Authors: Thio-Henestrosa, S.

and
Comas, M.

CoDaPack v2 USER’s GUIDE

April 20, 2016
 V

Dr. Santiago Thio-Henestrosa

Professor Titular d’Universitat (full professor)

University of Girona
Dept. of Computer Science and Applied Mathematics
Campus Montilivi — P-1, E-17071 Girona, Spain
santiago.thio@udg.edu

Mr. Marc Comas

Professor Associat (adjunct professor)

University of Girona
Dept. of Computer Science and Applied Mathematics
Campus Montilivi — P-1, E-17071 Girona, Spain
marc.comas@udg.edu
Preface

The program CoDaPack v2, together with the present manual, can be down-
loaded for free from the web at
http://ima.udg.edu/CoDaPack
There is also available a whole library of subroutines for Matlab, developed
mainly by John Aitchison, which can be obtained from John Aitchison himself
or from anybody of the compositional data analysis group at the University of
Girona (www.udg.edu). Finally, those interested in working with R (or S-plus)
may either use the full-fledged packages “compositions” van den Boogaart
et al. (2010) or ”robCompositions” Hron et al. (2010).

Santiago Thio-Henestrosa
Girona, April 20, 2016
Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 General considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 File Menu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1 General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Data Menu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.1 General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.2 Data: Transformations: ALR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.3 Data: Transformations: CLR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.4 Data: Transformations: ILR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.5 Data: Centering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.6 Data: Subcomposition/Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.7 Data: Amalgamation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.8 Data: Perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.9 Data: Power Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.10 Data: Rounded Zero Replacement . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.11 Data: Numeric to categorical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.12 Data: Categoric to numeric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.13 Data: Add numeric variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.14 Data: Delete variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Statistics Menu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.1 General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.2 Statistics: Compositional statistics summary . . . . . . . . . . . . . . . . 17
4.3 Statistics: Classical statistics summary . . . . . . . . . . . . . . . . . . . . . 18
4.4 Statistics: Additive-Logistic normality test . . . . . . . . . . . . . . . . . . 19
4.5 Statistics: Atypicality Indices (Fig. 4.7) . . . . . . . . . . . . . . . . . . . . 19
X Contents

5 Graphs Menu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.1 General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.2 Graphs: Ternary Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5.3 Graphs: Principal Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5.4 Graphs: ALR Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5.5 Graphs: CLR Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5.6 Graphs: ILR Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5.7 Graphs: Biplot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5.8 Graphs. Balance Dendrogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1
Introduction

The software presented in this manual is still under construction. The idea
is to build a user-friendly application to be used in all fields of science and
technology which need to operate with compositional data. It is conceived
as a complement to other statistical software, which is perfectly appropriate
for working on coordinates, and not as standalone software for the statistical
analysis of compositional data. Probably in a future it will incorporate more
classical statistic features.
To use this application the Software Java Virtual Machine is needed (min-
imum version 1.5.0). To get it working, it suffices to install and open the file
CoDaPack.exe.
CoDaPack v2 is based on menus. Numerical results appear on the output
part of the window, while graphs appear in new graphical windows.

1.1 General considerations

The freeware package can be downloaded from the web site:
http://ima.udg.edu/CoDaPack.
The package comes with a setup file and requires having Java Virtual Machine
(minimum version 1.5.0) installed. When a new version is available, CoDaPack
v2 shows a message asking for the user to import the new version.
To use CoDaPack v2, one has to execute the file Codapack.exe. Data
could be imported from Excel files or recovered from previous sessions. The
observations are organized in rows and the variables in columns.
CoDaPack v2 main window 1.1 has four parts. On the very top there are
the menus, on the left the active data frame and the name of its variables.
The Bigger part is the right side. On top of this part there is the place where
alphanumerical results are placed, and on bottom there is the data.
Using menus, one can execute macros which return numerical results on
the output part of the window and graphical output as independent windows.
2 1 Introduction

Fig. 1.1. CoDaPack v2 main window.

