Increasingly, for many application areas, it is becoming important to include elements of nonlinearity and non-Gaussianity in order to model accurately the underlying dynamics of a physical system. Moreover, it is typically crucial to process data on-line as it arrives, both from the point of view of storage costs as well as for rapid adaptation to changing signal characteristics. In this paper, we review both optimal and suboptimal Bayesian algorithms for nonlinear/non-Gaussian tracking problems, with a focus on particle filters. Particle filters are sequential Monte Carlo methods based on point mass (or “particle”) representations of probability densities, which can be applied to any state-space model and which generalize the traditional Kalman filtering methods. Several variants of the particle filter such as SIR, ASIR, and RPF are introduced within a generic framework of the sequential importance sampling (SIS) algorithm. These are discussed and compared with the standard EKF through an illustrative example.

Bayesian estimation problems where the posterior distribution evolves over time through the accumulation of data arise in many applications in statistics and related fields. Recently, a large number of algorithms and applications based on sequential Monte Carlo methods (also known as particle filtering methods) have appeared in the literature to solve this class of problems; see (Doucet, de Freitas & Gordon, 2001) for a survey. However, few of these methods have been proved to converge rigorously. The purpose of this paper is to address this issue. We present a general sequential Monte Carlo (SMC) method which includes most of the important features present in current SMC methods. This method generalizes and encompasses many recent algorithms. Under mild regularity conditions, we obtain rigorous convergence results for this general SMC method and therefore give theoretical backing for the validity of all the algorithms that can be obtained as particular cases of it. Keywords: Bayesian...

We develop methods for performing smoothing computations in general state-space models. The methods rely on a particle representation of the filtering distributions, and their evolution through time using sequential importance sampling and resampling ideas. In particular, novel techniques are presented for generation of sample realizations of historical state sequences. This is carried out in a forwardfiltering backward-smoothing procedure which can be viewed as the non-linear, non-Gaussian counterpart of standard Kalman filter-based simulation smoothers in the linear Gaussian case. Convergence in the mean-squared error sense of the smoothed trajectories is proved, showing the validity of our proposed method. The methods are tested in a substantial application for the processing of speech signals represented by a time-varying autoregression and parameterised in terms of timevarying partial correlation coe#cients, comparing the results of our algorithm with those from a simple smoother based upon the filtered trajectories.

The stochastic filtering problem deals with the estimation of the current state of a signal process given the information supplied by an associate process, usually called the observation process. We describe a particle algorithm designed for solving numerically discrete filtering problems. The algorithm involves the use of a system of n particles which evolve (mutate) in correlation with each other (interact) according to law of the signal process and, at fixed times, give birth to a number of offsprings depending on the observation process. We present several possible branching mechanisms and prove, in a general context the convergence of the particle systems (as n tends to 1) to the conditional distribution of the signal given the observation. We then apply the result to the discrete filtering and give several example when the results can be applied. AMS Subject Classification (1991): 93E11, 60G57, 65U05 Keywords: Filtering, Particle Systems, Branching Algorithms, Interacting Algor...

Several random particle systems approaches were recently suggested to solve numerically non linear filtering problems. The present analysis is concerned with genetic-type interacting particle systems. Our aim is to study the fluctuations on path space of such particle approximating models. Keywords : Central Limit, Interacting random processes, Filtering, Stochastic approximation. code A.M.S : 60F05, 60G35, 93E11, 62L20. 1 Introduction 1.1 Background and motivations The Non Linear Filtering problem consists in recursively computing the conditional distributions of a non linear signal given its noisy observations. This problem has been extensively studied in the literature and, with the notable exception of the linear-Gaussian situation or wider classes of models (B`enes filters [2]) optimal filters have no finitely recursive solution (ChaleyatMaurel /Michel [7]). Although Kalman filtering ([26],[29]) is a popular tool in handling estimation problems its optimality heavily depends on...

The non linear filtering problem consists in computing the conditional distributions of a Markov signal process given its noisy observations. The dynamical structure of such distributions can be modelled by a measure valued dynamical Markov process. Several random particle approximations were recently suggested to approximate recursively in time the so-called non linear filtering equations. We present an interacting particle system approach and we develop large deviations principles for the empirical measures of the particle systems. We end this paper extending the results to an interacting particle system approach which includes branchings. Keywords : Large deviations, Interacting random processes, Filtering, Stochastic approximation. code A.M.S : 60F10, 60H10, 60G35, 93E11, 62L20. 1 Introduction 1.1 Background and motivations The non linear filtering problem consists in computing the conditional distribution of a signal given its noisy observation. Roughly speaking, a basic model...

In this paper, we investigate the convergence of empirical processes for a class of interacting particle numerical schemes arising in biology, genetic algorithms and advanced signal processing. The Glivenko-Cantelli and Donsker theorems presented in this work extend the corresponding statements in the classical theory and apply to a class of genetic type particle numerical schemes of the non-linear filtering equation. Keywords : Empirical processes, Interacting particle systems, Glivenko-Cantelli and Donsker theorems. code A.M.S : 60G35, 92D25 UMR C55830, CNRS, Univ. Paul-Sabatier, 31062 Toulouse, delmoral@cict.fr y ledoux@cict.fr 1 Introduction 1.1 Background and Motivations Let E be a Polish space endowed with its Borel oe-field B(E). We denote by M 1 (E) the space of all probability measures on E equipped with the weak topology. We recall that the weak topology is generated by the bounded continuous functions on E and we denote by C b (E) the space of these functions. Let O...

Interacting particle methods have been recently proposed for the approximation of nonlinear filters. These are efficient recursive Monte Carlo methods, which in principle could be implemented in high dimensional problems --- i.e. which could beat the curse of dimensionality --- and where the particles automatically concentrate in regions of interest of the state space. In this paper we show that it is sometimes necessary to add a regularization step, and we analyze the approximation error for the resulting regularized interacting particle methods. 1. Introduction We consider the following model, where the unobserved state process fX t ; t 0g satisfies the stochastic differential equation (SDE) on R m dX t = b(X t ) dt + oe(X t ) dW t ; X 0 ¸ ¯ 0 ; (1) with standard Wiener process fW t ; t 0g, and where d--dimensional observations fzn ; n 1g are available at discrete time instants 0 ! t 1 ! \Delta \Delta \Delta ! t n ! \Delta \Delta \Delta z n = h(X t n ) + v n ; in additional w...

In this paper we consider the continuous time filtering problem and we estimate the order of convergence of an interacting particle system scheme presented by the authors in previous works. We will discuss how the discrete time approximating model of the Kushner-Stratonovitch equation and the genetic type interacting particle system approximation combine. We present quenched error bounds as well as mean order converge results. Keywords: Non Linear Filtering, Filter approximation, error bounds, interacting particle systems, genetic algorithms. Code A.M.S: 60G35,93E11, 62L20. 1 Introduction 1.1 Background and Motivations The aim of this work is the design of an interacting particle system approach for the numerical solving of the continuous time non linear filtering problem. Recently intense interest has been aroused in the non linear filtering theory community concerning the connections between non linear estimation, measure valued processes and branching and interacting particle sy...

We present in this work a natural Interacting Particle System (IPS) approach for modeling and studying the asymptotic behavior of Genetic Algorithms (GAs). In this model, a population is seen as a distribution (or measure) on the search space, and the Genetic Algorithm as a measure valued dynamical system. This model allows one to apply recent convergence results from the IPS literature for studying the convergence of genetic algorithms when the size of the population tends to infinity. We first review a number of approaches to Genetic Algorithms modeling and related convergence results. We then describe a general and abstract discrete time Interacting Particle System model for GAs, an we propose a brief review of some recent asymptotic results about the convergence of the N-IPS approximating model (of finite N-sized-population GAs) towards the IPS model (of infinite population GAs), including law of large number theorems, IL p -mean and exponential bounds as well as large deviations...