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Exact Solution of Bayes and Minimax Change-Detection Problems

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Title: Exact Solution of Bayes and Minimax Change-Detection Problems
Author(s): Isom, Joshua D.
Doctoral Committee Chair(s): Braatz, Richard D.
Doctoral Committee Member(s): Meyn, Sean P.; LaBarre, Robert E.; Rao, Christopher V.
Department / Program: Chemical Engineering
Discipline: Chemical Engineering
Degree Granting Institution: University of Illinios at Urbana-Champaign
Degree: Ph.D.
Genre: Dissertation
Subject(s): change detection stochastic optimal control formal languages 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.
Issue Date: 2009-04-20
Citation Info: Dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Chemical Engineering in the Graduate College of the University of Illinois at Urbana-Champaign, 2009
Genre: Dissertation / Thesis
Type: Text
Publication Status: unpublished
Date Available in IDEALS: 2009-04-20

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