Thesis & HDR

Doctoral Thesis

Title: Étude d’un ensemble de stimuli présentant une structure géométrique simple
(Study of a set of stimuli presenting a simple geometric structure)

Degree: Doctorat de Troisième Cycle (speciality: Statistics)

Institution: Faculté des Sciences de Paris

Year: February 1970

Jury: J-P. Benzécri (supervisor), D. Dugué (president), H. Rouanet (examiner)

Habilitation à diriger des recherches (HDR)

Title: Analyse géométrique des données : Stabilité, Structuration et Inférence
(Geometric Data Analysis: Stability, Structuring and Inference)

Speciality: Applied Mathematics (CNU section 26)

Institution: Université Paris Dauphine

Year: June 2000

Jury: J-P. Raoult (president), P. Cazes (reviewer), B. Bru (reviewer), L. Lebart (reviewer), C. Hess (examiner)

Abstract

The research focuses on geometric data analysis, that is, multidimensional statistical analysis methods centred around the determination of principal axes. These methods are presented within a geometric-formal framework that emphasises the fundamental concepts of cloud (in an affine then Euclidean space) and duality between measures and variables (in linear algebra terms). The work is structured around three themes.

The first theme concerns stability problems of a Euclidean cloud. The approach consists in taking a reference Euclidean cloud and comparing it to a cloud derived from it through a perturbation acting either on the Euclidean structure (change of metric) or on the points (removal/addition of points, projection onto a subspace, grouping of points); the results are then applied to the various clouds considered in principal component analysis and in simple and multiple correspondence analysis.

The second theme deals with structured data analysis, that is, clouds endowed with classical analysis-of-variance structures (in particular nesting and crossing). In particular, the decomposition of a weighted cloud indexed by the crossing of two factors into an additive cloud and an interaction cloud is studied in detail.

The third theme addresses statistical inference, first within the framework of combinatorial inference with randomisation tests, then within Bayesian inference. The combinatorial method is treated in detail for multiple correspondence analysis; the Bayesian method is applied to structured geometric data arising from principal component analysis.

Finally, works of applied statistics are presented, arising from in-depth collaborations with researchers from several fields in the human sciences (political space, racism, the publishing field, and accident studies) and in biological sciences (nutrition and epilepsy).