François Desbouvries

A fixed-lag particle smoother for blind SISO equalization of time-varying channels

We introduce a new sequential importance sampling (SIS) algorithm which propagates in time a Monte Carlo approximation of the posterior fixed-lag smoothing distribution of the symbols under doubly-selective channels. We perform an exact evaluation of the optimal importance distribution, at a reduced computational cost when compared to other optimal solutions proposed for the same state-space model. The method is applied as a soft input-soft output (SISO) blind equalizer in a turbo receiver framework and simulation results are obtained to show its outstanding BER performance.

Marginalized particle PHD filters for multiple object Bayesian filtering

The Probability Hypothesis Density (PHD) filter is a recent solution to the multi-target filtering problem. Because the PHD filter is not computable, several implementations have been proposed including the Gaussian Mixture (GM) approximations and Sequential Monte Carlo (SMC) methods. In this paper, we propose a marginalized particle PHD filter which improves the classical solutions when used in stochastic systems with partially linear substructure.

Optimal SIR algorithm vs. fully adapted auxiliary particle filter: a non asymptotic analysis

Particle filters (PF) and auxiliary particle filters (APF) are widely used sequential Monte Carlo (SMC) techniques. In this paper we comparatively analyse, from a non asymptotic point of view, the Sampling Importance Resampling (SIR) PF with optimal conditional importance distribution (CID) and the fully adapted APF (FA). We compute the (finite samples) conditional second order moments of Monte Carlo (MC) estimators of a moment of interest of the filtering pdf, and analyse under which circumstances the FA-based estimator outperforms (or not) the optimal Sequential Importance Sampling (SIS)-based one. Our analysis is local, in the sense that we compare the estimators produced by one time step of the different SMC algorithms, starting from a common set of weighted points. This analysis enables us to propose a hybrid SIS/FA algorithm which automatically switches at each time step from one loop to the other. We finally validate our results via computer simulations.

A mixed GM/SMC implementation of the probability hypothesis density filter

The Probability Hypothesis Density (PHD) filter is a recent solution for tracking an unknown number of targets in a multi-object environment. The PHD filter cannot be computed exactly, but popular implementations include Gaussian Mixture (GM) and Sequential Monte Carlo (SMC) based algorithms. GM implementations suffer from pruning and merging approximations, but enable to extract the states easily; on the other hand, SMC implementations are of interest if the discrete approximation is relevant, but are penalized by the difficulty to guide particles towards promising regions and to extract the states. In this paper, we propose a mixed GM/SMC implementation of the PHD filter which does not suffer from the above mentioned drawbacks. Due to the SMC part, our algorithm can be used in models where the GM implementation is unavailable; but it also benefits from the easy state extraction of GM techniques, without requiring pruning or merging approximations. Our algorithm is validated on simulations.

A Class of Fast Exact Bayesian Filters in Dynamical Models With Jumps

We address the statistical filtering problem in dynamical models with jumps. When a particular application is adequately modeled by linear and Gaussian probability density functions with jumps, a usual method consists in approximating the optimal Bayesian estimate [in the sense of the minimum mean square error (MMSE)] in a linear and Gaussian jump Markov state space system (JMSS). Practical solutions include algorithms based on numerical approximations or on sequential Monte Carlo (SMC) methods. In this paper, we propose a class of alternative methods which consists in building statistical models which, locally, similarly model the problem of interest, but in which the computation of the MMSE estimate can be be computed exactly (without numerical nor SMC approximations) and at a computational cost which is linear in the number of observations.

Bayesian Multi-Object Filtering for Pairwise Markov Chains

Random finite sets (RFS) are recent tools for addressing the multi-object filtering problem. The probability hypothesis density (PHD) Filter is an approximation of the multi-object Bayesian filter, which results from the RFS formulation of the problem and has been used in many applications. In the RFS framework, it is assumed that each target and associated observation follow a hidden Markov chain (HMC) model. HMCs conveniently describe some physical properties of practical interest for practitioners, but they also implicitly imply restrictive independence properties which, in practice, may not be satisfied by data. In this paper, we show that these structural limitations of HMC models can somehow be relaxed by embedding them into the more general class of pairwise Markov chain (PMC) models. We thus focus on the computation of the PHD filter in a PMC framework, and we propose a practical implementation of the PHD filter for a particular class of PMC models.

Bayesian Conditional Monte Carlo Algorithms for Nonlinear Time-Series State Estimation

Bayesian filtering aims at estimating sequentially a hidden process from an observed one. In particular, sequential Monte Carlo (SMC) techniques propagate in time weighted trajectories which represent the posterior probability density function (pdf) of the hidden process given the available observations. On the other hand, conditional Monte Carlo (CMC) is a variance reduction technique which replaces the estimator of a moment of interest by its conditional expectation given another variable. In this paper, we show that up to some adaptations, one can make use of the time recursive nature of SMC algorithms in order to propose natural temporal CMC estimators of some point estimates of the hidden process, which outperform the associated crude Monte Carlo (MC) estimator whatever the number of samples. We next show that our Bayesian CMC estimators can be computed exactly, or approximated efficiently, in some hidden Markov chain (HMC) models; in some jump Markov state-space systems (JMSS); as well as in multitarget filtering. Finally our algorithms are validated via simulations.

Marginalized PHD Filters for multi-target filtering

Multi-target filtering aims at tracking an unknown number of targets from a set of observations. The Probability Hypothesis Density (PHD) Filter is a promising solution but cannot be implemented exactly. Suboptimal implementation techniques include Gaussian Mixture (GM) solutions, which hold only in linear and Gaussian models, and Sequential Monte Carlo (SMC) algorithms, which estimate the number of targets and their state parameters for a more general class of models. In this paper, we address the case of Gaussian models where the state can be decomposed into a linear component and a non-linear one, and we show that the use of SMC methods in such models can indeed be reduced. Our technique not only improves the estimate of the number of targets but also that of their state. We finally adapt the technique to linear and Gaussian jump Markov state space systems (JMSS) in order to reduce the intractability of existing solutions, and to JMSS with partially linear and partially non-linear state vector.

Fixed-Interval Kalman Smoothing Algorithms in Singular State---Space Systems

Fixed-interval Bayesian smoothing in state–space systems has been addressed for a long time. However, as far as the measurement noise is concerned, only two cases have been addressed so far : the regular case, i.e., with positive definite covariance matrix; and the perfect measurement case, i.e., with zero measurement noise. In this paper we address the smoothing problem in the intermediate case where the measurement noise covariance is positive semi definite with arbitrary rank. We exploit the singularity of the model in order to transform the original state–space system into a pairwise Markov model with reduced state dimension. Finally, the a posteriori Markovianity of the reduced state enables us to propose a family of fixed-interval smoothing algorithms.

Optimal SIR algorithm vs. fully adapted auxiliary particle filter: A matter of conditional independence

Particle filters (PF) and auxiliary particle filters (APF) are widely used sequential Monte Carlo (SMC) techniques. In this paper we comparatively analyse the Sampling Importance Resampling (SIR) PF with optimal conditional importance distribution (CID) and the fully adapted APF (FA-APF). Both algorithms share the same Sampling (S), Weighting (W) and Resampling (R) steps, and only differ in the order in which these steps are performed. The order of the operations is not unsignificant: starting at time n − 1 from a common set of particles, we show that one single updated particle at time n will marginally be sampled in both algorithms from the same probability density function (pdf), but as a whole the full set of particles will be conditionally independent if created by the FA-APF algorithm, and dependent if created by the SIR algorithm, which results in support degeneracy.