,A-1
Fitting, Visual 2-4 Hidden Peaks 1-1,2-1
Font, Graph 3-2 Deconvolution 1-4

I-3
Index

Residuals 1-2 L
Second Derivative 1-3 Labels, Peak 3-2,3-19
Hidden Peaks Labels, State 3-9
Residuals 2-3 Laplace Function 7-43
Hints, XY 3-9 Layout, Graph 3-8
HVL Function 7-21 - 7-22 Levels, Peak Placement
1-14,2-9,6-7,6-23,6-26,6-34
I Levels, Peak Placement 2-14 6-12
IF Function A-2 List Peak Estimates 6-18,6-29,6-40
Implicit Peak Functions 3-24 Listing ASCII Files 4-14
Import Local Minima 1-15 - 1-16
File Formats 4-2 Loess Smoothing 1-11 - 1-12,5-18
Import Clipboard 4-12 Log Normal 4-Parameter 7-38
Import Digital Filter 4-3 Log Normal Cumulative Function 7-62
Import Guidelines 4-6 Log Normal Function 7-37
Importing AIA Files 4-7 Log Scaling 3-11
Importing ASCII Data 4-8 - 4-10 Logistic Dose Response Function 7-61
Multi-Column 4-10 Logistic Dose Response Peak 7-56
Single Column 4-10 Logistic Function 7-42
Importing Clipboard Data 4-12 Logistic Power Function 7-57
Importing Data 4-2 Lorentzian Cumulative Function 7-61
Importing dBASE Data 4-7 Lorentzian Function 7-4
Importing DIF Data 4-11 Lotus 123 Export 2-11,6-58 - 6-59
Multi-Column 4-11 Lotus 123 Import 4-4 - 4-6
Single Column 4-11
XY 4-11 M
Importing File Data 4-3 Maximum Iterations 6-42
Importing Scan 6-18,6-29,6-40 Metafiles 3-6
Importing SigmaPlot Data 4-7 Moving Points 5-14
Importing Worksheet Data 4-4 - 4-6
Inactive Points 1-12,4-4,4-16,5-13 N
Clearing 5-12 Natural Line Broadening 7-4
Zero Values 4-4 Negative Data, Zeroing 5-8
Inactive Points Scaling 3-2,6-5,6-24,6-35 NLC Function 7-25 - 7-27
Included Points 1-12,4-4,4-16 Noise Functions A-9
Inspect Function(X) 5-37 - 5-38 Noise, Adding 5-5
Functions A-1 - A-15 Non-Linear Fitting 1-15 - 1-16
Instrument Response Fn 1-5,1-7,5-24 - 5-31,6-35 Non-Parametric Digital Filter 1-10,1-12,4-12,5-10
Instrumental Broadening 7-2 - 5-11
Intermediate Peak Function 7-58 Normal Distribution Function 7-1
Intervals, Confidence & Prediction Normalize, Unit Area 5-9
2-5,3-3,3-26,6-56 Numeric Adjustment 1-14
Invert Plots 3-2,3-9 Numeric Adjustment 2-14 6-47
Inverted Gamma Function 7-49 Numeric Fitting 2-7,6-49 - 6-51
Italic, Titles 3-15 Numeric Graph Copy 3-6
Iteration, Graphical Update 6-54 Numeric Placement 6-8,6-17,6-27,6-38
Iteration, Stopping at Current 6-50,6-54 Numeric Summary 2-5 - 2-6,6-60
Analysis of Variance 6-63