Each routine asks the user for the data to be used. Some of the menus have
some options buttons to modify default values.
To execute a routine of CoDaPack, it has to be selected from the menus
with only one mouse click. A new window appears, like the one shown in Figure
1.2, which is standard for all CoDaPack routines, asking which columns of the
data to select. The left side contains the Available Data structure, the middle
part the Selected Data structure, the Groups structure and a Reset button. On
the right side the secific options are available. Finally on bottom of the form
there are the Accept and Cancel buttons. Between the left and the middle
part there are two arrows to pass information between them. You can also
pass information by double clicking on the item to move.
First of all the user should select the parts to be used in a routine, in
the example (Fig. 1.2) the parts which are to be plotted in a dendrogram.
To do it, mark one or more of the variables of the Available Data list and
then click on the arrow > or double clicking over its name. The name of
the selected column will appear in the middle structure, i.e. in the Selected
Data structure. This operation should be repeated in order to select all the
parts involved in the routine. To unselect some part already included in the
Selected Data structure, select the corresponding part inside the Selected Data
structure and click on the arrow < or double clicking over its name.
If a routine has the groups option only categorical variables could be se-
lected to perform a ”by groups” execution of the routine.
 1.1 General considerations 3

Fig. 1.2. Menu: Dendrogram.

CoDaPack has three different outputs: 1) New variables. They are placed
at the end of the data window. 2) Alfanumerical output. It is placed on the
output part on top of the window. And 3) Graphical output. It appears on
independent window.

CoDaPack has five main menus: File, Data, Statistics, Graphs and Help.
2
File Menu

2.1 General remarks

This menu (Fig. 2.1) manages with files. CoDaPack stores a set of data on
Data Frames or Tables. It is possible to have opened more than one Data
frame. A set of Data frames could be saved as a Workspace and also it could
be recovered by means of the item button Open Workspace. Workspace files
have the extension .cdp by default. Internally the cdp file uses the JSON
standard format.

Fig. 2.1. Menu: File.

Each Data frame contains the name of variables and its numerical values.
There are two kind of Missing values, non-detected or non-available data and
there should be an specific symbol to distinguish them. Non-detected data
should begin with a character prefix, for example <, followed by the value of
low detection limit while Non-Available data should use a symbol, for example
”NA”.
6 2 File Menu

Data frames could be imported and exported from Excel files. In case of
an importation (Fig. 2.2) user should indicate in which row starts the data, if
there are labels, non-available symbol and non-detected prefix. At any time,
the user can delete a Data Frame from the active workspace. The exportation
saves names variable into the first row of an Excel file and the data in rows
below variable names.

Fig. 2.2. Import Data Frame form.

At this moment, Export/Import features are compatible with Excel 97/2003

files. In future versions more formats are going to be supported (csv, rdata,etc)
3
Data Menu

3.1 General remarks

This menu (Fig. 3.1) manages three kind of routines: 1) transformations of

the data from the simplex to the real space or vice versa, 2) operations inside
the simplex, i.e. operations where both the input data and the output are in
the simplex, and 3) management of variables.

Fig. 3.1. Menu: Data.

3.2 Data: Transformations: ALR

With this option (Fig 3.2)
the data is transformed with the additive logratio transformation (alr)
(Aitchison, 1986) from the simplex (raw data) to real space (alr coordinates),
 
x1 x2 xD−1
y = alr(x) = ln , ln , . . . , ln ,
xD xD xD
8 3 Data Menu

Fig. 3.2. Data: Transformations: ALR.

where y ∈ RD−1 , the (D − 1)-dimensional real space, or with its inverse, the
generalised additive logistic transformation (agl),
" #
exp(y1 ) exp(yD−1 ) 1
x = agl(y) = PD−1 ,..., PD−1 , PD−1 ,
1 + i=1 exp yi 1 + i=1 exp yi 1 + i=1 exp yi
from the real space (alr coordinates) to the simplex (raw data).
In the alr transformation the divisor is taken to be the last part according
to the sequence selected by the user. The interface allows the user to reorder
the variable once they have been selected just by dragging its name in the
selected data list. Recall that alr coordinates depend on the divisor, and that
they conform an oblique basis (Egozcue and Pawlowsky-Glahn, 2005).

