Homogeneity and change-point detection tests for multivariate data using rank statistics

Lung-Yut-Fong, Alexandre; Lévy-Leduc, Céline; Cappé, Olivier

Journal de la société française de statistique, Tome 156 (2015) no. 4, pp. 133-162.

La détection et la localisation de changements dans des données de très grande dimension est un problème majeur dans plusieurs domaines d’applications. Dans ce contexte, la première contribution de notre papier est un nouveau test d’homogénéité non-paramétrique à deux échantillons pour des données multivariées fondé sur la statistique de rang de Wilcoxon. Le test d’homogénéité à deux échantillons que nous proposons peut être étendu au cas de données censurées et pour proposer un test d’homogénéité pour plus de deux échantillons. Nous proposons également une analyse détaillée du calcul de la puissance de notre statistique de test vis à vis de certaines alternatives locales. La seconde contribution de notre papier concerne l’utilisation de notre statistique de test pour faire de la détection rétrospective de ruptures. Nous montrons que notre méthode peut-être implémentée de façon efficace d’un point de vue algorithmique grâce à la programmation dynamique et nous proposons une méthode pour calculer les $p$-valeurs. Nous recommandons particulièrement notre approche dans les situations suivantes : lorsque les corrélations entre les coordonnées des observations sont modérées, lorsque les lois marginales ne peuvent pas être modélisées par les lois paramétriques usuelles ou lorsque les changements n’affectent qu’une partie des coordonnées des observations.

Detecting and locating changes in highly multivariate data is a major concern in several current statistical applications. In this context, the first contribution of the paper is a novel non-parametric two-sample homogeneity test for multivariate data based on the well-known Wilcoxon rank statistic. The proposed two-sample homogeneity test statistic can be extended to deal with ordinal or censored data as well as to test for the homogeneity of more than two samples. We also provide a detailed analysis of the power of the proposed test statistic (in the two sample case) against asymptotic local shift alternatives. The second contribution of the paper concerns the use of the proposed test statistic to perform retrospective change-point detection. It is first shown that the approach is computationally feasible even when looking for a large number of change-points thanks to the use of dynamic programming. Computable asymptotic $p$-values for the test are available in the case where a single potential change-point is to be detected. The proposed approach is particularly recommendable in situations where the correlations between the coordinates of the data are moderate, the marginal distributions are not well modelled by usual parametric assumptions (e.g., in the presence of outliers) and when faced with highly variable change patterns, for instance, if the potential changes only affect subsets of the coordinates of the data.

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