I-4
 Index

Chromatographic Analysis 6-62 Peaks, Hidden 1-1

Details of Fit 6-63 Pearson IV Function 7-52 - 7-54
Fitted Parameters 6-61 Pearson VII Function 7-10
Measured Values 6-61 Point Format 3-2,3-13
Overlap Areas 6-63 Prediction Intervals 2-5,3-26,6-57,B-2
Parameter Statistics 6-62 Preferences, Fit
6-15,6-19,6-26,6-29,6-37,6-40,6-42 - 6-46
O Pre-Smoothing 1-11
OLE2 Previous Files 4-15
Support for Excel Import 4-5 PRF Files 6-46
Outliers, Toggling in Fit 6-54 Printer Setup 3-4,3-20,3-23
Printing Graphs 2-7,3-3
P Printing Text 3-20,3-23
Parameter Priority, Scaling 3-12
Common Estimates 6-48 Process Points Mode 3-1
Locking 6-47 - 6-48 Pulse 2nd Width Function 7-58
Sharing 6-47 - 6-48 Pulse Cumulative Function 7-62
Parameter Adjustment 1-14,6-17,6-27,6-38,6-47 Pulse Peak Function 7-55
Parameter Constraints 1-15 Pulse Power Function 7-57
Peak Analysis 2-6,6-60
Peak Anchors 2-9,3-2 Q
Peak Fit, Graphical 6-52 - 6-55 Quattro Pro Export 2-11,6-58 - 6-59
Peak Fit, Numeric 6-49 - 6-51 Quattro Pro Import 4-4 - 4-6
Peak Functions
Built-In A-10 - A-12 R
Peak Labels 3-2,3-19,6-25,6-36 r2 B-1
Peak Placement, Levels 1-14,6-17,6-23,6-26,6-34 r2, DOF Adjusted B-1
Peak, Add 6-16,6-26,6-37 Reference, Comparing 5-3
Peak, Change Function 6-16,6-27,6-38,6-47 Reset AutoScan 6-18,6-39
Peak, Delete 2-10,6-16,6-26,6-38,6-47 Reset Scaling 3-2,3-12
Peak, Graphically Adjust 6-16,6-27,6-38 Residuals 1-2
Peak, Toggle 6-16,6-27,6-38 Residuals Graphs 2-8,3-3,6-65
PeakFit Editor 4-16 - 4-19 Delta SNP 2-8,6-67
Active and Inactive Points 4-16 Distribution 2-8,6-66
Auto-Entry 4-17 Maximum Likelihood 6-68
Basic Titles 4-19 Reverse Scaling 3-12
Calculation 4-17 Review 6-56 - 6-68
Cancelling Revisions 4-19 Robust Fitting 1-16,6-79 - 6-83
Clearing Data 4-18 Least Absolute Deviation 6-79 - 6-80
Copy, Cut, Paste 4-17 Least-Squares 6-79 - 6-80
Data Input 4-16 Lorentzian 6-79 - 6-80
Deleting Rows 4-17 Pearson VII Limit 6-79,6-81
Functions and Operators 4-19 Robust Minimization 6-42
Inserting Rows 4-17 Roots 3-25
Reversing X,Y 4-19
Saving Data 4-18 S
Sorting Data 4-19 SAMPLE.XLS 2-1
Weighting Data 4-20 Saving
PeakFit Graphs 3-1 - 3-19 Data Table 4-13

