Linköping University Postprint A Basic Convergence Result for Particle Filtering Xiao-Li Hu, Thomas B. Schön and Lennart Ljung N.B.: When citing this work, cite the original article. Original publication: Xiao-Li Hu, Thomas B. Schön and Lennart Ljung, A Basic Convergence Result for Particle Filtering, 2008, IEEE Transactions on Signal Processing, (56), 4, 1337-1348. http://dx.doi.org/10.1109/TSP.2007.911295. Copyright: IEEE, http://www.ieee.org Postprint available free at: Linköping University E-Press: http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-11748 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 4, APRIL 2008 1337 A Basic Convergence Result for Particle Filtering Xiao-Li Hu, Thomas B. Schön, Member, IEEE, and Lennart Ljung, Fellow, IEEE Abstract—The basic nonlinear filtering problem for dynamical where and is the func- systems is considered. Approximating the optimal filter estimate tion of the state that we want to estimate. We are inter- by particle filter methods has become perhaps the most common ested in estimating a function of the state, such as and useful method in recent years. Many variants of particle filters have been suggested, and there is an extensive literature on the the- from observed output data . An especially common oretical aspects of the quality of the approximation. Still a clear-cut case is of course when we seek an estimate of the state itself result that the approximate solution, for unbounded functions, con- , where . verges to the true optimal estimate as the number of particles tends In order to compute (3) we need the filtering probability density to infinity seems to be lacking. It is the purpose of this contribution function . It is well known that this density function to give such a basic convergence result for a rather general class of unbounded functions. Furthermore, a general framework, in- can be expressed using multidimensional integrals [1]. The cluding many of the particle filter algorithms as special cases, is problem is that these integrals only permits analytical solu- given. tions in a few special cases. The most common special case Index Terms—Convergence of numerical methods, nonlinear es- is of course when the model (2) is linear and Gaussian and timation, particle filter, state estimation. the solution is then given by the Kalman filter [2]. However, for the more interesting nonlinear/non-Gaussian case we are forced to approximations of some kind. Over the years there I. INTRODUCTION has been a large amount of ideas suggested on how to perform these approximations. The most popular being the extended HE nonlinear filtering problem is formulated as follows. Kalman filter (EKF) [3], [4]. Other popular ideas include TThe objective is to recursively in time estimate the state in the Gaussian-sum approximations [5], the point-mass filters the dynamic model, [6], [7], the unscented Kalman filter (UKF) [8] and the class of multiple model estimators [9]. See, e.g., [10] for a brief (1a) overview of the various approximations. In the current work (1b) we will discuss a rather recent and popular family of methods, commonly referred to as particle filters (PFs) or sequential where denotes the state, denotes the Monte Carlo methods. measurement, and denote the stochastic process and mea- The key idea underlying the particle filter is to approxi- surement noise, respectively. Furthermore, the dynamic equa- mate the filtering density function using a number of particles tions for the system are denoted by according to and the equations modelling the sensors are denoted by . Most applied signal processing problems can be written in the following special case of (1): (4) (2a) (2b) where each particle has a weight associated to it, and denotes the delta-Dirac mass located in . Due to the delta-Dirac with and independent and identically distributed (i.i.d.) and form in (4), a finite sum is obtained when this approximation is mutually independent. Note that any deterministic input signal passed through an integral and hence, multidimensional inte- is subsumed in the time-varying dynamics. The most com- grals are reduced to finite sums. All the details of the particle monly used estimate is an approximation of the conditional ex- filter were first assembled by Gordon et al. in 1993 in their sem- pectation inal paper [11]. However, the main ideas, save for the crucial (3) resampling step, have been around since the 1940s [12]. Whenever an approximation is used it is very important to ad- dress the issue of its convergence to the true solution and more Manuscript received March 16, 2007; revised September 11, 2007. The as- sociate editor coordinating the review of this manuscript and approving it for specifically, under what conditions this convergence is valid. publication was Dr. Subhrakanti Dey. This work was supported by the strategic An extensive treatment of the currently existing convergence research center MOVIII, funded by the Swedish Foundation for Strategic Re- results can be found in the book [13] and the excellent survey search, SSF. X.