IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 50, NO. 2, FEBRUARY 2002 411 Importance Sampling for Error Event Analysis of HMM Frequency Line Trackers M. Sanjeev Arulampalam, Rob J. Evans, Member, IEEE, and Khaled Ben Letaief, Senior Member, IEEE Abstract—This paper considers the problem of designing effi- cient and systematic importance sampling (IS) schemes for the per- formance study of hidden Markov model (HMM) based trackers. Importance sampling (IS) is a powerful Monte Carlo (MC) vari- ance reduction technique, which can require orders of magnitude fewer simulation trials than ordinary MC to obtain the same speci- fied precision. In this paper, we present an IS technique applicable to error event analysis of HMM based trackers. Specifically, we use conditional IS to extend our work in another of our papers to estimate average error event probabilities. In addition, we derive upper bounds on these error probabilities, which are then used to verify the simulations. The power and accuracy of the proposed method is illustrated by application to an HMM frequency tracker. Index Terms—Error events, HMM, importance sampling, Monte Carlo. I. INTRODUCTION F OR many complex tracking and communication systems, detailed system performance cannot be evaluated by pure analysis. For this reason, one resorts to the Monte Carlo (MC) simulation method. The popularity of MC is mainly due to its generality along with its ability to include system characteristics that may not be amenable to analytically based methods. Un- fortunately, for many practical systems, this technique requires a very large number of simulation trials in order to achieve ac- curate estimates. For example, it is easily shown that the MC estimation of an error probability with 10% precision re- quires more than independent simulation runs [2]. Thus, if is of order 10 , we need more than 10 independent trials. This is clearly too large a number to be practical, even for the most powerful computers. Hence, it is desirable to develop efficient simulation techniques that retain the ability to simu- late complex systems, yet require substantially less computation time than standard MC. One such technique is importance sam- pling (IS) [1]–[12]. The basic principle of IS is as follows. The input probability density 1 from which the random inputs for the simulation are Manuscript received February 5, 2001; revised October 16, 2001. The work of M. S. Arulampalam was supported by the Royal Academy of Engineering, London, U.K., with an Anglo–Australian post-doctoral research fellowship. The associate editor coordinating the review of this paper and approving it for pub- lication was Prof Simon J. Godsill. M. S. Arulampalam is with the Defence Science Technology Organization, Adelaide, Australia. R. J. Evans is with the University of Melbourne, Parkville, Australia. K. B. Letaief is with the Hong Kong University of Science and Technology, Hong Kong. Publisher Item Identifier S 1053-587X(02)00555-X. 1 We use “density” in the generic sense to mean either probability density func- tion for continuous variables or probability mass function for discrete variables. generated is biased so that the variance of the MC estimator is reduced. Consequently, an accurate IS estimate can be obtained with fewer simulations than required by ordinary MC. An un- biased estimate is obtained by weighting the simulation data by the a posteriori likelihood ratio, that is, the ratio of the true un- derlying input density to the simulation density. In this paper, we investigate the application of IS to the performance study of hidden Markov model (HMM) based frequency line trackers. An HMM consists of an underlying Markov chain whose states are observed indirectly through a series of noise-corrupted measurements [17], [18]. In recent years, HMM-based tracking algorithms have become popular, particularly in frequency line tracking [13], [15]. However, due to difficulty of analysis, currently, only limited theoretical results, based on asymptotic analyses, exist on the performance of these algorithms [22]–[24]. As a result, their performance has been mainly studied via simulations [13], [15], [19]. In this paper, we apply the principles of IS to a simulation-based performance study technique known as error event simulation. An error event refers to a brief divergence of the estimated track from the true path, and error event simulation involves es- timation of the probabilities of such errors using MC methods. This has been a useful technique in the performance study of many communication and HMM tracking systems [1], [9]. However, standard MC estimation requires a prohibitive amount of computation time, particularly at large SNR, where the error probabilities are very small. This necessitates the design of efficient estimation schemes. The development of such schemes for the study of HMM trackers is precisely the goal of this work. In [1], the authors developed an efficient IS scheme for esti- mating the probability of any specific error event, given a spe- cific true path. In this paper, this is extended to estimating av- erage error event probabilities. Specifically, the true and error paths are considered random, and conditional IS is employed to develop a systematic IS approach for this problem. In addition, upper bounds on these error probabilities are derived, which are then used to verify the simulations. Moreover, the bounds prove to be useful in the design of the conditional IS estimator. The utility of our IS approach is demonstrated by application to a HMM frequency tracker. The organization of the paper is as follows. Section II reviews the basics of HMM theory and presents an HMM frequency line tracker. In Section III, we discuss error events and derive an upper bound on the probability of such errors. Section IV reviews IS theory and presents a systematic approach to esti- mating the average error event probabilities in HMM tracking. This is applied to a specific HMM frequency tracker in Sec- 1053–587X/02$17.00 © 2002 IEEE