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 = TRUE, 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 = TRUE, 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 = TRUE)
```

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

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