Canonical Dependency Analysis Using a Bias-Corrected $\chi ^2$ Statistics Matrix

Abstract

Canonical correlation and canonical covariance analyses are popular dimensional reduction methods when managing two datasets. Because these methods seek a subspace maximizing correlation (or covariance), it is not suitable to apply them to datasets that have a non-linear relationship. To tackle this issue, some researchers have proposed canonical dependency analysis (CDA), which seeks a subspace maximizing dependency. However, applying this method to datasets with categorical variables may not be appropriate because CDA does not consider categorical variables directly. Moreover, some methods are time consuming for hyper-parameter tuning. We therefore propose a quantification method that includes a CDA that minimizes the distance between the dependency matrix and the estimated matrix and a calculation method for a bias-corrected $\chi ^2$ statistics matrix in a moderate amount of time. We derived the explicit updated formula of the parameter estimation using the majorization technique and applied a method to calculate a bias-corrected $\chi ^2$ statistics matrix without hyper-parameters. We then applied this method to both simulated and real datasets. From the simulated data, the proposed method shows the best performance when the data include some categorical variables. We obtain a reasonable result from the application to real datasets.