On clustering procedures and nonparametric mixture estimation

Abstract

This paper deals with nonparametric estimation of conditional densities in mixture models in the case when additional covariates are available. The proposed approach consists of performing a preliminary clustering algorithm on the additional covariates to guess the mixture component of each observation. Conditional densities of the mixture model are then estimated using kernel density estimates applied separately to each cluster. We investigate the expected L 1 -error of the resulting estimates and derive optimal rates of convergence over classical nonparametric density classes provided the clustering method is accurate. Performances of clustering algorithms are measured by the maximal misclassification error. We obtain upper bounds of this quantity for a single linkage hierarchical clustering algorithm. Lastly, applications of the proposed method to mixture models involving electricity distribution data and simulated data are presented.

Statistical analysis of a hierarchical clustering algorithm with outliers

“…ε = 0), the clustering risk of both SL and OSL is roughly zero. This result is consistent with Theorem 4.1, as well as with the results stated in Arias-Castro (2011) and Auray et al (2015). In these papers, the authors prove that, under assumptions close to (A1)-(A6) and when ε = 0, SL is consistent and its clustering risk tends quickly to zero.…”

“…This algorithm is known to be consistent without outliers (i.e. if ε = 0) and under assumptions close to ours (Arias-Castro, 2011, Auray et al, 2015).…”

“…, M are sufficiently separated. The single linkage hierarchical clustering algorithm has also been investigated by Arias-Castro (2011) and Auray et al (2015). However, it is well known that this algorithm is sensitive to outliers.…”

“…In this paper, we consider a mathematical framework close to the one used in Maier et al (2009), , Auray et al (2015). The data are generated according to a mixture of several distributions whose supports are assumed to be disjoint in order to identify the groups.…”

Simultaneous semi-parametric estimation of clustering and regression

“…The estimation of a regression model with a fixed group effect is generally performed using a two-step approach as for instance in epidemiology or in economics (Auray et al, 2015;Ando and Bai, 2016;Zhang et al, 2019). As a first step, a clustering on the individual based on the group related variables is performed to obtain an estimator of the group.…”