| The well-known "curse of dimensionality" makes high-dimensional data analysis unusually challenging. Dimension reduction plays a valuable role in enabling certain statistical analyses performed in a parsimonious way. The canonical correlation (CANCOR) method developed by Fung et al. (2002) is asymptotically equivalent to the sliced inverse regression (SIR) method, and reduces dimensionality by replacing the explanatory variables with a small number of composite directions without losing much information. However, the estimates by CANCOR are sensitive to outliers. In this dissertation, a weighted canonical correlation method (WCANCOR) is developed to robustify the CANCOR estimates. To simplify the composite directions estimated by CANCOR or WCANCOR, a constrained CANCOR or WCANCOR method is also proposed in this dissertation. By the constrained WCANCOR method, each composite direction consists of only a subset of the explanatory variables for easier interpretation. When the estimated covariance matrix of the explanatory variables is singular, which occurs frequently for certain types of high-dimensional data, the constrained WCANCOR method seeks among many equivalent directions a linear combination of explanatory variables with the smallest number of nonzero coefficients. |
