NDLI: The Realization Problem for Hidden Markov Models: The Complete Realization Problem

Content Provider	IEEE Xplore Digital Library
Author	Vidyasagar, M.
Copyright Year	2005
Description	Author affiliation: Tata Consultancy Service, No. 1 Software Units Layout, Madhapur, Hyderabad 500 081, India, sagar@atc.tc.co.n (Vidyasagar, M.)
Abstract	Suppose m is a positive integer, and let M:= {1,...,m}. Suppose {Yt} is a stationary stochastic process assuming values in M. IN this paper we study the question: When does there exist a hidden Markov model (HMM) that perfectly reproduces the complete statistics of this process? Though HMM's are more than forty years old, no complete solution to this problem is available. It is known that a necessary condition for the process to have a HMM is that an assoicated `Hankel' matrix should have finite rank. It is also known that the condition is not sufficient in general. In subsequent work, an alogrithm for constructing a HMM for a finite rank process has been given, assuming at the outset that the process has a HMM. Hence, to date there are no conditions, either necessary or sufficient, for a process to have a HMM that can be stated in terms of the process alone, and nothing else. Against this background, in the present paper we show the following: (i) Suppose a process has finite Hankel rank. Then there always exists a `regular quasi-realization' of the process. Moreover, two regular quasi-realizations are related through a similarity transformation. (ii) If in addition the process is á- mixing, every regular quasi-realization has additional features. Specifically, the `state transition' matrix associated with the quasi-realization satisfies the `quasi-strong Perron property' (its spectral radius is one, the spectral radius is a simple eigenvalue, and there are no other eigenvalues on the unit circle).
Starting Page	6632
Ending Page	6637
File Size	156230
Page Count	6
File Format	PDF
ISBN	0780395670
DOI	10.1109/CDC.2005.1583227
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2005-12-15
Publisher Place	Spain
Access Restriction	Subscribed
Rights Holder	IEEE/EUCA
Subject Keyword	Hidden Markov models Statistics Application software Stochastic processes Eigenvalues and eigenfunctions Markov processes Speech processing Source coding Computational biology Organisms
Content Type	Text
Resource Type	Article

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4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
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