[Réseaux Bayésiens Gaussiens Linéaires Croisés, des modèles parcimonieux]
Dans cet article, nous proposons de substituer aux régressions linéaires multivariées classiques des sous modélisations plus parcimonieuses construites à l’aide de réseaux bayésiens gaussiens. L’idée est d’améliorer la prédiction de variables par des covariables, grâce à une réduction sensible de la dimension paramétrique de la matrice de variance-covariance. Une mise en œuvre est développée par l’utilisation de DAG (graphe orienté sans circuit) structurés lorsque l’ensemble des nœuds à modéliser est un produit cartésien de deux ensembles. Un certain nombre de propriétés intéressantes de ces DAG et des réseaux bayésiens associés en découle. Une expérimentation numérique basée sur des données simulées est réalisée pour vérifier la faisabilité de la proposition à partir de données lorsque la structure du DAG n’est pas connue. Enfin, la proposition est appliquée à la prédiction de la composition corporelle à partir de covariables faciles à obtenir. Les résultats obtenus par une recherche systématique de cette classe de réseaux bayésiens sont comparés avec la prédiction du modèle saturé de regression multiple multivariée.
Linear Gaussian Bayesian networks can dramatically reduce the parametric dimension of the covariance matrices in the framework of multivariate multiple regression models. This idea is developed using structured, crossed directed acyclic graphs (DAGs) when node sets can be interpreted as the cartesian product of two sets. Some interesting properties of these DAGs are shown as well as the probability distributions of the associated Bayesian networks. A numerical experiment on simulated data was performed to check that the idea could be applied in practice. This modelling is applied to the prediction of body composition from easily measurable covariates and compared with the results of a saturated regression prediction.
Mot clés : réseau bayésien, DAG croisé, régression multiple multivariée, prédiction
@article{JSFS_2014__155_3_1_0,
author = {Tian, Simiao and Scutari, Marco and Denis, Jean-Baptiste},
title = {Crossed {Linear} {Gaussian} {Bayesian} {Networks,} parsimonious models},
journal = {Journal de la soci\'et\'e fran\c{c}aise de statistique},
pages = {1--21},
publisher = {Soci\'et\'e fran\c{c}aise de statistique},
volume = {155},
number = {3},
year = {2014},
mrnumber = {3272707},
zbl = {1316.62103},
language = {en},
url = {http://www.numdam.org/item/JSFS_2014__155_3_1_0/}
}
TY - JOUR AU - Tian, Simiao AU - Scutari, Marco AU - Denis, Jean-Baptiste TI - Crossed Linear Gaussian Bayesian Networks, parsimonious models JO - Journal de la société française de statistique PY - 2014 SP - 1 EP - 21 VL - 155 IS - 3 PB - Société française de statistique UR - http://www.numdam.org/item/JSFS_2014__155_3_1_0/ LA - en ID - JSFS_2014__155_3_1_0 ER -
%0 Journal Article %A Tian, Simiao %A Scutari, Marco %A Denis, Jean-Baptiste %T Crossed Linear Gaussian Bayesian Networks, parsimonious models %J Journal de la société française de statistique %D 2014 %P 1-21 %V 155 %N 3 %I Société française de statistique %U http://www.numdam.org/item/JSFS_2014__155_3_1_0/ %G en %F JSFS_2014__155_3_1_0
Tian, Simiao; Scutari, Marco; Denis, Jean-Baptiste. Crossed Linear Gaussian Bayesian Networks, parsimonious models. Journal de la société française de statistique, Tome 155 (2014) no. 3, pp. 1-21. http://www.numdam.org/item/JSFS_2014__155_3_1_0/
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