knitr::opts_chunk$set(echo = TRUE)
library(NNS) library(data.table) require(knitr) require(rgl) require(meboot) require(plyr) require(tdigest) require(dtw)
The limitations of linear correlation are well known. Often one uses correlation, when dependence is the intended measure for defining the relationship between variables. NNS dependence NNS.dep
is a signal:noise measure robust to nonlinear signals.
Below are some examples comparing NNS correlation NNS.cor
and NNS.dep
with the standard Pearson's correlation coefficient cor
.
Note the fact that all observations occupy the co-partial moment quadrants.
x = seq(0, 3, .01) ; y = 2 * x
NNS.part(x, y, Voronoi = TRUE, order = 3)
cor(x, y) NNS.dep(x, y)
Note the fact that all observations occupy the co-partial moment quadrants.
x=seq(0, 3, .01) ; y = x ^ 10
NNS.part(x, y, Voronoi = TRUE, order = 3)
cor(x, y) NNS.dep(x, y)
Note the fact that all observations occupy only co- or divergent partial moment quadrants for a given subquadrant.
set.seed(123) df <- data.frame(x = runif(10000, -1, 1), y = runif(10000, -1, 1)) df <- subset(df, (x ^ 2 + y ^ 2 <= 1 & x ^ 2 + y ^ 2 >= 0.95))
NNS.part(df$x, df$y, Voronoi = TRUE, order = 3, obs.req = 0)
NNS.dep(df$x, df$y)
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.
## p-values for [NNS.dep] x <- seq(-5, 5, .1); y <- x^2 + rnorm(length(x))
NNS.part(x, y, Voronoi = TRUE, order = 3)
NNS.dep(x, y, p.value = TRUE, print.map = TRUE)
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)
Analogous to an empirical copula transformation, we can generate new data
from the dependence structure of our original data
via the following steps:
This is accomplished using LPM.ratio(1, x, x)
for continuous variables, and LPM.ratio(0, x, x)
for discrete variables, which are the empirical CDFs of the marginal variables.
new data
:new data
must be of equal dimensions to original data
. new data
does not have to be of the same distribution as the original data
, nor does each dimension of new data
have to share a distribution type.
new data
:We then utilize LPM.VaR(...)
to ascertain new data
values corresponding to original data
position mappings, and return a matrix of these transformed values with the same dimensions as original.data
.
LPM.VaR(..., degree = 0, ...)
is the discrete transformation and significantly faster than the continuous transformation LPM.VaR(..., degree = 1, ...)
.
# Add variable x to original data to avoid total independence (example only) original.data <- cbind(x, y, z, x) # Determine dependence structure dep.structure <- apply(original.data, 2, function(x) LPM.ratio(1, x, x)) # Generate new data of equal dimensions to original data with different mean and sd (or distribution) new.data <- sapply(1:ncol(original.data), function(x) rnorm(dim(original.data)[1], mean = 10, sd = 20)) # Apply dependence structure to new data new.dep.data <- sapply(1:ncol(original.data), function(x) LPM.VaR(dep.structure[,x], 1, new.data[,x]))
NNS.copula(original.data) NNS.copula(new.dep.data)
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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