3.3 Data: Transformations: CLR

With this feature (Fig 3.3) the data is transformed from the simplex (raw data)
to real space (clr-coefficients) according to the centred logratio transformation
(clr) (Aitchison, 1986),
   
x x1 x2 xD
y = clr(x) = ln = ln , ln , . . . , ln ,
gD (x) gD (x) gD (x) gD (x)
where y ∈ RD−1 and gD (x) is the geometric mean of the parts involved, i.e.
D
!1/D D
!
Y 1 X
gD (x) = xi = exp ln xi ,
i=1
D i=1

or with the inverse transformation (clr−1 ), from real space (clr coefficients) to
the simplex (raw data) (Aitchison, 1986),
 3.4 Data: Transformations: ILR 9

Fig. 3.3. Data: Transformations: CLR.

" #
−1 exp(y1 ) exp(y2 ) exp(yD )
x = clr (y) = PD , PD ,..., PD , .
i=1 exp yi i=1 exp yi 1 + i=1 exp yi
Recall that the clr coordinates represent a generating system, not a basis,
and therefore clr coordinates sum up to zero (Egozcue and Pawlowsky-Glahn,
2005). As a consequence, covariances and correlations between clr-parts have
the same drawbacks as covariances and correlations between compositional
parts: they are not subcompositionally coherent.

3.4 Data: Transformations: ILR

With this feature (Fig 3.4) the data is transformed from the simplex (raw
data) to real space (ilr coordinates) with the isometric logratio transformation
(ilr), or from real space to the simplex applying the inverse isometric logratio
transformation (ilr−1 ), both defined by a sequential binary partition (Egozcue
et al., 2003; Egozcue and Pawlowsky-Glahn, 2005).
The ilr transformation consists on: y = ilr(x) = (y1 , y2 , . . . , yD−1 ) ∈
PD
RD−1 , where yi = j=1 ψij ln xj , i = 1, 2, . . . , D − 1 , and
q
si
ri (si +ri ) , if at step i the part j is coded in the SBP as +1;


 q
ψij = − r ,
si (si +ri ) , if at step i the part j is coded in the SBP as −1;
i


0, if at step i the part j is coded in the SBP as 0;


with ri the number of parts coded at step i in the SBP as +1, and si the
number of parts coded at step i in the SBP as −1.
And The ilr−1 transformation consists on: x = ilr−1 (y) = (x1 , x2 , . . . , xD ) ∈
D
PD−1
S , where [x1 , x2 , . . . , xD ] = C exp [z1 , z2 , . . . , zD ] , zj = j=1 ψij yi , C
stands for the closure operation (Aitchison, 1986)
10 3 Data Menu

Fig. 3.4. Data: Transformations: ILR.

" #
z1 z2 zD
C [z1 , z2 , . . . , zD ] = PD , PD , . . . , PD , (3.1)
j=1 zj j=1 zj j=1 zj

Definition of the partition (Egozcue and Pawlowsky-Glahn, 2005):

A partition is a hierarchical grouping of parts of the original compositional
vector, starting with the whole composition as a group and ending with each
part in a single group. First, the compositional vector is divided into two
non-overlapping groups of parts. In a similar way, each of these two groups
is divided again, and so on until all groups contain only a single part. If D is
the number of parts of the original composition, the number of steps of the
partition is D − 1. CoDapack includes two different ways to define a partition:
1. Default partition. The default partition is defined by the Haar basis. It
consists in separating, at each step, the parts approximately in the middle.
Figure 3.4 shows the corresponding +1, −1 and 0 codification.
2. Defined manually by the user. Activating this option, a new button
appears and clicking on it to show a new window. This window, shown in
Figure 3.5, has a grid where rows represents parts and columns the steps

Fig. 3.5. Auxiliary window for defining a partition.

 3.5 Data: Centering 11

of the partition. To define the partition, every time the user marks with a
single click one part, a + sign appears in the grid at the cell corresponding
to this part in the current step. At each step of partition, a + sign means
that the part is assigned to the first group, a − sign to the second, and
it remains blank if this part is not in the group which is divided at this
order. To remove a + sign from the current step it is necessary to mark
the cell of the current step of the partition grid that contains this + sign
with a single click. To finish a step, press the Next Step button. At each
step it is only possible to divide one group. This group is marked with a
green color on the partition grid. In order to facilitate this task, when the
Next Step button is pressed, all the information (labels and partition) is
reordered in such a way that the next parts to divide appear in a sequence.
To eliminate some steps of the partition, press the Previous Step button
as many times as required.

