Correlation and Dependence

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.

x = seq(0, 3, .01) ; y = 2 * x

cor(x, y)

## [1] 1

NNS.dep(x, y)

## $Correlation
## [1] 1
## 
## $Dependence
## [1] 1

x = seq(0, 3, .01) ; y = x ^ 10

cor(x, y)

## [1] 0.6610183

NNS.dep(x, y)

## $Correlation
## [1] 0.9325653
## 
## $Dependence
## [1] 0.9325653

x = seq(0, 12*pi, pi/100) ; y = sin(x)

cor(x, y)

## [1] -0.1297766

NNS.dep(x, y)

## $Correlation
## [1] -0.0002077384
## 
## $Dependence
## [1] 0.8938585

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)

## $Correlation
## [1] 0.05741863
## 
## $Dependence
## [1] 0.2282347

p-values for 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.dep(x, y, p.value = TRUE, print.map = TRUE)

## $Correlation
## [1] -0.03697032
## 
## $`Correlation p.value`
## [1] 0.23
## 
## $`Correlation 95% CIs`
##       2.5%      97.5% 
## -0.1649756  0.3423353 
## 
## $Dependence
## [1] 0.8373177
## 
## $`Dependence p.value`
## [1] 0
## 
## $`Dependence 95% CIs`
##      2.5%     97.5% 
## 0.3414161 0.5663028

Multivariate Dependence 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

References

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