Solved – Detecting non-linear relationships between distributions

correlationnonlinear

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?

Best Answer

Were you looking for something like $Distance Correlation$?

http://en.wikipedia.org/wiki/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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