State Estimation Schemes for Independent Component Coupled Hidden Markov Models
William P. Malcolm, Novi Quadrianto, Lakhdar Aggoun · Stochastic Analysis and Applications · 2010
Conventional Hidden Markov models generally consist of a Markov chain observed through a linear map corrupted by additive noise. This general class of model has enjoyed a huge and diverse range of applications, for example, speech processing, biomedical signal processing and more recently quantitative finance. However, a lesser known extension of this general class of model is the so-called Factorial Hidden Markov Model (FHMM). FHMMs also have diverse applications, notably in machine learning, artificial intelligence and speech recognition [13 Ghahramani , Z. , and Jordan , M. 1997 . Factorial hidden Markov models . Machine Learning 29 : 245 – 273 .[Crossref], [Web of Science ®] , [Google Scholar], 17 Logan , B. , and Moreno , P.J. 1997 . Factorial Hidden Markov Models for Speech Recognition: Preliminary Experiments. Cambridge Research Laboratory, Technical Report CRL 97/7, September. [Google Scholar]]. FHMMs extend the usual class of HMMs, by supposing the partially observed state process is a finite collection of distinct Markov chains, either statistically independent or dependent. There is also considerable current activity in applying collections of partially observed Markov chains to complex action recognition problems, see, for example, [6 Brand , M. , Oliver , N. , and Pentland , A. 1997 . Coupled hidden Markov models for complex action recognition . IEEE Conference on Computer Vision and Pattern Recognition , San Juan , Puerto Rico .[Crossref] , [Google Scholar]]. In this article we consider the Maximum Likelihood (ML) parameter estimation problem for FHMMs. Much of the extant literature concerning this problem presents parameter estimation schemes based on full data log-likelihood EM algorithms. This approach can be slow to converge and often imposes heavy demands on computer memory. The latter point is particularly relevant for the class of FHMMs where state space dimensions are relatively large. The contribution in this article is to develop new recursive formulae for a filter-based EM algorithm that can be implemented online. Our new formulae are equivalent ML estimators, however, these formulae are purely recursive and so, significantly reduce numerical complexity and memory requirements. A computer simulation is included to demonstrate the performance of our results.