Gaussian proposal density using moment matching in SMC methods

Abstract

In this article we introduce a new Gaussian proposal distribution to be used in conjunction with the sequential Monte Carlo (SMC) method for solving non-linear filtering problems. The proposal, in line with the recent trend, incorporates the current observation. The introduced proposal is characterized by the exact moments obtained from the dynamical system. This is in contrast with recent works where the moments are approximated either numerically or by linearizing the observation model. We show further that the newly introduced proposal performs better than other similar proposal functions which also incorporate both state and observations.

A semi-exact sequential Monte Carlo filtering algorithm in Hidden Markov Chains

“…However, techniques such as local linearizations [6], Taylor series expansion [7], or UT [8] have been proposed for approximating p(y n |x n−1 ) and p(x n |x n−1 , y n ). More precisely, starting from an SMC approximation {w…”

“…Derive an SMC approximation of p n|n using a classical SMC algorithm, such as an SIR or APF algorithm [1]. Previous approximations can possibly be used to derive some sampling importance distributions [7] [9]. Other SMC algorithms why optimize a given criterion are described in [10] [11].…”

A particle smoothing implementation of the fully-adapted auxiliary particle filter: An alternative to auxiliary particle filters

“…Let us thus briefly remind some approximation techniques ofp(y n |x n−1 ) for the APF andp(x n |x n−1 , y n ) for both algorithms. Roughly speaking, most of techniques consist in first approximating locally p(x n , y n |x i n−1 ) by a Gaussian pdf, the moments of which are approximated bu using a n-th degree Taylor polynomial [6] [7] or the Unscented Transformation for particle filters [8] applied to the state and/or observation equation(s), then in deducing a local approximation of p(y n |x i n−1 ) and p(x n |x i n−1 , y n ). The approximation of p(y n |x i n−1 ) byp(y n |x i n−1 ) can be used as a choice of the first stage weights τ i n in the APF.…”

“…J is averaged over P = 1000 realizations, and T = 50 time indices. Exact first and second moments of distributions concerned are calculable [7], so when it is necessary, same approximations are used for the PS-APF and the APF and we note for simplicityp EMM (.) =p(.).…”

Topics in particle filteriing and smoothing

“…The performances of different importance functions are also compared in terms of Kullback-Leibler divergence. This chapter is based on (Saha et al (2009b(Saha et al ( , 2007(Saha et al ( , 2006; Aihara et al (2008)). …”