I-5
Index

Scan 6-19,6-29,6-40 Double Exponential Function 7-43

Saving Data as WK3 4-13 Error Function 7-44
Saving Text as WK3 3-20,3-23 Extreme Value -4 Fronted 7-41
Savitzky-Golay Smoothing 1-11,5-19 Extreme Value -4 Tailed 7-40
Scaling, Default 3-2,3-12 Extreme Value Function 7-39
Scaling, Graph 3-11 F-Variance Function 7-50
Scaling, Manual 3-12 Gamma Function 7-46
SCL Files 3-12 Gaussian Function 7-1
SCN Files 6-18,6-29,6-40 Inverted Gamma Function 7-49
Second Derivative 1-3,2-6 - 2-8 Laplace Function 7-43
AutoFit Peaks 6-20 - 6-30 Log Normal 4-Parameter 7-38
Inspect 5-33 - 5-34 Log Normal Function 7-37
Sectioning 1-12 - 5-15 Logistic Function 7-42
Global 1-12 Normal Function 7-1 - 7-3
Graphical 5-14 Pearson IV Function 7-52 - 7-54
Local 1-12,5-15,6-17,6-28,6-39 Student t Function 7-45
Numeric 5-13 Weibull Function 7-47
Sectioning Mode 3-1 Statistics, Fit B-1 - B-2
SigmaPlot Status Bar, Graph 3-2,3-9
Importing Data 4-7 Status Bar, Main Window 4-23 - 4-24
SigmaPlot Export 2-11,6-58 - 6-59 Student t Function 7-45
SigmaPlot Import 4-7 Subscripts 3-15
Sigmoid Function 7-61 Subtract Baseline 1-13
Smoothing Data 1-11 Imported 1-13
Space Delimited 3-24 Subtract Baseline 2-13
Spectral Fitting Considerations 7-2,7-5,7-9 Imported 5-32
Spectroscopy Summation Functions A-7
Compton Edge Function 7-16 Superscripts 3-15
Constrained Gaussian Function 7-13 Supplementary Functions
Deconvolving Instrument Response 1-6 Built-In A-13 - A-15
Gamma Ray Function 7-14 - 7-15 Symbols, Titles 3-15
Gaussian Function 7-1 - 7-3 Symmetric Double Gaussian Cumulative 7-60
Gaussian-Lorentzian Sum Function 7-11 Symmetric Double Sigmoidal 7-59
Gaussian-Lorentzian XP Function 7-12
Lorentzian Function 7-4 - 7-6 T
Pearson VII Function 7-10 Tab Delimited 3-24
Voigt Function 7-7 TableCurve Editor 4-16 - 4-19
Voigt G/L Function 7-8 - 7-9 Text Windows 3-20 - 3-21
Split Plots 3-2,3-9 Titles
Spreadsheets New X-Y 5-4
Column Selection 4-4 Using Previous 4-6
SSE B-1 Titles, Custom Graph 3-2,3-14
SSM B-1 Titles, Size 3-14 - 3-15
Standard Error, Fit B-2 Titles, State 3-9
Statistical Functions A-4 Toggling Points 5-14
Statistics Tool Bar 4-23
Beta Function 7-48 Transforming Data 5-5 - 5-6
Cauchy Function 7-4 Transition Functions
Chi-Squared Function 7-51 Built-In A-13 - A-15

I-6
 Index

Transmission to Absorbance 5-5 Z

Trigonometric Functions A-2 - A-3 Zero Values
TTL Files 3-15 Management 4-4
Zeroing FFT Frequencies 5-22
U Zeroing Negative Data 5-8,6-6
Uniformly-Spaced X-Values 1-12,5-11 Zoom Mode 3-1,5-15
User Functions 6-69 - 6-77 Zoom-In 2-9,6-12,6-23,6-34,6-54
Built-In Functions 6-72 Zoom-Out 3-12,5-38
Clearing 6-74
Derivatives and Integrals 6-72
Entering 6-71
Estimates and Constraints 6-73
Functions A-1 - A-15
Graphical Adjustment 6-76
Importing 6-74
In PeakFit 6-69
Last Session 6-78
Libraries 6-75
Partial Derivatives 6-77
Saving 6-74
Selecting 6-47,6-70

V
V2D Files 3-10
Visual Adjustment 1-14,2-9,6-16,6-26,6-37
Visual Fitting 2-4,2-10
Voigt Function 1-5 - 7-7
Voigt Function, Approximation 7-9
Voigt Function, G/L Widths 7-8

W
Wave Number Calculation 5-5
Weibull Function 7-47
Weighting Data
PeakFitEditor 4-20
Standard Deviations 4-20
Weights 4-9
Widths, Varying 2-10
WMF Files 3-7

X
XY ASCII Files 4-8
XY Data
Clearing 5-12
XY Data Table Constants A-8 - A-9
XY DIF Files 4-11
XY Hints 3-1,3-9

I-7