Regularization Methods for Canonical Correlation

Regularization Methods for Canonical Correlation Analysis, Rank Correlation Matrices and Renyi Correlation Matrices by Ying Xu Doctor of Philosophy in Statistics University of California, Berkeley Professor Peter J. Bickel, Chair In multivariate analysis, canonical correlation analysis is a method that enable us to gain insight into the relationships between the two sets of variables. It determines linear combinations of variables of each type with maximal correlation between the two linear combinations. However, in high dimensional data analysis, insufficient sample size may lead to computational problems, inconsistent estimates of parameters. In Chapter 1, three new methods of regularization are presented to improve the traditional CCA estimator in high dimensional settings. Theoretical results have been derived and the methods are evaluated using simulated data. While the linear methods are successful in many circumstances, it certainly has some limitations, especially in cases where strong nonlinear dependencies exist. In Chapter 2, I investigate some other measures of dependence, including the rank correlation and its extensions, which can capture some non-linear relationship between variables. Finally the Renyi correlation is considered in Chapter 3. I also complement my analysis with simulations that demonstrate the theoretical results.