3.5 Data: Centering

With this feature (Fig. 3.6) the data is centered, that is, it is perturbed by the

Fig. 3.6. Data: Centring.

center or closed geometric mean of the data (Aitchison, 1986). This routine
centers the data set, that is, it returns the data set Y formed by the D-part
compositions y = gN (X)−1 ⊕ X, where

N
!1/N N
!1/N 
Y Y
gN (X) = C  xk1 ,..., xkD 
k=1 k=1

is the closed geometric mean of the data set X. The center of the set Y is e,
the barycenter of the simplex; e.g. for D = 3 the geometric center of a ternary
12 3 Data Menu

diagram is [0.333, 0.333, 0.333]. If Show Center is activated this routine writes
the center of the parts selected on the output window.

3.6 Data: Subcomposition/Closure

With this feature (Fig. 3.7) the data is closed, i.e. data are converted into parts

Fig. 3.7. Data: Closure.

of some whole summing to a given constant, Y = C(X) . This constant is, by

default 1.0 but could be entered by the user by means of the Closure form.
If S parts, S < D , are selected, a subcomposition with S-parts is obtained
(Aitchison, 1986).

3.7 Data: Amalgamation

This feature (Fig. 3.8) amalgamates some columns of the data (Aitchison,
1986). The result of the amalgamation of some of the parts of a D-composition
selected by the user is the sum of those parts. Amalgamation should be used
only as a first step in preparing the data for further analysis, as this operation
is non-linear in the Aitchison geometry and might lead to inconsistent results if
compared to analysis made without amalgamation (Egozcue and Pawlowsky-
Glahn, 2005).

3.8 Data: Perturbation

With this feature (Fig. 3.9) a vector perturbs the data (Aitchison, 1986). The
 3.9 Data: Power Transformation 13

Fig. 3.8. Data: Amalgamation.

Fig. 3.9. Data: Perturbation.

output is a matrix of D-part compositions

y = p ⊕ x = C [p1 x1 , p2 x2 , . . . , pD xD ] ,

where C stands for the closure operation (Eq. 3.1), and p is a given D-part
composition. The user has to indicate on Perturbation box the vector p, which
has to be the same length as the compositions x.

3.9 Data: Power Transformation

This feature (Fig. 3.10) applies a power transformation to the data (Aitchison,
1986). For a ∈ R , the power transformation returns
14 3 Data Menu

Fig. 3.10. Data: Powering.

a ⊗ x = C [xa1 , xa2 , . . . , xaD ] .

The user has to indicate the constant of the operation on the Power box.

3.10 Data: Rounded Zero Replacement

This feature (Fig. 3.11) applies a transformation to the data to avoid zeros.

Fig. 3.11. Data: Rounded Zero Replacement.

Rounded zero replacement consists in substituting an observation x , with

zeros in some parts, by an observation y using the expression:
 3.11 Data: Numeric to categorical 15

 δi ,  P  if xi = 0;
yi = xj =0 δj
 xi 1 − Cx , if xi > 0.

where δi is the replacement value for the i-th part defined by the user and Cx
the components sum of observation x (Martı́n-Fernández et al., 2000).
CoDaPack v2 differentiates non-available and non-detected data. This rou-
tine applies to non-detected data. As it was seen on chapter there is an in-
dividual constant δi for each non-detected value, that is stored on the data
frame.

3.11 Data: Numeric to categorical

This feature (Fig. 3.12) transforms the selected variables into a strings, and

Fig. 3.12. Data: Numeric to categorical.

overwrites the result on the same variables.

16 3 Data Menu

3.12 Data: Categoric to numeric

This feature transforms the selected variables coded with a string into a nu-
merical ones, and overwrites the result on the same variables.

3.13 Data: Add numeric variables

This is a usefull feature to import data directly to the data set by a simple
copy-paste action.

Fig. 3.13. Data: Add numeric variables.

3.14 Data: Delete variables

This routine deletes the selected variables from the Workspace.
4
Statistics Menu

4.1 General remarks

This menu returns characteristic values for a data set from a compositional
or a classical point of view (Fig. 4.1).

Fig. 4.1. Menu: Statistics.

