```
library(NNS)
library(data.table)
require(knitr)
require(rgl)
require(meboot)
```

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`

`cor`

.Note the fact that all observations occupy the co-partial moment quadrants.

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

`cor(x, y)`

`## [1] 1`

`NNS.dep(x, y)`

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

Note the fact that all observations occupy the co-partial moment quadrants.

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

`cor(x, y)`

`## [1] 0.6610183`

`NNS.dep(x, y)`

```
## $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**.

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

`cor(x, y)`

`## [1] -0.1297766`

`NNS.dep(x, y)`

```
## $Correlation
## [1] -0.002982095
##
## $Dependence
## [1] 0.9999998
```

Note the fact that all observations occupy only co- or divergent partial moment quadrants for a given subquadrant.

```
set.seed(123)
<- 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)) df
```

`NNS.dep(df$x, df$y)`

```
## $Correlation
## [1] 0.02524717
##
## $Dependence
## [1] 0.9830499
```

`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]
<- seq(-5, 5, .1); y <- x^2 + rnorm(length(x)) x
```

`NNS.dep(x, y, p.value = TRUE, print.map = TRUE)`

```
## $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)
<- rnorm(1000); y <- rnorm(1000); z <- rnorm(1000)
x NNS.copula(cbind(x, y, z), plot = TRUE, independence.overlay = TRUE)
```

`## [1] 0.05942998`

Analogous to an empirical copula transformation, we can generate
`new data`

from the dependence structure of our
`original data`

via the following steps:

**Determine the dependence structure:**

This is accomplished using
** LPM.ratio(1, x, x)** for continuous
variables, and

`LPM.ratio(0, x, x)`

**Generate or supply**`new data`

:

`new data`

does not have to be of the same distribution or
dimension as the `original data`

, nor does each dimension of
`new data`

have to share a distribution type.

**Apply dependence structure to**`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
`new.data`

.```
# Add variable x to original data to avoid total independence (example only)
<- cbind(x, y, z, x)
original.data
# Determine dependence structure
<- apply(original.data, 2, function(x) LPM.ratio(1, x, x))
dep.structure
# Generate new data with different mean, sd and length (or distribution type)
<- sapply(1:ncol(original.data), function(x) rnorm(dim(original.data)[1]*2, mean = 10, sd = 20))
new.data
# Apply dependence structure to new data
<- sapply(1:ncol(original.data), function(x) LPM.VaR(dep.structure[,x], 1, new.data[,x])) new.dep.data
```

`NNS.copula(original.data)`

`## [1] 0.4379469`

`NNS.copula(new.dep.data)`

`## [1] 0.4390599`

If the user is so motivated, detailed arguments and proofs are provided within the following: