Finding non-linear correlations in R

You can use nlcor package in R. This package finds the nonlinear correlation between two data vectors. There are different approaches to estimate a nonlinear correlation, such as infotheo. However, nonlinear correlations between two variables can take any shape.

nlcor is robust to most nonlinear shapes. It works pretty well in different scenarios.

At a high level, nlcor works by adaptively segmenting the data into linearly correlated segments. The segment correlations are aggregated to yield the nonlinear correlation. The output is a number between 0 to 1. With close to 1 meaning high correlation. Unlike a pearson correlation, negative values are not returned because it has no meaning in nonlinear relationships.

More details about this package here

To install nlcor, follow these steps:

install.packages("devtools") 
library(devtools)
install_github("ProcessMiner/nlcor")
library(nlcor)

After you install it,

# Implementation 
x <- seq(0,3*pi,length.out=100)
y <- sin(x)
plot(x,y,type="l")

# linear correlation is small
cor(x,y)
# [1] 6.488616e-17
# nonlinear correlation is more representative
nlcor(x,y, plt = T)
# $cor.estimate
# [1] 0.9774
# $adjusted.p.value
# [1] 1.586302e-09
# $cor.plot

As shown in the example the linear correlation was close to zero although there was a clear relationship between the variables that nlcor could detect.

Note: The order of x and y inside the nlcor is important. nlcor(x,y) is different from nlcor(y,x). The x and y here represent 'independent' and 'dependent' variables, respectively.