A little program of correspondence analysis that I make with Mathematica 9 André Dauphiné Geographer University of Nice Sophia-antipolis

(*Analyse des Correspondances*)
(*Importation des données, \
suppression du nom des variables*)
(*les noms des lignes sont placées \
dans le fichier nom*)

ClearAll["Global`*"]
d = Import[SystemDialogInput["FileOpen"], "XLS"];
d = Flatten[d, 1];
nomcol = Take[d, 1];
nomcol = Drop[Flatten[nomcol], 1];
d1 = Transpose[d];
nomligne = Take[d1, 1];
nomligne = Drop[Flatten[nomligne], 1];
d2 = Drop[d, 1, 1];
touttot = Total[Flatten[d2]];
totcol = Total[d2]/touttot;
totlig = Total[Transpose[d2]]/touttot;
ncol = Length[totcol];
nlig = Length[totlig];
d3 = d2/touttot;
d4 = Table[d3[[i, All]]/totlig[[i]], {i, 1, nlig}];
d5 = Table[d4[[All, i]]*(1/Sqrt[totcol[[i]]]), {i, 1, ncol}];
d5 = Transpose[d5];
dcent = Table[d5[[All, i]] - Sqrt[totcol[[i]]], {i, 1, ncol}];
dcor = Covariance[Transpose[dcent]];
Print["Matrice des covariances"]
dcotmat = dcor;
Grid[dcotmat, Frame -> All, Alignment -> "."]
{vals, vecs} = Eigensystem[N[dcor]];
recvals = Sqrt[vals];
trac = Total[vals];
Print["Pourcentage de variance par chaque composante"]
pcvals = (vals/trac)*100;
Column[pcvals, Frame -> All, Alignment -> "."]
ListPlot[pcvals, Filling -> Axis]
stur = recvals*vecs;
Print["Saturations composantes variables"]
saturations = Transpose[stur];
sturt = N[saturations];
Grid[sturt, Frame -> All, Alignment -> "."]
proj = vecs.dcent;
projt = Transpose[proj];
Print["Saturations composantes objets"]
projc = N[projt];
Grid[projc, Frame -> All, Alignment -> "."]
ny = DialogInput[
  DynamicModule[{name = ""}, Column[{"Choisir la composante 1",
     InputField[Dynamic[name], String],
     ChoiceButtons[{DialogReturn[name], DialogReturn[]}]}]]]
ny = ToExpression[ny];
nx = DialogInput[
  DynamicModule[{name = ""}, Column[{"Choisir la composante 2",
     InputField[Dynamic[name], String],
     ChoiceButtons[{DialogReturn[name], DialogReturn[]}]}]]]
nx = ToExpression[nx];
Print[]
Print["Graphiques des saturations composantes variables"]
dcomp = Partition[
   Riffle[saturations[[All, ny]], saturations[[All, nx]]], 2];
dcompespace = Partition[Riffle[projt[[All, ny]], projt[[All, nx]]], 2];
a1 = ListPlot[dcomp, PlotStyle -> Directive[PointSize[Large], Red], 
  AxesOrigin -> {0, 0}, AxesLabel -> {ny, nx}]
Print[]
Print["Graphiques des saturations composantes objets"]
a2 = ListPlot[dcompespace, 
  PlotStyle -> Directive[PointSize[Large], Blue], 
  AxesOrigin -> {0, 0}, AxesLabel -> {ny, nx}]
Print[]
Print["Graphiques des saturations composantes variables et \
composantes objets"]
Show[a1, a2]

Hi Leo,

First and foremost thank you for brining this topic in this forum!

Below are responses to some of your questions and remarks.

1. Correspondence analysis is just SVD applied over contingency matrices with entries modified by Chi^2 related formulas. So yes, Mathematica supports correspondence analysis "in some way" because these steps are easy to program and apply.

2. I read the referenced article from The Mathematica Journal. You can download the notebook and use MatrixPlot over the contingency matrices to get something more visually appealing than the print outs with TableForm.

3. I find you saying that Figure 5 in that article does not make Mathematica justice meaningless. What is plotted corresponds to what are the background explanations of the plot. What do you mean with that remark?

4. I understand that you want an out of the box function or package that does correspondence analysis. Take a look at Mathematica's function PrincipalComponents. Its functionality, guide, and demonstrations are probably very close to what you are asking for.

"[...] I am sure it is, but I doubt "these steps are easy to program and apply" otherwise someone would have done it already, [...]"

Yes, you are correct. The TMJ article referenced above ("An Introduction to Correspondence Analysis") has these steps. See the definition and application of the function chisqd and the application of SingularValueDecomposition.

"[...] or am I wrong? Certainly I would not know how to start .."

I think a good start is to download and interactively read the notebook of the article "An Introduction to Correspondence Analysis": http://www.mathematica-journal.com/data/uploads/2010/09/Yelland.nb .

"[...] in correspondence analysis, creating the output is just half the work. Interpreting the output is difficult, to say the least."

This probably means that correspondence analysis as a technique is not mature enough and / or a different paradigm should be used.

Several suggestions come to mind.

1. Use Non-Negative Matrix Factorization (NNMF) instead of SVD (in correspondence analysis). SVD produces orthogonal basis vectors with coordinates that have mixed signs, and hence are hard to interpret if SVD is applied over contingency matrices of categorical data. The basis vectors of NNMF have positive coordinates and can be easily interepretted, but they are not orthogonal.

2. If you can phrase your problem as a classification problem, then the applications of Decision Trees and Naive Bayesian Classifiers might give nice, easy to interpret insights into your data. (For example, see Mathematica's function Classify.) These classifiers have several methods and techniques to evaluate their prediction strength. (Hence you can evaluate your insight derived by using them.)

3. Abandon the vector space representation and use probabilistic models, like Random walks, N-grams, Markov chains, Hidden Markov models. (All these apply to the text classification problem.)

Very many thanks, excellent link. More burning of midnight oil ...

Hi David,

I do appreciate your effort and help.

As a matter of fact, I came across this article, but look at the output on Figure 5 (just before 8. Conclusion): it doesn't do Mathematica justice, I am afraid. (I know I am difficult ...)

It does appear as there is a white spot on the map, somewhere.

Many thanks for the swift reply, David, and thank your for the R-links. I may have to resort to R, after all. However, my maths / stats expertise, including programming, is limited to that of the average social science postgraduate.

Basically correspondence analysis is used to determine correlation (better: contingency) between categorical variables. The presentation of this fact is the tricky part. I have discovered some dated Mathematica Solutions (e.g., 5.0), using line printer output.

The author of the text in the link below (Greenacre, he appears to be the present-day guru in this field) gives an example at page 26ff, with a sample Output on p 29 using an Excel Add-in. Surely Mathematica should offer better solutions.

http://statmath.wu.ac.at/courses/CAandRelMeth/CARME1.pdf

The following link illustrates why the presentation of output is so important (e.g., p. 14): interpreting correspondence Analysis results is difficult. My hope is to create interactive output, as available in Mathematica, to be able to inspect 3-d graphs from several angles, etc.

http://www.skeptron.uu.se/broady/sec/p-gda-0609-III.pdf