Lumpable hidden Markov models-model reduction and reduced complexity filtering

Abstract

Abstract-This paper is concerned with filtering of hidden Markov processes (HMPs) which possess (or approximately possess) the property of lumpability. This property is a generalization of the property of lumpability of a Markov chain which has been previously addressed by others. In essence, the property of lumpability means that there is a partition of the (atomic) states of the Markov chain into aggregated sets which act in a similar manner as far as the state dynamics and observation statistics are concerned. We prove necessary and sufficient conditions on the HMP for exact lumpability to hold. For a particular class of hidden Markov models (HMMs), namely finite output alphabet models, conditions for lumpability of all HMPs representable by a specified HMM are given. The corresponding optimal filter algorithms for the aggregated states are then derived.The paper also describes an approach to efficient suboptimal filtering for HMPs which are approximately lumpable. By this we mean that the HMM generating the process may be approximated by a lumpable HMM. This approach involves directly finding a lumped HMM which approximates the original HMM well, in a matrix norm sense. An alternative approach for model reduction based on approximating a given HMM by an exactly lumpable HMM is also derived. This method is based on the alternating convex projections algorithm. Some simulation examples are presented which illustrate the performance of the suboptimal filtering algorithms.

On the Relationships Between Lumpability and Filtering of Finite Stochastic Systems

“…Conditions (d) and (f) were proven in [21] for n = 2. Condition (d) for any n ≥ 2 was derived in [10] from a very general criterion for a function of a Markov chain to be a Markov chain.…”

“…Indeed, if any of the four conditions in Theorem 3.1 is satisfied then {(Y t , g(X t ))} t∈N is an FSS with characteristic matrices ( D y , y ∈ Y ) whatever the probability distribution of (Y 0 , X 0 ). Thus the unnormalised filter satisfies a linear equation of the type (9), that is equation (21). A direct calculation may give some insight into this fact.…”

“…The basic idea was to interpret the process {Y t } t∈N as the function f (Y t , X t ) = Y t of the Markov chain {(Y t , X t )} t∈N and to use criteria for lumpability of Markov chains. A similar idea, though not explicitly stated, was used in [21] to discuss the problem that we are interested in. Indeed, we are concerned here by the function f (Y t , X t ) = (Y t , g(X t )) of the Markov chain {(Y t , X t )} t∈N where g is some lumping map for the Markov chain {X t } t∈N .…”

“…In [21] a procedure was derived that yielded a 2-lumpable approximation to a given MC, that is an approximation of a given MC by a strongly lumpable MC with two lumps (n = 2). By extension, 2-lumpable approximations to HMCs were obtained.…”

“…A direct way is to look for conditions under which {(Y t , g(X t ))} t∈N is an FSS, so that the filter { π t } t∈N and the unnormalised filter { ρ t } t∈N ) satisfy a recursive equation as in (6) and, respectively, in (9). A related problem was discussed in [21] for HMCs. In [21], a concept of strong lumpability for HMCs was defined.…”