Augmented quantization : a general approach to mixture models

Charlie Sire (1, 2, 3) , Rodolphe Le Riche (2, 4, 5, 6, 7) , Didier Rullière (2, 4, 5, 6) , Jérémy Rohmer (1) , Lucie Pheulpin (3) , Yann Richet (8)

1 BRGM - Bureau de Recherches Géologiques et Minières (BRGM)
2 LIMOS - Laboratoire d'Informatique, de Modélisation et d'Optimisation des Systèmes
3 IRSN/PSE-ENV/SCAN/BEHRIG - Bureau d'expertise en hydrogéologie, risques Inondation, météorologiques et géotechniques
4 Mines Saint-Étienne MSE - École des Mines de Saint-Étienne
5 FAYOL-ENSMSE - Institut Henri Fayol
6 FAYOL-ENSMSE - Département Génie mathématique et industriel
7 CNRS - Centre National de la Recherche Scientifique
8 IRSN - Institut de Radioprotection et de Sûreté Nucléaire

Quantization methods classically provide a discrete representation of a continuous set. This type of representation is relevant when the objective is the visualisation of weighted prototype elements representative of a continuous phenomenon. Nevertheless, more complex descriptions may be investigated. In this sense, mixture models identify subpopulations in a sample, arising from different distributions. The Gaussian mixture model is particularly popular and relies on the Expectation-Maximisation (EM) algorithm [1] for maximum likelihood estimation. The computation of the likelihood limits the type of distributions in the mixture; more specifically, for the Dirac distributions and even uniform components despite their high interest in practice for processing computer experiments. Their visualization is convenient and can lead to a sensitivity analysis where variables with largest marginals are least sensitive and vice versa, as shown by our application to a flooding real case in [2]. The objective of our study is to build a very general method to provide a mixture model that approximates a sample (xi) n i=1 ∈ X n ⊂ R n from a random variable X. The representatives of the sample are the calculated components of the mixture, chosen in a parameterized family of laws denoted R. We investigate, for a given number of representatives ℓ ∈ N, the mixture X˜ ℓ = R(J) approximating X. The representatives (R(j) ) ℓ j=1 and the discrete random variable J ∈ {1, . . . , ℓ} need to be optimised.

Domaines: Mathématiques [math]; Statistiques [stat]

Charlie Sire, Rodolphe Le Riche, Didier Rullière, Jérémy Rohmer, Lucie Pheulpin, et al.. Augmented quantization : a general approach to mixture models. MASCOT-NUM2023, UQ@Paris-Saclay, Apr 2023, LE CROISIC, France. ⟨hal-03975978⟩