Review of: Brigitte Le Roux and Henry Rouanet, Geometric Data Analysis, From
Correspondence Analysis to Structured Data Analysis, Kluwer, Dordrecht, 2004,
xi+475 pp

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Geometric Data Analysis, From Correspondence Analysis to Structured Data  Analysis (book review)

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Global Perspectives in Youth Work

Review of: Brigitte Le Roux and Henry Rouanet, Geometric Data Analysis, From Correspondence Analysis to Structured Data Analysis, Kluwer, Dordrecht, 2004, xi+475 pp

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arXiv:0804.1244v1  [cs.AI]  8 Apr 2008Review of: Brigitte Le Roux and Henry Rouanet, Geomet-ric Data Analysis, From Correspondence Analysis to Struc-tured Data Analysis, Kluwer, Dordrecht, 2004, xi+475 pp.The term “Geometric Data Analysis” is due to Patrick Suppes (Stanford) whowrites a Foreword for this encyclopedic view of Correspondence Analysis. Theuniqueness of this work lies in the detailed conceptual framework, and in showinghow, where and why statistical inference methods come into play.As a data analysis methodology, Correspondence Analysis is formal andgeometric. The former follows from the fact that mathematical “structuresgovern procedures”, the mathematics carry the burden of demonstration, andultimately “this means a fantastic saving in intellectual investment”. The lat-ter is due to the focus on clouds of points in geometric spaces. Unlike thesampling-oriented approach of a good deal of statistical data analysis, includ-ing multivariate data analysis, for Correspondence Analysis, “description comesfirst” since statistics is not reducible to probability.The word “model” is used, in general, in lots and lots of senses (statisti-cal, mathematical, physical models; mixture model; linear model; noise model;neural network model; sparse decomposition model; even data model). So it ishelpful that the methodological models under discussion in this book are termedframe models. As a geometric frame model, it is also nice to see due emphasisgiven to “Euclidean classification” as a “companion method” of CorrespondenceAnalysis. Euclidean classification is most in harmony with CorrespondenceAnalysis when minimum variance (or Ward’s) agglomerative hierarchical clus-tering is used. The data aggregates are defined in the same CorrespondenceAnalysis metric spaces. The clustering is Euclidean because it is constructed inthe Correspondence Analysis factor space.In social sciences, there are “two distinct conceptions of the role of statistics,namely sustaining a sociology of variables versus constructing a social space”.The former goes hand in hand with traditional statistical analysis, whereas thelatter is part and parcel of Geometric Data Analysis. We will return to theconstruction of social space below with reference to Bourdieu.As a “cornerstone” of Correspondence Analysis, one has the “measure versusvariable duality”, or the duality of the associated (row, column) vector spaces.When aggregated, the measures are summed, whereas the variables, when theyare aggregated, are averaged. A transition notation where subscripts denotemeasures and superscripts denote variables is handy – much handier than matrixnotation – for important aspects of the analysis. Among the latter are: howone’s data departs from the “reference” marginal frequencies; how one’s data isexpressed as a “transition” from one to the other of these marginal frequencies;and how the χ2 metric is defined on the dual, complementary clouds.Discussing the suitability of Correspondence Analysis for structured data(that is to say, structured around such variables as age and gender characteristicsin a questionnaire), firstly having such structuring factors as supplementaryvariables allows for their incorporation into the analysis. Later, other analysessuch as Analysis of Variance (ANOVA) could be very appropriate. Statistical1inference is a natural complement to descriptive analysis – as Benze´cri wrote,“The model should follow the data, not the reverse!” – and both phases togethercomprise a powerful way to inductively carry out data analysis.Chapter 2 presents a most accessible and readable introduction to Corre-spondence Analysis, based on the view of the measure versus variable duality.This leads to Correspondence Analysis being viewed as analysis of a measureover the Cartesian product with strictly positive marginals. All of the essentialsof Correspondence Analysis are introduced, going as far as aids to interpreta-tion (e.g. contributions of points to the inertia of axes; quality of representation;data coding; supplementary elements; linkages with matrix expression, Fisher’slinear discriminant analysis, regression, and canonical correlation analysis; linkswith multidimensional scaling; and how or where CA links up with probabilisticmodel frames). A range of questions end the chapter, with solutions and withcomments.Chapter 3 dealing with analysis of a Euclidean cloud of points proceeds byway of Huyghens, and spectral decomposition (i.e., eigenvalue, eigenvector de-composition) to concentration (hyper)ellipsoids, and following discussion of thepartition of a cloud the reader is led to clustering. Pride of place is accorded ag-glomerative hierarchical clustering with the minimum variance criterion (Ward’smethod; “Euclidean classification” since the input consists of Euclidean projec-tions on factors). The reciprocal nearest neighbors algorithm is described butnot the nearest neighbor chain algorithm (which is more manageable from acomputational complexity point of view, since it has an O(n2) computationalbound for most widely used agglomerative criteria). For the nearest neighborchain and reciprocal nearest neighbor algorithms, which came out of Benze´cri’slab in the early 1980s and are now in R and Clustan among other packages, seeMurtagh (2005). The ultrametric is not discussed. For the view of Correspon-dence Analysis being a “tale of three metrics”, χ2, Euclidean and ultrametric,see Murtagh (2005). Chapter 3 ends with exercises and solutions.Chapter 4 deals comprehensively with Principal Components Analysis, viewedfrom the points of view of formal description and also interpretation of results,through the prism of Correspondence Analysis. In