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Hidden Markov ProcessesTheory and Applications to Biology$
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M. Vidyasagar

Print publication date: 2014

Print ISBN-13: 9780691133157

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691133157.001.0001

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Hidden Markov Processes: The Complete Realization Problem

Hidden Markov Processes: The Complete Realization Problem

Chapter:
(p.177) Chapter Seven Hidden Markov Processes: The Complete Realization Problem
Source:
Hidden Markov Processes
Author(s):

M. Vidyasagar

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691133157.003.0007

This chapter considers hidden Markov processes (HMPs), focusing on the so-called complete realization problem. It is quite easy to prove a universal necessary condition for the given process to have a hidden Markov model (HMM). However, this condition is not sufficient in general. In principle, one can derive a “necessary and sufficient condition,” but the “necessary and sufficient condition” is virtually a restatement of the problem to be solved and does not shed any insight into the solution. The chapter first introduces a very useful matrix known as the “Hankel” matrix before discussing the nonsufficiency of the finite Hankel rank condition, an abstract necessary and sufficient condition, and the existence of regular quasi-realizations. It also describes the spectral properties of alpha-mixing processes and goes on to analyze ultra-mixing processes and a sufficient condition for the existence of HMMs.

Keywords:   hidden Markov processes, complete realization problem, hidden Markov model, Hankel matrix, Hankel rank condition, quasi-realization, alpha-mixing process, ultra-mixing process

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