Exact filtering and smoothing in short or long memory stochastic switching systems

Abstract

Abstract-Let X be a hidden real stochastic chain, R be a discrete finite Markov chain, Y be an observed stochastic chain. In this paper we address the problem of filtering and smoothing in the presence of stochastic switches where the problem is to recover both R and X from Y. In the classical conditionally Gaussian state space models, exact computing with polynomial complexity in the time index is not feasible and different approximations are used. Different alternative models, in which the exact calculations are feasible, have been recently proposed since 2008. The core difference between these models and the classical ones is that the couple (R, Y) is a Markov one in the recent models, while it is not in the classical ones. Another extension deals with the case in which the observed chain Y is not necessarily Markovian conditionally on (X, R) and, in particular, the long-memory distributions can be considered. The aim of this paper is to show that, in the context of these different recent models, it is possible to calculate any moments of the posterior marginal distribution, which makes it feasible to know these distributions with any desired precision.

New trends in Markov models and related learning to restore data

“…Thus different approximations, deterministic [22] or stochastic [3,16,64], have to be used. More recently, it has been shown that considering a TMC T = (X, U, Y ) in which (U, Y ) is Markovian makes both computations feasible with complexity linear in time ( [58] and references therein). Such models can be extended in numerous directions; in particular, copulas could be used in the PMC (U, Y ).…”

Filtrage statistique optimal rapide dans des systèmes linéaires à sauts non stationnaires

“…Plus récemment il a été proposé différents modèles « à sauts », dont certains très simples, dans lesquels le problème peut être résolu de manière exacte avec une complexité linéaire en n . Leur synthèse est présentée dans (Pieczynski et Abbassi, 2009 ;Abbassi, 2012). Ensuite, un modèle général, appelé « modèle caché conditionnellement linéaire à sauts markoviens » (MCCLSM), unifiant et étendant les différent modèles précédents, a été proposé dans (Pieczynski, 2011a).…”