# KdCor: Kernel Distance Correlation Statistics In GiniDistance: A New Gini Correlation Between Quantitative and Qualitative Variables

## Description

Computes Kernel distance correlation statistics, in which Xs are quantitative, Y are categorical, sigma is kernel standard deviation and returns the measures of dependence.

## Usage

 1  KdCor(x, y, sigma) 

## Arguments

 x data y label of data or univariate response variable sigma kernel standard deviation

## Details

KdCor compute distance correlation statistics. The sample size (number of rows) of the data must agree with the length of the label vector, and samples must not contain missing values. Arguments x, y are treated as data and labels.

The kernel distance correlation is defined as follow.

{dCor}_{κ_X,κ_Y}(\mathbf{X}, Y) = \frac{\mbox{ dCov}_{κ_X,κ_Y}(\mathbf{X}, Y)}{ √{\mbox{ dCov}_{κ_X,κ_X}(\mathbf{X},\mathbf{X})} √{\mbox{ dCov}_{κ_Y,κ_Y}(Y,Y)}}

where

\begin{array}{c} \mathrm{dCov}_{κ_X,κ_Y}(X,Y) = {E}d_{κ_X}(X,X')d_{κ_Y}(Y,Y') + {E}d_{κ_X}(X,X'){E}d_{κ_Y}(Y,Y') \\ - 2{E}≤ft[{E}_{X'}d_{κ_X}(X,X') {E}_{Y'}d_{κ_Y}(Y,Y')\right].\label{dCovkk} \end{array}

## Value

KdCor returns the sample kernel distance correlation

## References

Sejdinovic, D., Sriperumbudur, B., Gretton, A. and Fukumizu, K. (2013). Equivalence of Distance-based and RKHS-based Statistics in Hypothesis Testing, The Annals of Statistics, 41 (5), 2263-2291.

Zhang, S., Dang, X., Nguyen, D. and Chen, Y. (2019). Estimating feature - label dependence using Gini distance statistics. IEEE Transactions on Pattern Analysis and Machine Intelligence (submitted).

KgCov KgCor dCor
 1 2 3  x<-iris[,1:4] y<-unclass(iris[,5]) KdCor(x, y, sigma=1)