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
.
##Linear Equivalence Note the fact that all observations occupy the co-partial moment quadrants.
## [1] 1
## $Correlation
## [1] 1
##
## $Dependence
## [1] 1
##Nonlinear Relationship Note the fact that all observations occupy the co-partial moment quadrants.
## [1] 0.6610183
## $Correlation
## [1] 0.9413457
##
## $Dependence
## [1] 0.9413457
##Dependence Note the fact that all observations occupy only co- or divergent partial moment quadrants for a given subquadrant.
set.seed(123)
df <- data.frame(x = runif(10000, -1, 1), y = runif(10000, -1, 1))
df <- subset(df, (x ^ 2 + y ^ 2 <= 1 & x ^ 2 + y ^ 2 >= 0.95))
NNS.dep(df$x, df$y, print.map = TRUE)
## $Correlation
## [1] -0.007630343
##
## $Dependence
## [1] 0.9963612
#References If the user is so motivated, detailed arguments and proofs are provided within the following: