NDLI: Quickest Change Detection and Kullback-Leibler Divergence for Two-State Hidden Markov Models

Content Provider	IEEE Xplore Digital Library
Author	Cheng-Der Fuh Yajun Mei
Copyright Year	1991
Abstract	In this paper, the quickest change detection problem is studied in two-state hidden Markov models (HMM), where the vector parameter θ of the HMM changes from θ₀ to θ₁ at some unknown time, and one wants to detect the true change as quickly as possible while controlling the false alarm rate. It turns out that the generalized likelihood ratio (GLR) scheme, while theoretically straightforward, is generally computationally infeasible for the HMM. To develop efficient but computationally simple schemes for the HMM, we first discuss a subtlety in the recursive form of the generalized likelihood ratio (GLR) scheme for the HMM. Then we show that the recursive CUSUM scheme proposed in Fuh (Ann. Statist., 2003) can be regarded as a quasi-GLR scheme for pseudo post-change hypotheses with certain dependence structure between pre- and postchange observations. Next, we extend the quasi-GLR idea to propose recursive score schemes in the scenario when the postchange parameter θ₁ of the HMM involves a real-valued nuisance parameter. Finally, the Kullback-Leibler (KL) divergence plays an essential role in the quickest change detection problem and many other fields, however it is rather challenging to numerically compute it in HMMs. Here we develop a non-Monte Carlo method that computes the KL divergence of two-state HMMs via the underlying invariant probability measure, which is characterized by the Fredholm integral equation. Numerical study demonstrates an unusual property of the KL divergence for HMM that implies the severe effects of misspecifying the postchange parameter for the HMM.
Sponsorship	IEEE Signal Processing Society
Starting Page	4866
Ending Page	4878
Page Count	13
File Size	3043043
File Format	PDF
ISSN	1053587X
Volume Number	63
Issue Number	18
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2015-01-01
Publisher Place	U.S.A.
Access Restriction	One Nation One Subscription (ONOS)
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Hidden Markov models Markov processes Joints Standards Density functional theory Integral equations sequential detection Change-point CUSUM hidden Markov model (HMM) Kullback-Leibler (KL) divergence score test
Content Type	Text
Resource Type	Article
Subject	Signal Processing Electrical and Electronic Engineering

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