Nonparametric estimation of the mixing density using polynomials

We consider the problem of estimating the mixing density $f$ from $n$ i.i.d. observations distributed according to a mixture density with unknown mixing distribution. In contrast with finite mixtures models, here the distribution of the hidden variable is not bounded to a finite set but is spread out over a given interval. We propose an approach to construct an orthogonal series estimator of the mixing density $f$ involving Legendre polynomials. The construction of the orthonormal sequence varies from one mixture model to another. Minimax upper and lower bounds of the mean integrated squared error are provided which apply in various contexts. In the specific case of exponential mixtures, it is shown that the estimator is adaptive over a collection of specific smoothness classes, more precisely, there exists a constant A\textgreater{}0 such that, when the order $m$ of the projection estimator verifies $m\sim A \log(n)$, the estimator achieves the minimax rate over this collection. Other cases are investigated such as Gamma shape mixtures and scale mixtures of compactly supported densities including Beta mixtures. Finally, a consistent estimator of the support of the mixing density $f$ is provided

Model-based graph clustering of a collection of networks using an agglomerative algorithm

Graph clustering is the task of partitioning a collection of observed networks into groups of similar networks. Here similarity means networks have a similar structure or graph topology. To this end, a model-based approach is developed, where the networks are modelled by a finite mixture model of stochastic block models. Moreover, a computationally efficient clustering algorithm is developed. The procedure is an agglomerative hierarchical algorithm that maximizes the so-called integrated classification likelihood criterion. The bottom-up algorithm consists of successive merges of clusters of networks. Those merges require a means to match block labels of two stochastic block models to overcome the label-switching problem. This problem is addressed with a new distance measure for the comparison of stochastic block models based on their graphons. The algorithm provides a cluster hierarchy in form of a dendrogram and valuable estimates of all model parameters

A semiparametric extension of the stochastic block model for longitudinal networks

To model recurrent interaction events in continuous time, an extension of the stochastic block model is proposed where every individual belongs to a latent group and interactions between two individuals follow a conditional inhomogeneous Poisson process with intensity driven by the individuals' latent groups. The model is shown to be identifiable and its estimation is based on a semiparametric variational expectation-maximization algorithm. Two versions of the method are developed, using either a nonparametric histogram approach (with an adaptive choice of the partition size) or kernel intensity estimators. The number of latent groups can be selected by an integrated classification likelihood criterion. Finally, we demonstrate the performance of our procedure on synthetic experiments, analyse two datasets to illustrate the utility of our approach and comment on competing methods

OMP-type Algorithm with Structured Sparsity Patterns for Multipath Radar Signals

A transmitted, unknown radar signal is observed at the receiver through more than one path in additive noise. The aim is to recover the waveform of the intercepted signal and to simultaneously estimate the direction of arrival (DOA). We propose an approach exploiting the parsimonious time-frequency representation of the signal by applying a new OMP-type algorithm for structured sparsity patterns. An important issue is the scalability of the proposed algorithm since high-dimensional models shall be used for radar signals. Monte-Carlo simulations for modulated signals illustrate the good performance of the method even for low signal-to-noise ratios and a gain of 20 dB for the DOA estimation compared to some elementary method

Adaptive Density Estimation in the Pile-up Model Involving Measurement Errors

International audienceMotivated by fluorescence lifetime measurements this paper considers the problem of nonparametric density estimation in the pile-up model. Adaptive nonparametric estimators are proposed for the pile-up model in its simple form as well as in the case of additional measurement errors. Furthermore, oracle type risk bounds for the mean integrated squared error (MISE) are provided. Finally, the estimation methods are assessed by a simulation study and the application to real fluorescence lifetime data

An MCMC approach for estimating a fluorescence lifetime with pile-up distortion

Ce travail présente un nouvel estimateur de la distribution de la durée de vie en fluorescence. Un échantillonneur de Gibbs est développé pour estimer les paramètres quand le minimum d'un nombre aléatoire de variables distribuées selon un mélange exponentiel est observé. L'algorithme est testé avec des données simulées, et une comparaison avec des méthodes utilisées en pratique est faite. Nos résultats indiquent que la méthode proposée requiert moins d'observations que des méthodes classiques pour obtenir la même qualité statistique de l'estimation