4.2 Statistics: Compositional statistics summary

This menu (Fig. 4.2) produces two types of descriptive statistics: the first
related to logratios (Variation Array, CLR variance and Total Variance) and
the second related to compositional descriptive statistics (Centre, Min, Max
and quartiles).
1. Variation Array: Returns a matrix where the upper diagonal contains
the logratio variances and the lower diagonal contains the logratio means.
That is, the ij-th component of the upper diagonal is var [ln(Xi /Xj )] ,
and the ij-th component of the lower diagonal is E[ln(Xi /Xj )] , where
i, j = 1, 2, . . . , D .
2. CLR Variances: Returns, for each part, the sum of logratio variances
that involve it. Thus, for the i-th clr component ξi we have
D
1 X
var(ξi ) = var [ln (Xi /Xj )] .
2D
j=1,i6=j
18 4 Statistics Menu

Fig. 4.2. Compositional Statistics Summary.

3. Total Variance: The sum of all clr Variances is the Total Variance totvar.
4. Centre: Returns the centre of the data set, that is, ξˆ = C[g1 , g2 , . . . , gD ],
Q 1/N
N
where gi = k=1 x ki stands for the geometric mean of part Xi in
data set X. The data set X has been previously closed.
5. Minimum and Maximum: For each part of the data set X it returns
the maximum and the minimum of the closed data set.
6. Quartiles: For each part of the data set X it returns the first quartile
Q1 , the median Q2 and the third quartile Q3 of the closed data set. The
user has to select the columns to close and where to put the results. There
are two buttons in this routine:
The output of the routine (Fig. 4.3) is placed on the output part. It includes
a color classification of the logratio variances (elements of the upper diagonal
of Variation Array). It is assumed that the logarithm of the logratio variances
follow a t-student distribution, then dark blue colores those elements below
percentile 5, light blue from percentile 5 to 25, light red form percentiles 75
to 95 and dark red up to percentile 95.

4.3 Statistics: Classical statistics summary

This menu (Fig. 4.4) produces an standard descriptive statistics, including
mean (arithmetic), standard deviation, covariance matrix, Min, Max and
quartiles). The output of the routine (Fig. 4.5) is placed on the output part.
 4.5 Statistics: Atypicality Indices (Fig. 4.7) 19

Fig. 4.3. Numerical output for Compositional Statistics Summary.

4.4 Statistics: Additive-Logistic normality test

This alows the user to perform a test for logistic normality of a D-part compo-
sition (Fig. 4.6) (Aitchison, 1986, p. 143). It includes all marginal, univariate
distributions (with a total of (D − 1) tests); all bivariate angle distributions
(with a total of D(D − 1)/2 tests); and the (D − 1)-dimensional radius distri-
bution. For each kind of test the Anderson-Darling, Cramer-von Misses and
Watson statistics are computed and their significance is given.

4.5 Statistics: Atypicality Indices (Fig. 4.7)

With this feature (Fig. 4.7) the user obtains the atypical observations and their
indices under the assumption of Additive Logistic Normal distribution of the
selected parts. The user has to select the columns to calculate its atypical
observations and the threshold of atypicality (usually 0.95) has to be given.
20 4 Statistics Menu

Fig. 4.4. Classical Statistics Summary.

 4.5 Statistics: Atypicality Indices (Fig. 4.7) 21

Fig. 4.5. Numerical output for Classical Statistics Summary.

22 4 Statistics Menu

Fig. 4.6. Logistic Normality Tests.

Fig. 4.7. Atypicality indices.

5
Graphs Menu

5.1 General remarks

These menus enable the user to create graphs in independent windows. The
user can customise the appearance of each graph and, in some cases, plot the
observations in the graph according to a previous classification. These graphs
can be zoomed and, in 3D, rotated.

Fig. 5.1. Menu: Graphs.

To perform a zoom in a graph it is possible to use the slider scroll at the

bottom of the graph or just using the scroll wheel of the mouse.
It is also possible to rotate a figure by means of the left button of the mouse.
Holding the left mouse button and moving it the graph rotates following the
direction of the mouse. If the graph is 2D then the figure just moves inside
the windows without rotation. To move the graph inside the window holding
the left mouse button and simultaneously holding the ALT key.
Furthermore, the graphs can be saved by means of snapshots of what
windows have each moment. This can be done with the menu File-Snapshot
and the files produced could be in jpeg, eps, png and bitmap formats.
24 5 Graphs Menu

The same menu File includes a submenu Configuration that allows to

customize the elements of the graph like lines and labels by means of changing
size and colors.

