Didier Chauveau

An EM-Like Algorithm for Semi- and Nonparametric Estimation in Multivariate Mixtures

We propose an algorithm for nonparametric estimation for finite mixtures of multivariate random vectors that strongly resembles a true EM algorithm. The vectors are assumed to have independent coordinates conditional upon knowing from which mixture component they come, but otherwise their density functions are completely unspecified. Sometimes, the density functions may be partially specified by Euclidean parameters, a case we call semiparametric. Our algorithm is much more flexible and easily applicable than existing algorithms in the literature; it can be extended to any number of mixture components and any number of vector coordinates of the multivariate observations. Thus it may be applied even in situations where the model is not identifiable, so care is called for when using it in situations for which identifiability is difficult to establish conclusively. Our algorithm yields much smaller mean integrated squared errors than an alternative algorithm in a simulation study. In another example using a real dataset, it provides new insights that extend previous analyses. Finally, we present two different variations of our algorithm, one stochastic and one deterministic, and find anecdotal evidence that there is not a great deal of difference between the performance of these two variants. The computer code and data used in this article are available online.

A stochastic EM algorithm for a semiparametric mixture model

Recently, there has been a considerable interest in finite mixture models with semi-/non-parametric component distributions. Identifiability of such model parameters is generally not obvious, and when it occurs, inference methods are rather specific to the mixture model under consideration. Hence, a generalization of the EM algorithm to semiparametric mixture models is proposed. The approach is methodological and can be applied to a wide class of semiparametric mixture models. The behavior of the proposed EM type estimators is studied numerically not only through several Monte-Carlo experiments but also through comparison with alternative methods existing in the literature. In addition to these numerical experiments, applications to real data are provided, showing that the estimation method behaves well, that it is fast and easy to be implemented.

Semiparametric mixtures of regressions

We present an algorithm for estimating parameters in a mixture-of-regressions model in which the errors are assumed to be independent and identically distributed but no other assumption is made. This model is introduced as one of several recent generalizations of the standard fully parametric mixture of linear regressions in the literature. A sufficient condition for the identifiability of the parameters is stated and proved. Several different versions of the algorithm, including one that has a provable ascent property, are introduced. Numerical tests indicate the effectiveness of some of these algorithms.

Bandwidth Selection in an EM-Like Algorithm for Nonparametric Multivariate Mixtures

In this paper we describe a method to select the bandwidth used in the nonparametric EM (npEM) algorithm of Benaglia et al. (2008). This method is a generalization of the Silverman’s rule of thumb used to select a bandwidth in kernel density estimation, and it results in one bandwidth for each mixture component and each block of conditionally independent and identically distributed repeated measures.

Semi-Parametric Estimation for Conditional Independence Multivariate Finite Mixture Models

The conditional independence assumption for nonparametric multivariate finite mixture models, a weaker form of the well-known conditional independence assumption for random effects models for longitudinal data, is the subject of an increasing number of theoretical and algorithmic developments in the statistical literature. After presenting a survey of this literature, including an in-depth discussion of the all-important identifiability results, this article describes and extends an algorithm for estimation of the parameters in these models. The algorithm works for any number of components in three or more dimensions. It possesses a descent property and can be easily adapted to situations where the data are grouped in blocks of conditionally independent variables. We discuss how to adapt this algorithm to various location-scale models that link component densities, and we even adapt it to a particular class of univariate mixture problems in which the components are assumed symmetric. We give a bandwidth selection procedure for our algorithm. Finally, we demonstrate the effectiveness of our algorithm using a simulation study and two psychometric datasets.

Changes in the epigenome and transcriptome of the poplar shoot apical meristem in response to water availability affect preferentially hormone pathways

The adaptive capacity of long-lived organisms such as trees to the predicted climate changes, including severe and successive drought episodes, will depend on the presence of genetic diversity and phenotypic plasticity. Here, the involvement of epigenetic mechanisms in phenotypic plasticity toward soil water availability was examined in Populus×euramericana. This work aimed at characterizing (i) the transcriptome plasticity, (ii) the genome-wide plasticity of DNA methylation, and (iii) the function of genes affected by a drought-rewatering cycle in the shoot apical meristem. Using microarray chips, differentially expressed genes (DEGs) and differentially methylated regions (DMRs) were identified for each water regime. The rewatering condition was associated with the highest variations of both gene expression and DNA methylation. Changes in methylation were observed particularly in the body of expressed genes and to a lesser extent in transposable elements. Together, DEGs and DMRs were significantly enriched in genes related to phytohormone metabolism or signaling pathways. Altogether, shoot apical meristem responses to changes in water availability involved coordinated variations in DNA methylation, as well as in gene expression, with a specific targeting of genes involved in hormone pathways, a factor that may enable phenotypic plasticity.

Nonparametric mixture models with conditionally independent multivariate component densities

Models and algorithms for nonparametric estimation of finite multivariate mixtures have been recently proposed, where it is usually assumed that coordinates are independent conditional on the subpopulation from which each observation is drawn. Hence in these models the dependence structure comes only from the mixture. This assumption is relaxed, allowing for independent multivariate blocks of coordinates, conditional on the subpopulation from which each observation is drawn. Otherwise the density functions of these blocks are completely multivariate and nonparametric. An EM-like algorithm for this model is proposed, and some strategies for selecting the bandwidth matrix involved in the nonparametric estimation step of it are derived. The performance of this algorithm is evaluated through several numerical simulations. A real dataset of reasonably large dimension is experimented on this new model and algorithm to illustrate its potential from the model based, unsupervised clustering perspective.

Winter-dormant shoot apical meristem in poplar trees shows environmental epigenetic memory

The winter-dormant shoot apical meristem of the tree species Populus keeps an epigenetic memory of environmental variations that arose during the preceding vegetative period.

Some Algorithms to Fit some Reliability Mixture Models under Censoring

Estimating the unknown parameters of a reliability mixture model may be a more or less intricate problem, especially if durations are censored. We present several iterative methods based on Monte Carlo simulation that allow to fit parametric or semiparametric mixture models provided they are identifiable. We show for example that the well-known data augmentation algorithm may be used successfully to fit semiparametric mixture models under right censoring. Our methods are illustrated by a reliability example.