Distributions and the Bootstrap Method of Some Statistics in Principal Canonical Correlation Analysis

Abstract

We consider two sets of variables with a joint distribution and analyze the canonical correlations between the variables in the two sets. One of the analyses used is the canonical correlation analysis, which finds linear combinations of variables in the sets that have the maximum correlation, and these linear combinations are the first coordinates in new systems. Then, a second linear combination in each set is obtained such that the linear combination is uncorrelated with the first linear combination. The procedure is continued until two new coordinate systems are specified completely. This theory was developed by Hotelling (1935, 1936). In this paper, we first determine the principal components of the two sets and then calculate the canonical correlation between the two principal components. Principal components analysis is a procedure used for analyzing multivariate data that transforms the original variables into new ones that are uncorrelated and account for decreasing proportions of the variance in the data. This analysis attempts to characterize or explain the variability in a vector variable by replacing it with a new variable with fewer components with large variance. We know that the interpretation of principal components is easier than the canonical variate. Therefore, comparing canonical correlation analysis with principal component analysis, we can say that the canonical correlations of two principal components are more useful for understanding the relationships of the given data sets. This paper derives the limiting distribution of the canonical correlation