5.2 Graphs: Ternary Diagram

This feature displays a ternary diagram of three or four selected parts (Fig.
5.2).

Fig. 5.2. Graphs: Ternary Diagram form.

Once the graph is produced it is possible to center de data into the ternary
diagram and to draw a grid, the latest only on 2D graph (Fig. 5.3).
Also by means of two buttons (Fig. 5.4) is possible to interchange the parts
of the vertices of the ternary.

5.3 Graphs: Principal Components

This feature calculates the two (or three) compositional principal components
for a 3-part (or 4-part) composition and displays the result in a ternary dia-
gram (Fig. 5.5). Also it has the same features of grid and centering as ternary
diagram.
Also this routine returns, as a numerical result, the Principal Components
and the cumulative proportion explained with each component (Fig. 5.6).

5.4 Graphs: ALR Plot

This feature displays a plot of three (four in 3D) alr-transformed parts (Fig.
5.7). The new variables obtained with the ALR transformation are displayed
 5.4 Graphs: ALR Plot 25

Fig. 5.3. 2D Ternary Diagram display with or without centering and grid.

Fig. 5.4. 2D Ternary Diagram display with or without centering and grid.

in an orthogonal coordinate system to visualise how the plot changes when

permuting the components or initial columns. Nevertheless, care is required
when interpreting the plot, as the axis are not really orthogonal, but at 60o
(Egozcue and Pawlowsky-Glahn, 2005).
26 5 Graphs Menu

Fig. 5.5. 3D Ternary Principal Component Analysis.

Fig. 5.6. Numerical output of Ternary Principal Components.

The 3D ALR Plot allows to change the 2D view by changing which of the
two axes is displayed.

5.5 Graphs: CLR Plot

This feature displays a plot in an orthogonal coordinate system of the data
after the centred logratio transformation (clr) of two (three in 3D) selected
parts. It has the same capabilities than ALR Plot.

5.6 Graphs: ILR Plot

This feature displays a plot in an orthogonal coordinate system of the data
after the isometric logratio transformation (ilr) of three (four in 3D) selected
parts according to a sequential binary partition. The way to selec the partition
is the same as in Transformation-ILR routine.
 5.8 Graphs. Balance Dendrogram 27

Fig. 5.7. ALR Plot window after some rotation with groups.

The figure obtained has the same capabilities than ALR and CLR Plot.

5.7 Graphs: Biplot

This feature performs a biplot (Aitchison, 1997; Aitchison and Greenacre,
2002) of selected parts. Once performed the graph (Fig. 5.8) the user could
choose 1) which 2D view prefers (axes XY, YZ or XZ), 2) to display obser-
vations or not, and 3) which biplot display depending of α value (??). α = 0
corresponds to a Covariance Biplot, α = 1 Form Biplot, and α = 0.5 Sym-
metric Scaling Biplot, which is the default value.
Also this routine returns, as a numerical result, the Principal Components
and the cumulative proportion explained with each component (Fig. 5.9).
Biplot consists on the decomposition of clr matrix, X = U DV 0 . If numer-
ical output is desired the routine writes three matrices: U D, D and V . U D
are the ilr coordinates of the original data.

5.8 Graphs. Balance Dendrogram

This feature performs a Balance Dendrogram (Egozcue and Pawlowsky-Glahn,
2005) by means of a sequential binary partititon of selected parts. The way to
selec the partition is the same as in Transformation-ILR routine (Fig. 5.10).
As a numerical output it this routine returns on the output window the
sequential binary partition used, the mean and the variance of each balance
(Fig. 5.11). Also on the Data window are the ilr coordinates produced with
this partition.
28 5 Graphs Menu

Fig. 5.8. Window of Graphs: Biplot.

Fig. 5.9. Numerical output of Biplot.

Fig. 5.10. Graphs: Balance Dendrogram.

 5.8 Graphs. Balance Dendrogram 29

Fig. 5.11. Numerical Output of Balance Dendrogram routine.

References

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Aitchison, J. (1997). The one-hour course in compositional data analysis
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