-L. Hu is with the Department of Mathematics, College of Science, China papers [14], [15]. They consider stability, uniform convergence Jiliang University, 310018 Hangzhou China (e-mail: [email protected]). (see also [16] and [17]), central limit theorems (see also [18]) T. B. Schön and L. Ljung are with the Division of Automatic Control, Depart- and large deviations (see also [19] and [20]). The previous re- ment of Electrical Engineering, Linköping University, SE–581 83 Linköping, Sweden (e-mail: [email protected]; [email protected]). sults prove convergence of probability measures and only treat Digital Object Identifier 10.1109/TSP.2007.911295 bounded functions , effectively excluding the most commonly 1053-587X/$25.00 © 2008 IEEE 1338 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 4, APRIL 2008 used state estimate, the mean value. To the best of our knowl- The state process is a Markov process with initial state edge there are no results available for unbounded functions . obeying an initial distribution . The dynamics, de- The main contribution of this paper is that we prove convergence scribing the state evolution over time, is modelled by a Markov of the particle filter for a rather general class of unbounded func- transition kernel such that tions, applicable in many practical situations. This contribution will also describe a general framework for particle filtering al- (6) gorithms. It is worth stressing the key mechanisms that enables us to for all , where denotes the Borel -algebra study unbounded functions in the particle filtering context. on . Given the states , the observations are conditionally 1) The most important idea, enabling the contribution in the independent and have the following marginal distribution present paper, is that we consider the relation between the function and the density functions for noises. This im- plies that the class of functions will depend on the in- (7) volved noise densities. 2) We have also introduced a slight algorithm modification, For convenience we assume that and required to complete the proof. It is worth mentioning that have densities with respect to a Lebesgue measure, allowing us this modification is motivated from the mathematics in the to write proof. However, it is a useful and reasonable modification of the algorithm in its own right. Indeed, it has previously been used to obtain a more efficient algorithm [21]. (8a) In Section II we provide a formal problem formulation and introduce the notation we need for the results to follow. A brief (8b) introduction to particle filters is given in Section III. In an at- tempt to make the results as available as possible the particle In the following example it is explained how a model in the form filter is discussed both in an application oriented fashion and in (2) relates to the more general framework introduced above. a more general setting. The algorithm modification is discussed 1) Example 2.1: Let the model be given by (2), where the and illustrated in Section IV. Section V provides a general ac- probability density functions of and are denoted by count of convergence results and in Section VI we state the main and , respectively. Then we have the following relations: result and discuss the conditions that are required for the result to hold. The result is then proved in Section VII. Finally, the (9a) conclusions are given in Section VIII. (9b) II. PROBLEM FORMULATION B. Conceptual Solution In practice, we are most interested in the marginal distribu- The problem under consideration in this work is the fol- tion , since the main objective is usually to estimate lowing. For a fixed time , under what conditions and for which and the corresponding conditional covariance. This functions does the approximation offered by the particle filter section is devoted to describing the generally intractable form converge to the true estimate of . By the total probability formula and Bayes’ for- mula, we have the following recursive form for the evolution of (5) the marginal distribution: In order to give the results in the most simple form possible we are only concerned with -convergence in this paper. The more general case of -convergence for is also under (10a) consideration, using a Rosenthal-type inequality [22]. A. Dynamic Systems (10b) We will now represent model (1) in a slightly different frame- where we have defined and as transformations between work, more suitable for a theoretical treatment. Let ( ) probability measures on . be a probability space on which two real vector-valued Let us now introduce some additional notation, commonly stochastic processes and used in this context. Given a measure , a function , and a are defined. The -dimensional Markov transition kernel , denote process describes the evolution of the hidden state of a dynamic system, and the -dimensional process denotes (11) the available observation process of the same system.