Exact Solution of Bayes and Minimax Change-Detection Problems

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.

A study of computational methods to analyze gene expression data

The recent advent of new technologies has led to huge amounts of genomic data. With these data come new opportunities to understand biological cellular processes underlying hidden regulation mechanisms and to identify disease related biomarkers for informative diagnostics. However, extracting biological insights from the immense amounts of genomic data is a challenging task. Therefore, effective and efficient computational techniques are needed to analyze and interpret genomic data. In this thesis, novel computational methods are proposed to address such challenges: a Bayesian mixture model, an extended Bayesian mixture model, and an Eigen-brain approach. The Bayesian mixture framework involves integration of the Bayesian network and the Gaussian mixture model. Based on the proposed framework and its conjunction with K-means clustering and principal component analysis (PCA), biological insights are derived such as context specific/dependent relationships and nested structures within microarray where biological replicates are encapsulated. The Bayesian mixture framework is then extended to explore posterior distributions of network space by incorporating a Markov chain Monte Carlo (MCMC) model. The extended Bayesian mixture model summarizes the sampled network structures by extracting biologically meaningful features. Finally, an Eigen-brain approach is proposed to analyze in situ hybridization data for the identification of the cell-type specific genes, which can be useful for informative blood diagnostics. Computational results with region-based clustering reveals the critical evidence for the consistency with brain anatomical structure.