Optimal Filter Approximations in Conditionally Gaussian Pairwise Markov Switching Models

Abstract

Abstract-We consider a general triplet Markov Gaussian linear system (X, R, Y), where X is an hidden continuous random sequence, R is an hidden discrete Markov chain, Y is an observed continuous random sequence. When the triplet (X, R, Y) is a classical "Conditionally Gaussian Linear State-Space Model" (CGLSSM), the mean square error optimal filter is not workable with a reasonable complexity and different approximate methods, e.g. based on particle filters, are used. We propose two contributions. The first one is to extend the CGLSSM to a new, more general model, called the "Conditionally Gaussian Pairwise Markov Switching Model" (CGPMSM), in which X is not necessarily Markov given R. The second contribution is to consider a particular case of CGPMSM in which (R, Y) is Markov and in which an exact filter, optimal in the sense of mean square error, can be performed with linear-time complexity. Some experiments show that the proposed method and the suited particle filter have comparable efficiency, while the second one is much faster.Index Terms-Conditionally Gaussian linear state-space model, conditionally Gaussian pairwise markov switching model, exact optimal filtering, Gaussian switching system, hidden Markov models.

Evolutionary Computing and Particle Filtering: A Hardware-Based Motion Estimation System

“…Examples of particle filters implemented using these more general models as mathematical foundation can be found in the literature [23]. However, and since the objective of this work is to show the benefits of both evolutionary computing and hardware acceleration in particle filtering, the classical HMMs approach is sufficient.…”

“…At the end of each estimation step, the current state is obtained from the whole particle population using the expression (23 (23) Xk is the estimated state at time k, x x k are the state variables of the particle i at time k, and w l k are the normalized weights for each particle, computed as in (24).…”

Parameter estimation in conditionally Gaussian pairwise Markov switching models and unsupervised smoothing

“…The distribution of is defined by , Gaussian distributions transitions , and the system (1), in which represents the noises independent of and Thus, system parameters , depend on the switches , and denotes the item about the mean values, where and are the means of the two processes respectively, which are only decided by the value of and without dependence on . Such a system, named "conditionally Gaussian pairwise Markov switching model" CGPMSM [1], extends the wellknown "conditionally Gaussian linear state-space model" (CGLSSM) [2][3]. The latter in which and in (1) are set to be zero, is considered as the "natural" switching Gaussian system, however, its application is quite limited as it does not allow fast optimal filters [4][5].…”

“…Such a system, named "conditionally Gaussian pairwise Markov switching model" CGPMSM [1], extends the wellknown "conditionally Gaussian linear state-space model" (CGLSSM) [2][3]. The latter in which and in (1) are set to be zero, is considered as the "natural" switching Gaussian system, however, its application is quite limited as it does not allow fast optimal filters [4][5]. Another recent particular case of CGPMSM, called "conditionally Gaussian observed Markov switching model" (CGOMSM), consists in taking CGPMSM with .…”

“…This paper proposes a novel algorithm to automatically estimate the parameters and the parameters defining the distribution of the switches of CGPMSM described above from a limited-size set of observations only, with assumed to be known as zero. The interest of such a general algorithm relies on the fact that it is theoretically possible to approximate any non-linear and non-Gaussian system by the linear switching system (1).…”