knitr::opts_chunk$set(echo = TRUE) require(NNS) require(knitr) require(rgl)
The limitations of linear correlation are well known. Often one uses correlation, when dependence is the intended measure for defining the relationship between variables. NNS dependence NNS.dep
is a signal:noise measure robust to nonlinear signals.
Below are some examples comparing NNS correlation NNS.cor
and NNS.dep
with the standard Pearson's correlation coefficient cor
.
Note the fact that all observations occupy the co-partial moment quadrants.
x = seq(0, 3, .01) ; y = 2 * x cor(x, y) NNS.dep(x, y, print.map = TRUE, order = 3)
Note the fact that all observations occupy the co-partial moment quadrants.
x=seq(0, 3, .01) ; y = x ^ 10 cor(x, y) NNS.dep(x, y, print.map = TRUE, order = 3)
Note the fact that all observations occupy only co- or divergent partial moment quadrants for a given subquadrant.
set.seed(123) df <- data.frame(x = runif(10000, -1, 1), y = runif(10000, -1, 1)) df <- subset(df, (x ^ 2 + y ^ 2 <= 1 & x ^ 2 + y ^ 2 >= 0.95)) NNS.dep(df$x, df$y, print.map = TRUE)
If the user is so motivated, detailed arguments and proofs are provided within the following:
Deriving Nonlinear Correlation Coefficients from Partial Moments
Beyond Correlation: Using the Elements of Variance for Conditional Means and Probabilities
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