EFFICIENT PARTICLE FILTERING FOR JUMP MARKOV SYSTEMS Christophe Andrieu - Manuel Davy - Arnaud Doucet University of Bristol, Statistics Group, Bristol BS8 1TW, UK University of Cambridge, Signal Processing Group, Trumpington Street, Cambridge CB2 1PZ, UK IRCCyN/CNRS, 1 rue de la Noe, BP 92101, 44321 Nantes cedex 3, France University of Melbourne, Department of Electrical Engineering, Parkville, Victoria 3010, Australia. Email: c.andrieu@bris.ac.uk - md283@eng.cam.ac.uk - a.doucet@ee.mu.oz.au Abstract We address here the problem of developing efficient particle filtering techniques in order to estimate the state of Jump Markov Systems (JMS). These processes are often met in signal pro- cessing (target tracking, communication...). Our algorithm takes advantage of the structure of the process. We apply our algo- rithm to time varying autoregressive processes. 1 Introduction 1.1 Background Many estimation problems arising in signal processing and re- lated fields can be cast in the following form. A unique dy- namic model represents the signal of interest ( and ), which is not observed but statistically related to the observations ( and ). More formally the process is typically described with a Markov transition density , where, for a set of variables , we denote . The observations are usually assumed to be independent conditional upon the signal process , and marginally distributed according to . One is then interested in estimat- ing the sequence of posterior densities and typ- ically their marginals . This so-called optimal fil- tering problem usually does not admit a closed-form solution and many approaches have been proposed in the last 35 years in order to approximate these distributions. Particle filtering techniques are simulation-based methods that have recently revolutionized this field. They allow for recursive state estima- tion of virtually any dynamical system, even when elements of non-linearity and non-Gaussianity are present. Spectacular re- sults can be found for real applications in [2], and the field is currently growing at a steady pace. One of the great interest of particle filtering techniques, and more generally of Monte Carlo methods, is that they allow for heuristic approaches pre- viously developed to be embedded in a rigourous framework. The problem addressed in this paper is that of the development of efficient particle filtering techniques to perform on-line de- tection and estimation for a very important class of dynami- cal systems, named Jump Markov Systems (JMS) or Markov Switching State Space Models. JMS are a class of models ap- pearing in signal processing, target tracking and econometrics among others [1], [4]. This class extends significantly the class of models described above and specific particle filtering meth- ods have to be developed to perform optimal estimation. 1.2 Jump Markov Systems 1.2.1 Statistical Model Let be a stationary, finite, discrete, first order homoge- neous Markov chain taking its values in a set , with transi- tion probabilities We define the finite number of elements of . Now con- sider a family of densities where and , and define the state transition conditional den- sities, (1) The initial state is distributed according to a distribution . Note that the dimension, or nature, of might be a function of the sequence , but we do not make this dependence explicit in order to alleviate notation. Neither the process nor are observed. Instead, we observe where (2) with (the number of observations can vary over time). It is possible to add exogenous variables in the equations, i.e. and can also depend on , but we omit them to simplify notation. Example. Bearings-Only Tracking for a Maneuvering Source [1]. Let us consider the following standard tracking model. A single target evolves in a 2-D plan . Conditional upon a maneuver of the target ( is modeled as a Markov chain) the state consists of the location and velocity , which evolve according to a standard linear Gaussian model (3) We observe (4) The nonlinearity appearing in the observation equation (4) pre- cludes the use of standard suboptimal filtering techniques de- veloped for JMLS. Other examples include multiple target track- ing. 1.2.2 Estimation Objectives The aim of optimal filtering is to estimate sequentially in time the unknown “hidden” states and more precisely the series of posterior distributions . Their margi- nals, and in particular the filtering densities , are of interest in practice. A simple application of Bayes’ rule al- lows for an easy formulation of the recursion that updates the posterior distributions: There is no closed form solution to this recursion and for state estimates of the form which include the Minimum Mean Square Estimate (MMSE) of the state, and its covariance for example. For the sake of simplicity in notation, finite sums will be replaced further on by integrals whenever it is convenient.