Transposable Regularized Covariance Models with Applications to High-dimensional Data a Dissertation Submitted to the Department of Statistics and the Committee on Graduate Studies of Stanford University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

High-dimensional data is becoming more prevalent with new technologies in biomedical sciences, imaging and the Internet. Many examples of this data often contain complex relationships between and among sets of variables. When arranged in the form of a matrix, this data is transposable, meaning that either the rows, columns or both can be treated as features. To model transposable data, we present a modification of the matrix-variate normal, the mean-restricted matrix-variate normal, and introduce Transposable Regularized Covariance Models by placing penalties on inverse covariance matrices. We give theoretical results exploiting the structure of our transposable models that give computationally feasible algorithms for parameter estimation and calculation of conditional expectations. These contributions make the matrix-variate normal accessible for application to high-dimensional data. We apply our model to two applications: missing data imputation and large-scale inference with the matrix-variate normal distribution. Examples, simulations and results are given using the Netflix movie-rating data and microarrays, demonstrating the flexibility and functionality of our transposable models.