Solved – Detecting non-linear relationships between distributions

X,Y are r.v's exactly related by some unknown non-linear relationship.

Does there exist a neat analog of Correlation, that gives some information about a this relationship?

Were you looking for something like $Distance Correlation$?

This will be non-zero for any sort of relationship between your $x$ and $y$. Therefore this can trap arbitrary non-linear relationships though interpreting the values is harder than interpreting correlation.

If this is what you need try the "energy" library in R.

set.seed(1234) x<-rnorm(1000,0,1) y<-x^2 cor(x,y) [1] -0.03908369 library(energy) dcor(x,y, R=500)) [1] 0.5478997  #to get a p-value for the distance correlation: dcov.test(x,y)          dCov test of independence  data:  index 1, replicates 500 nV^2 = 140.74, p-value = 0.001996 sample estimates:      dCov  0.3751596  

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