# Getting Started with NNS: Correlation and Dependence" In NNS: Nonlinear Nonparametric Statistics

```knitr::opts_chunk\$set(echo = TRUE)
```
```require(NNS)
require(knitr)
require(rgl)
require(data.table)
```

# 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`.

## Linear Equivalence

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

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

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

## Nonlinear Relationship

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

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

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

## Dependence

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)
```

# References

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

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NNS documentation built on May 15, 2019, 1:05 a.m.