Maximum Projection Gini Correlation (MaGiC) for mixed categorical and numerical data

Abstract

We propose a projection correlation for measure of dependence between numerical multivariate variables and categorical variables. The projection correlation, defined as the maximum of the Gini correlations (i.e., MaGiC) between the categorical variable and the univariate projections of the multivariate vector, is non-parametric, and intuitively produces a high coefficient when the two variables are dependent, and zero when they are independent. We show that MaGiC possesses the property of nestedness, in that it is non-decreasing with the increasing number of features in the numerical vector, while remaining unchanged if additional numerical features are independent of the categorical variable and original features. We establish $\sqrt{n}$-consistency of the sample projection correlation. A powerful $K$-sample test can be carried out via the MaGiC-based independence test. When compared with related correlation definitions for multivariate variables, MaGiC also enjoys a faster implementation, with the computational complexity $O\left(mn(d+logn)\right)$ where $d$ is the dimension of the numerical variable, $n$ is the sample size, and $m$ is the number of projections performed, as opposed to $O\left(d{n}^2\right)$ for Gini correlation. We demonstrate these properties through simulation and application to real datasets.