Generalized correlation coefficients

[DominiqueMakowski]

We could add that as a method.

Implemented in the generalCorr package.

generalCorr::gmcmtx0(mtcars[1:4])   
#>             mpg        cyl       disp         hp
#> mpg   1.0000000 -0.8557900 -0.9508994 -0.9379374
#> cyl  -0.9433125  1.0000000  0.9759183  0.9583212
#> disp -0.8941676  0.9151419  1.0000000  0.9306311
#> hp   -0.8530474  0.8446589  0.8170031  1.0000000

^{Created on 2021-05-29 by the reprex package (v1.0.0)}

This function runs a kernel regression for each pair (x on y, and conversely y on x) and returns the generalized correlation coefficient (R*), which corresponds to the square roots of the R2 with the observed sign of the Pearson correlation coefficient.

Among some of the interesting properties, max(|R*xy|, |R*yx|) measures (nonlinear) dependence. For example, let x=1:20 and y=sin(x). This y has a perfect (100%) nonlinear dependence on x, and yet Pearson correlation coefficient r(xy) -0.0948372 is near zero. "This shows a miserable failure of traditional r(x,y) to measure dependence when nonlinearities are present".

generalCorr::gmcmtx0(data.frame(x=1:20, y=sin(1:20)))
                
#>    x           y
#> x  1 -0.04847292
#> y -1  1.00000000

^{Created on 2021-05-29 by the reprex package (v1.0.0)}