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.
## [1] 1
## $Correlation
## [1] 1
##
## $Dependence
## [1] 1
Note the fact that all observations occupy the co-partial moment quadrants.
## [1] 0.6610183
## $Correlation
## [1] 0.9880326
##
## $Dependence
## [1] 0.9998937
Even the difficult inflection points, which span both the co- and
divergent partial moment quadrants, are properly compensated for in
NNS.dep
.
## [1] -0.1297766
## $Correlation
## [1] -0.0003237856
##
## $Dependence
## [1] 1
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))
## $Correlation
## [1] 0.02403114
##
## $Dependence
## [1] 0.9831656
NNS.dep()
p-values and confidence intervals can be obtained from sampling
random permutations of \(y \rightarrow
y_p\) and running NNS.dep(x,$y_p$)
to compare against a null hypothesis of 0 correlation, or independence
between \((x, y)\).
Simply set
NNS.dep(..., p.value = TRUE, print.map = TRUE)
to run 100 permutations and plot the results.
## $Correlation
## [1] 0.01943686
##
## $`Correlation p.value`
## [1] 0.34
##
## $`Correlation 95% CIs`
## 2.5% 97.5%
## -0.1391829 0.1246556
##
## $Dependence
## [1] 0.7206435
##
## $`Dependence p.value`
## [1] 0
##
## $`Dependence 95% CIs`
## 2.5% 97.5%
## 0.1131909 0.2852342
NNS.copula()
These partial moment insights permit us to extend the analysis to multivariate instances and deliver a dependence measure \((D)\) such that \(D \in [0,1]\). This level of analysis is simply impossible with Pearson or other rank based correlation methods, which are restricted to bivariate cases.
set.seed(123)
x <- rnorm(1000); y <- rnorm(1000); z <- rnorm(1000)
NNS.copula(cbind(x, y, z), plot = TRUE, independence.overlay = TRUE)
## [1] 0.1231362
If the user is so motivated, detailed arguments and proofs are provided within the following: