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Units, Functions, Etc: Polynomial Surface Fit

This document describes fitting polynomial surfaces to multivariate data. It defines functions for determining the number of polynomial terms, ordering the coefficients, and fitting surfaces to data. The data is fitted with a quadratic polynomial using two different response variables, resulting in surfaces F1 and F2.

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pravin
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83 views6 pages

Units, Functions, Etc: Polynomial Surface Fit

This document describes fitting polynomial surfaces to multivariate data. It defines functions for determining the number of polynomial terms, ordering the coefficients, and fitting surfaces to data. The data is fitted with a quadratic polynomial using two different response variables, resulting in surfaces F1 and F2.

Uploaded by

pravin
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Units, functions, etc

Data
0 1 2 3
0 0.375 0.25 0.859 1.011
1 0.375 0.25 0.935 1.124
2 0.25 0.5 0.888 ...

0 1 2 3
X Data 100 Y Data 100 Z1 Data Z2 Data

( X YZ1)

Polynomial surface fit

0 0 1 0
XY X XY Y

Polynomial degree: Degree 2

coeffs1 regress( XY Z1 Degree )


Get rid of the initiaal tersm that are only needed by interp

0
coeffs1 submatrix coeffs1 3 last coeffs1 0 0
T
coeffs1 2.613 10
5
7.5108 10
5
0.0074 0.7396 0.0026 0.0001
Functions for coefficient order

( 2 deg)
Nterms( deg)
deg 2

Step( v Nvar deg) for i 0 deg if Nvar = 1


v v i
i 0 i 0
for i 0 Nvar 1 if deg = 1
v v 1
i i i i

otherwise
inc Nterms( deg 1 )
for i 0 inc 1
v v 1
i Nvar 1 i Nvar 1
v stack( Step( submatrix( v 0 inc 1 0 cols( v ) 1 ) Nvar deg 1 ) Step( sub
v

COrder( deg) v 0
Nterms( deg) 1 1
Step( v 2 deg)
Functions for coefficient order

The number of terms: Nterms( Degree ) 6

The powers of x and y. The first column is the power of x, the second column the power of y

1 1
0 2

0 1
I COrder( Degree ) I
0 0
1 0


2 0

This is then the expression for the multidimensional polynomial






last ( Coeffs)
I I
Coeffs X Y 1

i 0 i
Fit( X Y Coeffs I)
i
i 0
Fit1( x y ) Fit( x y coeffs1 I)

Create a fitted surface using the CreateMesh function

For viewing, extrapolate surface by: E 3

x s min( X) E x e max( X) E y s min( Y) E y e max( Y) E


F1 CreateMesh Fit1 x s x e y s y e 10 10

( X Y Z1 ) F1
This data has a lot of scatter!

( X Y Z2 )

coeffs2 regress( XY Z2 Degree )


0
coeffs2 submatrix coeffs2 3 last coeffs2 0 0
T
coeffs2 4.1329 10
5
3.6324 10
5
0.0014 0.9215 0.0195 0.0004

bmatrix( v inc rows( v ) 1 0 cols( v ) 1 ) Nvar 1 deg))


Fit2( x y ) Fit( x y coeffs2 I)


F2 CreateMesh Fit2 x s x e y s y e 10 10

( X Y Z2 ) F2

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