Exact Solution of Bayes and Minimax Change-Detection Problems

Isom, Joshua D.

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https://hdl.handle.net/2142/11062 Copy

Description

Title

Exact Solution of Bayes and Minimax Change-Detection Problems

Author(s)

Isom, Joshua D.

Issue Date

2009-04-20

Doctoral Committee Chair(s)

Braatz, Richard D.

Committee Member(s)

• Meyn, Sean P.
• LaBarre, Robert E.
• Rao, Christopher V.

Department of Study

Chemical Engineering

Discipline

Chemical Engineering

Degree Granting Institution

University of Illinios at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• change detection
• Stochastic optimal control
• Formal languages
• Cumulative sum (CUSUM)
• Bayes
• Minimax
• partially observable Markov decision process

Abstract

The challenge of detecting a change in the distribution of data is a sequential decision problem that is relevant to many engineering solutions, including quality control and machine and process monitoring. This dissertation develops techniques for exact solution of change-detection problems with discrete time and discrete observations. Change-detection problems are classified as Bayes or minimax based on the availability of information on the change-time distribution. A Bayes optimal solution uses prior information about the distribution of the change time to minimize the expected cost, whereas a minimax optimal solution minimizes the cost under the worst-case change-time distribution. Both types of problems are addressed. The most important result of the dissertation is the development of a polynomial-time algorithm for the solution of important classes of Markov Bayes change-detection problems. Existing techniques for epsilon-exact solution of partially observable Markov decision processes have complexity exponential in the number of observation symbols. A new algorithm, called constellation induction, exploits the concavity and Lipschitz continuity of the value function, and has complexity polynomial in the number of observation symbols. It is shown that change-detection problems with a geometric change-time distribution and identically- and independently-distributed observations before and after the change are solvable in polynomial time. Also, change-detection problems on hidden Markov models with a fixed number of recurrent states are solvable in polynomial time. A detailed implementation and analysis of the constellation-induction algorithm are provided. Exact solution methods are also established for several types of minimax change-detection problems. Finite-horizon problems with arbitrary observation distributions are modeled as extensive-form games and solved using linear programs. Infinite-horizon problems with linear penalty for detection delay and identically- and independently-distributed observations can be solved in polynomial time via epsilon-optimal parameterization of a cumulative-sum procedure. Finally, the properties of policies for change-detection problems are described and analyzed. Simple classes of formal languages are shown to be sufficient for epsilon-exact solution of change-detection problems, and methods for finding minimally sized policy representations are described.

Type of Resource

text

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