terms of number of pages itis more than half the length of Dunteman (1989) (but admittedly shorter thanthe 500 page length Jolliffe, 2002). The treatment of PCA in Le Roux andRouanet is unique and in the spirit of Geometrical Data Analysis.In chapter 5, attention is given to Multiple Correspondence Analysis. Aparadigmatic case is questionnaire analysis, where each question has a numberof response modalities, and just one among these response modalities is indicatedby a subject. Much that is theoretical (e.g., Burt table; viewpoints linked todiscriminant analysis, and canonical correlation analysis) and practical (e.g.,choice of active questions; data coding for handling infrequent modalities) iscovered. Before ending the chapter with questions and solutions, there is adetailed case study, relating to a French Government survey of cultural andleisure activities (3002 respondents, carried out in 1997).We have already mentioned “structured data” above. Examples of suchstructuring context, available in addition to the essential data being studied,2include gender, age and education variables in the cultural and leisure question-naire; or authors and years in textual-based authorship assessment. Due to theoverall aim of studying the effect of experimental (independent, under analystcontrol) variables on the dependent variables (response variables), and also theneed often to take nesting and crossing structures of variables into account, thereis linkage in chapter 6, “Structured Data Analysis”, with ANOVA, MANOVAand regression. These techniques are discussed within the model frame of Cor-respondence Analysis: they are “grafted” into the Geometric Data Analysisframework. A case study is provided based on visually-based annotated videoof basketball players, with the aim of selecting potential high achievers.Chapter 7 on stability analysis first touches on alternatives, functional anal-ysis (Krzanowski, Critchley, Tanaka, Pack and Jolliffe, Be´nasse´ni), and boot-strapping (Diaconis and Efron, Daudin et al., Lebart et al.), before returningto the approach of Escofier and Le Roux that perturbs a cloud relative to areference cloud, and checks what are the effects of doing this. A range of casestudies is used to investigate an outcome without a given group being present,or to investigate an outcome following deletion of a single point, or to investi-gate the discarding of variables or modalities. A range of stability results foreigenvalues is also considered.Often in applications inference is needed, based on statistical modeling, andthe authors encourage a comprehensive approach to “Inductive Data Analysis”in chapter 8. The point is made that descriptive statistics are based on relativefrequencies and do not depend on sample size, whereas test statistics (for induc-tion or inference) do combine an erstwhile descriptive statistic with the samplesize. There is much to reflect on in this chapter. Traditional significance testing,frequentist and combinatorial inference, Bayesian inference, – all are discussedin a way that is well-grounded geometrically and eminently readable.Chapter 9, 85 pages in length, deals with three major case studies: a medica-tion treatment study of Parkinson’s patients; French political attitudes before,and voting in, elections in 1997; and part of a large “Education Program forGifted Youth (EPGY)” study carried out at Stanford University. There is agreat wealth of experience etched deeply into these application studies.An overview of applicable mathematics rounds off the book. We have notedearlier the authors’ eschewing of matrix formalism in favor of transitions. Matrixdescription is to be found throughout, but the appropriateness and powerfulnessof the transition based approach is given pride of place. The transition formalismuses quite lightly the Einstein tensor notation that is used more extensively inBenze´cri (1973). Van Rijsbergen (2004) provides another very commendablesurvey of geometric data analysis in information retrieval, using Dirac notationthat is now standard in quantum physics.The transition formalism is nicely expressed by mapping, or morphism,diagrams (as were used a great deal in Cailliez and Page`s, 1976; and whichhave been taken further in McCullagh, 2002, who appraises statistical modelingthrough the prism of category theory). Le Roux and Rouanet are unique intheir succinct introduction to homomorphisms and endomorphisms, the appli-cation of spectral decomposition in this context, and discussion of extremality3properties.A major feature of this book, following the example of Benze´cri, is theclose practical and experienced attention throughout paid to the epistemologyof data analysis, i.e. the interface between data and reality. This is nowherestronger than in the authors’ continuation of Bourdieu’s (see e.g. Bourdieu,1984) approach to mapping out of a social field or space. In chapters 5, 6 and9, there is appreciable discussion of Bourdieu’s use of Geometric Data Analysis.In conclusion, this book constitutes essential background material on Geo-metric Data Analysis, and, for the seasoned professional, a most valuable sourceof reference.ReferencesBENZE´CRI, J.P. et Coll. (1973). L’Analyse des Donne´es. Vol. 1: Taxinomie.Vol. 2: Correspondances, Paris: Dunod. (1976, 2nd edn.)BOURDIEU, P. (1984). Distinction: A Social Critique of the Judgement ofTaste, translated by R. Nice. Cambridge, MA: Harvard University Press. (1979)La Distinction: Critique Sociale du Jugement, Paris: Les E´ditions de Minuit.CAILLIEZ, F. and PAGE`S, J.-P. (1976). Introduction a` l’Analyse des Donne´es.Paris: Socie´te´ de Mathe´matiques Applique´es et de Sciences Humaines. SMASH.DUNTEMAN, G.H. (1989). Principal Components Analysis. Thousand Oaks,CA: Sage Publications.JOLLIFFE, I.T. (2002). Principal Components Analysis, 2nd ed., New York:Springer.McCULLAGH, P. (2002). “What is a statistical model?”, Annals of Statistics,30, 1225–1310.MURTAGH, F. (2005). Correspondence Analysis and Data Coding with Javaand R, Boca Raton, FL: Chapman and Hall/CRC.VAN RIJSBERGEN, K. (2004). The Geometry of Information Retrieval, Cam-bridge: Cambridge University Press.Fionn MurtaghScience Foundation IrelandandDepartment of Computer Science, Royal Holloway, University of London4

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