Yohan Petetin

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.

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.

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.

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.

A non asymptotical analysis of the optimal SIR algorithm vs. the fully adapted auxiliary particle filter

Particle filters (PF) and auxiliary particle filters (APF) are widely used sequential Monte Carlo (SMC) techniques for estimating the a posteriori filtering probability density function (pdf) in a Hidden Markov Chain (HMC). These algorithms have been theoretically analysed from an asymptotical statistics perspective. In this paper we provide a non asymptotical, finite number of samples comparative analysis of two particular SMC algorithms : the Sampling Importance Resampling (SIR) PF with optimal conditional importance distribution (CID), and the fully adapted APF (FA). Starting from a common set of N particles, we compute closed form expressions of the mean and variance of the empirical Monte Carlo (MC) estimators of a moment of the a posteriori filtering pdf. Both algorithms have the same mean, but in the case where resampling is used, the variance of the SIR algorithm always exceeds that of the FA algorithm.

Further Rao-Blackwellizing an already Rao-Blackwellized algorithm for Jump Markov State Space Systems

Exact Bayesian filtering is impossible in Jump Markov State Space Systems (JMSS), even in the simple linear and Gaussian case. Suboptimal solutions include sequential Monte-Carlo (SMC) algorithms which are indeed popular, and are declined in different versions according to the JMSS considered. In particular, Jump Markov Linear Systems (JMLS) are particular JMSS for which a Rao-Blackwellized (RB) Particle Filter (PF) has been derived. The RBPF solution relies on a combination of PF and Kalman Filtering (KF), and RBPF-based moment estimators outperform purely SMC-based ones when the number of samples tends to infinity. In this paper, we show that it is possible to derive a new RBPF solution, which implements a further RB step in the already RBPF with optimal importance distribution (ID). The new RBPF-based moment estimator outperforms the classical RBPF one whatever the number of particles, at the expense of a reasonable extra computational cost.

A semi-exact sequential Monte Carlo filtering algorithm in Hidden Markov Chains

Bayesian filtering is an important issue in Hidden Markov Chains (HMC) models. In many problems it is of interest to compute both the a posteriori filtering pdf at each time instant n and a moment Θn thereof. Sequential Monte Carlo (SMC) techniques, which include Particle filtering (PF) and Auxiliary PF (APF) algorithms, propagate a set of weighted particles which approximate that filtering pdf at time n, and then compute a Monte Carlo (MC) estimate of Θn. In this paper we show that in models where the so-called Fully Adapted APF (FA-APF) algorithm can be used such as semi-linear Gaussian state-space models, one can compute an estimate of the moment of interest at time n based only on the new observation yn and on the set of particles at time n - 1. This estimate does not suffer from the extra MC variation due to the sampling of new particles at time n, and is thus preferable to that based on that new set of particles, due to the Rao-Blackwell (RB) theorem. We finally extend our solution to models where the FA-APF cannot be used any longer.

A particle smoothing implementation of the fully-adapted auxiliary particle filter: An alternative to auxiliary particle filters

The Fully Adapted Auxiliary Particle Filter (FA-APF) is a well known Sequential Monte Carlo (SMC) algorithm for computing recursively the filtering pdf in a Hidden Markov Chain (HMC) model. However, in most of cases, the FA-APF cannot be used directly because the required functions are unavailable. To cope with this issue, the Auxiliary Particle Filter (APF) uses Importance Sampling (IS) with two degrees of freedom. APF techniques need an importance distribution and also a reliable approximation of the predictive likelihood. In this paper, we propose a class of SMC algorithms which also try to mimic the FA-APF but which have the advantage not to require any approximation of the predictive likelihood. The performances of our solution as compared to the APF algorithm is provided by simulations.