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.

`## [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] 0.9325653
##
## $Dependence
## [1] 0.9325653
```

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

The asymmetrical analysis is critical for further determining a causal path between variables which should be identifiable, i.e., it is asymmetrical in causes and effects.

The previous cyclic example visually highlights the asymmetry of
dependence between the variables, which can be confirmed using
** NNS.dep(..., asym = TRUE)**.

`## [1] -0.1297766`

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

`## [1] -0.1297766`

```
## $Correlation
## [1] -0.07074914
##
## $Dependence
## [1] 0.07074914
```

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

`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] 0.2003584
##
## $`Correlation p.value`
## [1] 0.24
##
## $`Correlation 95% CIs`
## 2.5% 97.5%
## -0.1748792 0.3783630
##
## $Dependence
## [1] 0.8886629
##
## $`Dependence p.value`
## [1] 0
##
## $`Dependence 95% CIs`
## 2.5% 97.5%
## 0.3342770 0.5537688
```

`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: