Application of Singular Value Decomposition to DNA Microarray

Sclove and Professor Jan Verschelde, for kindly agreeing to be on my committee despite their busy schedules. I would like to thank Kari Dueball and Darlette Willis for their assistance during my stay at UIC, the UIC mathematics department and the Institute of Mathematics and its Applications (IMA) for their generous support and fellowship, Ali Shaker and Marcus Bishop for kindly helping me with La-Summary In chapter 1 we introduce the singular value decomposition (SVD) of matrices and its extensions. We then mention some applications of SVD in analyzing gene expression data, image processing and information retrieval. We also introduce the low rank approximation of matrices and present our Monte Carlo algorithm to achieve this approximation along with other algorithms. Chapter 2 deals with clustering methods and their application in analysing gene expression data. We introduce the most common clustering methods such as the KNN, SVD, and weighted least square methods. Chapter 3 deals with imputing missing data in gene expression of micro-arrays. We use some of the methods mentioned in part 2 for these purposes. Then we introduce our fixed rank approximation algorithm (FRAA) for imputing missing data in the DNA gene expression array. Finally, we use simulation to compare FRAA versus other methods and indicate the advantages and its shortcomings, and how to overcome the shortcomings of FRAA.