Multidimensional Data Analysis

Statistical data sets are becoming increasingly large. Before modeling them, it is essential to explore them and reduce their dimensionality while losing as little information as possible. This is the objective of this course on multidimensional exploratory statistics. Methodologically, the tools used in this course are primarily those of Euclidean geometry. Statistical problems and concepts will therefore be translated into the language of Euclidean geometry before being addressed within that framework. The two families of exploratory methods that will be covered in this course are:

1) automatic classification methods, which group observations into classes and reduce the variability among the observations to the variability between these classes;

2) principal component analysis methods, which identify the main directions of variation among the observations and enable the generation of interpretable, low-dimensional representations of this variation.

To bridge the gap between Euclidean geometry and multidimensional exploratory statistics. To develop comprehensive skills in exploring large datasets and analyzing them prior to statistical modeling.

Courses on Euclidean geometry, normed vector spaces, and endomorphism reduction.

Recommended prerequisites: Courses in univariate and bivariate descriptive statistics. A solid understanding of matrix algebra.

Ongoing assessment (homework / small projects) + final exam

I - Introduction:

a) Multidimensional data, observations, variables, coding; b) Mappings from point clouds to Euclidean metric spaces; c) The need for dimensionality reduction: components / classes.

II - Geometric Representations of Statistical Quantities

Univariate description:

a) Mean, frequency,

b) Variance and standard deviation.

c) Centering and reducing a variable.

Bivariate relationships:

a) Bivariate relationship & conditioning.

b) Covariance and correlation of two quantitative variables.

c) R² from an analysis of variance involving one quantitative variable and one qualitative variable. d) Phi² and T² for two qualitative variables. e) Standard notation for relationships. f) Limitations of bivariate analysis and how to overcome them.

III - Automatic Classification

Differences and similarities.

a) Measures.

b) Partial vs. overall similarity.

Partial similarity: logical/conceptual classification using Galois lattices.

Overall resemblance:

a) Partitioning in metric space: the K-means method and refinements.

b) Hierarchical classification: indices, CAH algorithm, partition selection criteria.

c) Mixed classification.

d) Interpretation of classes.

e) Classification by variables.

IV - Principal Component Analysis

Standardized ACP

a) Individual cloud, inertia, and direct PCA.

b) Variable cloud, inertia, and dual PCA.

c) Duality relationships and joint interpretation of the graphs.

d) Additional elements & duality relationship.

e) The first component as an estimate of a continuous latent variable.

General ACP (with any metrics)

a) Scatter plot and direct PCA.

b) Application to multidimensional scaling.

c) Which ACP of the columns, for which duality relations?

d) Interpretation aids.

e) Additional elements & duality relationship.

(f) Reconstruction formula (decomposition into singular components).

Binary Correspondence Analysis

a) Phi2 as the direct and dual inertia of the line-profile and column-profile clouds.

b) Which metrics for which duality relations: barycentric positioning.

c) Combined interpretation of the graphs.

d) The Guttman Effect. E) Additional Information.

Multiple Correspondence Analysis.

a) Application of CBA to a complete disjunctive logic table.

b) Application of CBA to a Burt matrix; equivalence.

c) Barycentric relationships between individuals and modalities. Barycentric relationships between modalities.

d) The Guttman effect. e) Additional elements.

e) The first component as an estimate of a continuous latent variable.

The Practice of ADM

a) Complementarity between AF and CA.

b) How to conduct a proper ADM.

Hourly volumes:

            CM: 21

            TD: 21

            TP: 0

            Land: 0