Particle Filtering for Large-Dimensional State Spaces With Multimodal Observation Likelihoods

Abstract

Abstract-We study efficient importance sampling techniques for particle filtering (PF) when either (a) the observation likelihood (OL) is frequently multimodal or heavy-tailed, or (b) the state space dimension is large or both. When the OL is multimodal, but the state transition pdf (STP) is narrow enough, the optimal importance density is usually unimodal. Under this assumption, many techniques have been proposed. But when the STP is broad, this assumption does not hold. We study how existing techniques can be generalized to situations where the optimal importance density is multimodal, but is unimodal conditioned on a part of the state vector.Sufficient conditions to test for the unimodality of this conditional posterior are derived. Our result is directly extendable to testing for unimodality of any posterior.The number of particles N to accurately track using a PF increases with state space dimension, thus making any regular PF impractical for large dimensional tracking problems. But in most such problems, most of the state change occurs in only a few dimensions, while the change in the rest of the dimensions is small. Using this property, we propose to replace importance sampling from a large part of the state space (whose conditional posterior is narrow enough) by just tracking the mode of the conditional posterior. This introduces some extra error, but it also greatly reduces the importance sampling dimension. The net effect is much smaller error for a given N, especially when the available N is small. An important class of large dimensional problems with multimodal OL is tracking spatially varying physical quantities such as temperature or pressure in a large area using a network of sensors which may be nonlinear and/or may have nonnegligible failure probabilities. Improved performance of our proposed algorithms over existing PFs is demonstrated for this problem.Index Terms-Importance sampling for multimodal posteriors, large dimensional sequential state estimation, particle filtering, posterior mode tracking, tracking spatially varying physical quantities.

Tracking sparse signal sequences from nonlinear/non-Gaussian measurements and applications in illumination-motion tracking

“…The state transition prior corresponding to the above state models can be written as: [23,22] and PF-MT [24]. PF-MT [24] splits the state vector Xt into Xt = [X t,s , X t,r ] where Xt,s denotes the coefficients of a small dimensional state vector, which can change significantly over time, while Xt,r refers to the rest of the states (large dimensional) which usually change much more slowly over time.…”

“…PF-MT [24] splits the state vector Xt into Xt = [X t,s , X t,r ] where Xt,s denotes the coefficients of a small dimensional state vector, which can change significantly over time, while Xt,r refers to the rest of the states (large dimensional) which usually change much more slowly over time. PF-MT importance samples only on Xt,s, while replacing importance sampling by deterministic posterior Mode Tracking (MT) for Xt,r and thus significantly reducing the importance sampling dimension.…”

“…Rao-Blackwellized PF (RB-PF) [22,23], PF with posterior mode tracking (PF-MT) algorithm [24,25,26] and later works such as [27,28,29,30] are possible solutions for large dimensional tracking problems. RB-PF requires that conditioned on the small dimensional state vector, the state space model be linear and Gaussian.…”

Improved Minimum Entropy Filtering for Continuous Nonlinear Non-Gaussian Systems Using a Generalized Density Evolution Equation

“…Although the MEE criteria can minimize both the probabilistic uncertainty and dispersion of the estimation error, it cannot guarantee the minimum error. In this work , an improved performance index J is considered: (13) where 2 …”

“…Although the filtering problem can be tackled using numerical integration [8,9], it is difficult to implement when the dimension of the state vector is higher. Sequential Monte Carlo simulation [10][11][12][13][14][15], which is also named particle filtering strategy, has shown its great advantages to deal with filtering problems for nonlinear non-Gaussian systems. Nevertheless, there are still many issues to be solved, for example, (1) random sampling may bring the accumulation of Monte Carlo error and even lead to filter divergence; (2) a large number of particles are needed to avoid degradation and to improve the estimation accuracy, which makes the calculation a sharp increase.…”