Information bounds and MCMC parameter estimation for the pile-up model

International audienceThis paper is concerned with the pile-up model defined as a nonlinear transformation of a distribution of interest. An observation of the pile-up model is the minimum of a random number of independent variables from the distribution of interest. One specific pile-up model is encountered in time-resolved fluorescence where only the first photon of a random number of photons is observed. In the first part of the paper the Cramér-Rao bound is studied to optimize the experimental conditions by choosing the best tuning parameter which is the average number of variables over which the minimum is taken. The implication is that the tuning parameter currently used in fluorescence does not minimize the acquisition time. However, data obtained at the optimal choice of the tuning parameter require an estimator adapted to the pile-up effect, therefore, an appropriate Gibbs sampler is presented. The covariance matrix of this estimator turns out to be close to the Cramér-Rao bound and hence the acquisition time may be reduced considerably

Estimation dans le modèle d'empilement avec application aux mesures de la fluorescence résolue en temps

This thesis studies the so-called pile-up model and proposes adequate estimators. An observation of the pile-up model is the minimum of a random number of variables from the target distribution. The pile-up distribution is the result of a non linear distortion of the target distribution. The goal is to identify the target distribution from observations of the pile-up model. The model is motivated by the application TCSPC in time-resolved fluorescence, where the extent of distortion is determined by a tuning parameter selected by the user. A study of the Cramér-Rao bound provides the best value of this parameter. Simulations with a Gibbs sampler confirm the theoretical results on a significant reduction of the variance compared to the current practice. Another estimator is proposed by a maximum likelihood approach based on a new contrast and whose computation time is satisfactory. In many cases the estimator can be computed by an EM-type algorithm. Furthermore, the consistence as well as the limit distribution is established. A comparison to the current practice in fluorescence shows that a reduction of the acquisition time by a factor 10 is possible. In the last part, a non parametric estimator of the mixing density of an infinite mixture of exponential densities is proposed. The estimator is based on orthogonal series and it is shown to be optimal in the sense that its mean integrated square error achieves the minimax rate on some specific smoothness spaces. Moreover, the estimator can be adapted to the pile-up model, when the target distribution is an infinite exponential mixture.Cette thèse étudie le modèle d’empilement et propose des estimateurs appropriés. Une observation de ce modèle est le minimum d’un nombre aléatoire de variables de la loi initiale. La distribution du modèle d’empilement est le résultat d’une distorsion non linéaire de la loi initiale. L’objectif est d’identifier la loi initiale à partir des observations du modèle d’empilement. Le modèle est motivé par l’application TCSPC en fluorescence, où l’ampleur de la distorsion est déterminée par un paramètre de réglage sélectionné par l’utilisateur. Une étude de la borne de Cramér-Rao fournit la meilleure valeur de ce paramètre. Des simulations avec un échantillonneur de Gibbs confirment les résultats théoriques sur une réduction significative de la variance en comparaison avec la pratique habituelle. Un autre estimateur est proposé par une approche de maximum de vraisemblance basé sur un nouveau contraste et dont le temps de calcul est satisfaisant. Dans des nombreux cas, l’estimateur peut se calculer par un algorithme de type E. M. Par ailleurs, la consistance ainsi que la loi limite de cet estimateur sont établies. Une comparaison avec la pratique actuelle en fluorescence montre qu’une réduction du temps d’acquisition d’un facteur 10 est envisageable. Finalement, un estimateur non paramétrique de la densité mélangeante d’un mélange infini de lois exponentielles est proposé. Celui-ci est basé sur des séries orthogonales et se montre optimal dans le sens que son erreur quadratique atteint la vitesse minimax dans des espaces de régularité bien choisis. Cet estimateur est aussi adapté au modèle d’empilement, lorsque la loi initiale est un mélange infini de lois exponentielles