In this article, we present an overview of methods for sequential simulation from posterior distributions. These methods are of particular interest in Bayesian filtering for discrete time dynamic models that are typically nonlinear and non-Gaussian. A general importance sampling framework is developed that unifies many of the methods which have been proposed over the last few decades in several different scientific disciplines. Novel extensions to the existing methods are also proposed. We show in particular how to incorporate local linearisation methods similar to those which have previously been employed in the determin-istic filtering literature; these lead to very effective importance distributions. Furthermore we describe a method which uses Rao-Blackwellisation in order to take advantage of the analytic structure present in some important classes of state-space models. In a final section we develop algorithms for prediction, smoothing and evaluation of the likelihood in dynamic models.

A new approach toward target representation and localization, the central component in visual tracking of non-rigid objects, is proposed. The feature histogram based target representations are regularized by spatial masking with an isotropic kernel. The masking induces spatially-smooth similarity functions suitable for gradient-based optimization, hence, the target localization problem can be formulated using the basin of attraction of the local maxima. We employ a metric derived from the Bhattacharyya coefficient as similarity measure, and use the mean shift procedure to perform the optimization. In the presented tracking examples the new method successfully coped with camera motion, partial occlusions, clutter, and target scale variations. Integration with motion filters and data association techniques is also discussed. We describe only few of the potential applications: exploitation of background information, Kalman tracking using motion models, and face tracking. Keywords: non-rigid object tracking; target localization and representation; spatially-smooth similarity function; Bhattacharyya coefficient; face tracking. 1

. In this report, we present an overview of sequential simulationbased methods for Bayesian filtering of nonlinear and non-Gaussian dynamic models. It includes in a general framework numerous methods proposed independently in various areas of science and proposes some original developments. Keywords: Bayesian estimation, optimal filtering, nonlinear non-Gaussian state space models, hidden Markov models, sequential Monte Carlo methods. 1. Introduction 1 Many problems in statistical signal processing, automatic control, applied statistics or econometrics can be stated as follows. A transition equation describes the prior distribution of the Markovian hidden signal of interest fx k ; k 2 Ng, the so-called hidden state process, and an observation equation describes the likelihood of the observations fy k ; k 2 Ng, k being the discrete time index. The aim is to estimate the hidden state process using the observations. In the Bayesian framework, all relevant information on fx 0 ; x ...

The Kalman filter provides an effective solution to the linear-Gaussian filtering problem. However, where there is nonlinearity, either in the model specification or the observation process, other methods are required. We consider methods known generically as particle filters, which include the condensation algorithm and the Bayesian bootstrap or sampling importance resampling (SIR) filter. These filters

Optimal filtering problems are ubiquitous in signal processing and related fields. Except for a restricted class of models, the optimal filter does not admit a closed-form expression. Particle filtering methods are a set of flexible and powerful sequential Monte Carlo methods designed to solve the optimal filtering problem numerically. The posterior distribution of the state is approximated by a large set of Dirac-delta masses (samples/particles) that evolve randomly in time according to the dynamics of the model and the observations. The particles are interacting; thus, classical limit theorems relying on statistically independent samples do not apply. In this paper, our aim is to present a survey of recent convergence results on this class of methods to make them accessible to practitioners.

this article then is to develop methodology for modeling the nonnormality of the ut, the vt, or both. A second departure from the model specification ( 1 ) is to allow for unknown variances in the state or observational equation, as well as for unknown parameters in the transition matrices Ft and Ht. As a third generalization we allow for nonlinear model structures; that is, X t = ft(Xt-l) q- Ut, and Yt = ht(xt) + vt, t = 1, ..., n, (2) whereft( ) and ht(. ) are given, but perhaps also depend on some unknown parameters. The experimenter may wish to entertain a variety of error distributions. Our goal throughout the article is an analysis for general state-space models that does not resort to convenient assumptions at the expense of model adequacy

Abstract—An overview of statistical and information-theoretic aspects of hidden Markov processes (HMPs) is presented. An HMP is a discrete-time finite-state homogeneous Markov chain observed through a discrete-time memoryless invariant channel. In recent years, the work of Baum and Petrie on finite-state finite-alphabet HMPs was expanded to HMPs with finite as well as continuous state spaces and a general alphabet. In particular, statistical properties and ergodic theorems for relative entropy densities of HMPs were developed. Consistency and asymptotic normality of the maximum-likelihood (ML) parameter estimator were proved under some mild conditions. Similar results were established for switching autoregressive processes. These processes generalize HMPs. New algorithms were developed for estimating the state, parameter, and order of an HMP, for universal coding and classification of HMPs, and for universal decoding of hidden Markov channels. These and other related topics are reviewed in this paper. Index Terms—Baum–Petrie algorithm, entropy ergodic theorems, finite-state channels, hidden Markov models, identifiability, Kalman filter, maximum-likelihood (ML) estimation, order estimation, recursive parameter estimation, switching autoregressive processes, Ziv inequality. I.

This paper theoretically explores the characteristics underpinning quadratic term structure models (QTSMs), which designate the yield on a bond as a quadratic function of underlying state variables. We develop a comprehensive QTSM, which is maximally exible and thus encompasses the features of several diverse models including the double square-root model of Longsta (1989), the univariate quadratic model of Beaglehole and Tenney (1992), and the Squared-Autoregressive-Independent-Variable Nominal Term Structure (SAINTS) model of Constantinides (1992). We document a complete classication of admissibility and empirical identication for the QTSM, and demonstrate that the QTSM can overcome limitations inherent in ane term structure models (ATSMs). Using the Ecient Method of Moments of Gallant and Tauchen (1996), we test the empirical performance of the model in determining bond prices and compare the performance to the ATSMs. The results of the goodness-of-t tests suggest that the QTSMs...

rapolation techniques, especially exponential smoothing and exponentially weighted moving average methods ([20, 71]). Developments of smoothing and discounting techniques in stock control and production planning areas led to formalisms in terms of linear, state-space models for time series with time-varying trends and seasonal patterns, and eventually to the associated Bayesian formalism of methods of inference and prediction. From the early 1960s, practical Bayesian forecasting systems in this context involved the combination of formal time series models and historical data analysis together with methods for subjective intervention and forecast monitoring, so that complete forecasting systems, rather than just routine and automatic data analysis and extrapolation, were in use at that time ([19, 22]). Methods developed in those early days are still in use now in some companies in sales forecasting and stock control areas. There have been major developments in models and methods since t

In this paper we present Monte Carlo methods for multi-target tracking and data association. The methods are applicable to general non-linear and non-Gaussian models for the target dynamics and measurement likelihood. We provide efficient solutions to two very pertinent problems: the data association problem that arises due to unlabelled measurements in the presence of clutter, and the curse of dimensionality that arises due to the increased size of the state-space associated with multiple targets. We develop a number of algorithms to achieve this. The first, which we will refer to as the Monte Carlo Joint Probabilistic Data Association Filter (MC-JPDAF), is a generalisation of the strategy proposed in [1], [2]. As is the case for the JPDAF, the distributions of interest are the marginal filtering distributions for each of the targets, but these are approximated with particles rather than Gaussians. We also develop two extensions to the standard particle filtering methodology for tracking multiple targets. The first, which we will refer to as the Sequential Sampling Particle Filter (SSPF), samples the individual targets sequentially by utilising a factorisation of the importance weights. The second, which we will refer to as the Independent Partition Particle Filter (IPPF), assumes the associations to be independent over the individual targets, leading to an efficient componentwise sampling strategy to construct new particles. We evaluate and compare the proposed methods on a challenging synthetic tracking problem.