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 Pearsonâ€™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)
```

`## [1] 1`

`NNS.dep(x,y,print.map = T,order=3)`

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

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

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

`## [1] 0.6610183`

`NNS.dep(x,y,print.map = T,order=3)`

```
## $Correlation
## [1] 0.9699069
##
## $Dependence
## [1] 0.9699069
```

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 = T)
```

```
## $Correlation
## [1] -0.007630343
##
## $Dependence
## [1] 0.9963612
```

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