knitr::opts_chunk$set(echo = TRUE) require(NNS) require(knitr) require(rgl)

The limitations of linear correlation are well known. Often one uses correlation, when dependence is the intended measure for defining the relationship between variables. NNS dependence `NNS.dep`

is a signal:noise measure robust to nonlinear signals.

Below are some examples comparing NNS correlation `NNS.cor`

and `NNS.dep`

with the standard Pearson's correlation coefficient `cor`

.

Note the fact that all observations occupy the co-partial moment quadrants.

x=seq(0,3,.01); y=2*x cor(x,y) NNS.dep(x,y,print.map = T,order=3)

Note the fact that all observations occupy the co-partial moment quadrants.

x=seq(0,3,.01); y=x^10 cor(x,y) NNS.dep(x,y,print.map = T,order=3)

Note the fact that all observations occupy only co- or divergent partial moment quadrants for a given subquadrant.

set.seed(123) df<- data.frame(x=runif(10000,-1,1),y=runif(10000,-1,1)) df<- subset(df, (x^2 + y^2 <= 1 & x^2 + y^2 >= 0.95)) NNS.dep(df$x,df$y,print.map = T)

If the user is so motivated, detailed arguments and proofs are provided within the following:

*Nonlinear Nonparametric Statistics: Using Partial Moments

*Nonlinear Correlation and Dependence Using NNS

*Deriving Nonlinear Correlation Coefficients from Partial Moments

*Beyond Correlation: Using the Elements of Variance for Conditional Means and Probabilities

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