Section: New Results

Statistical Learning and Bayesian Analysis

Non-parametric Methods for Function Approximation

Linear Regression with Random Projections [16]

We investigate a method for regression that makes use of a randomly generated subspace $G_P$ (of finite dimension $P$) of a given large (possibly infinite) dimensional function space $F$, for example, $L_2\left({[0,1]}^d\right)$. $G_P$ is defined as the span of $P$ random features that are linear combinations of a basis functions of $F$ weighted by random Gaussian i.i.d. coefficients. We show practical motivation for the use of this approach, detail the link that this random projections method share with RKHS and Gaussian objects theory and prove, both in deterministic and random design, approximation error bounds when searching for the best regression function in $G_P$ rather than in $F$, and derive excess risk bounds for a specific regression algorithm (least squares regression in $G_P$). This paper stresses the motivation to study such methods, thus the analysis developed is kept simple for explanations purpose and leaves room for future developments.

Nonparametric Bayesian Estimation

DPM pour l'inférence dans les modèles dynamiques non linéaires avec des bruits de mesure alpha-stable [50]

Stable random variables are often use to model impulsive noise; Recently it has be shown that communication at very high frequency suffer from such a noise. Stable noise cannot however be considered as usual noise in estimation processes because the variance does not usually exists nor an analytic expression for the probability density function. In this work we show how to manage such a problem using a bayesian nonparametric approach. We develop a Sequential Monte Carlo based algorithm to realize the estimation in a non linear dynamical system. The measurement noise is a non-stationnary stable process and it is modeled using a Dirichlet Process Mixture.

Random Finite Sets for Multisensor Multitarget Tracking

Multi-sensor PHD filtering with application to sensor management [2]

The aim of multi-object filtering is to address the multiple target detection and/or tracking problem. This thesis focuses on the Probability Hypothesis Density (PHD) filter, a well-known tractable approximation of the Random Finite Set (RFS) filter when the observation process is realized by a single sensor. The first part proposes the rigorous construction of the exact multi-sensor PHD filter and its simplified expression, without approximation, through a joint partitioning of the target state space and the sensors. With this new method, the exact multi-sensor PHD can be propagated in simple surveillance scenarii. The second part deals with the sensor management problem in the PHD framework. At each iteration, the Balanced Explorer and Tracker (BET) builds a prediction of the posterior multi-sensor PHD thanks to the Predicted Ideal Measurement Set (PIMS) and produces a multi-sensor control according to a few simple operational principles adapted to surveillance activities