---
title: "Getting Started with NNS: Clustering and Regression"
author: "Fred Viole"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Getting Started with NNS: Clustering and Regression}
  %\VignetteEngine{knitr::rmarkdown}
  \usepackage[utf8]{inputenc}
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```

```{r setup2, message=FALSE, warning=FALSE}
library(NNS)
library(data.table)
require(knitr)
require(rgl)
require(meboot)
```


# Clustering and Regression
Below are some examples demonstrating unsupervised learning with NNS clustering and nonlinear regression using the resulting clusters.  As always, for a more thorough description and definition, please view the References.

## NNS Partitioning <span style="color:red">`NNS.part()`</span>
**`NNS.part`** is both a partitional and hierarchical clustering method.  `NNS` iteratively partitions the joint distribution into partial moment quadrants, and then assigns a quadrant identification (1:4) at each partition.

**`NNS.part`** returns a `data.table` of observations along with their final quadrant identification.  It also returns the regression points, which are the quadrant means used in **`NNS.reg`**.
```{r linear}
x = seq(-5, 5, .05); y = x ^ 3

for(i in 1 : 4){NNS.part(x, y, order = i, Voronoi = TRUE, obs.req = 0)}
```


### X-only Partitioning
**`NNS.part`** offers a partitioning based on $x$ values only **`NNS.part(x, y, type = "XONLY", ...)`**, using the entire bandwidth in its regression point derivation, and shares the same limit condition as partitioning via both $x$ and $y$ values.
```{r x part,results='hide'}
for(i in 1 : 4){NNS.part(x, y, order = i, type = "XONLY", Voronoi = TRUE)}
```

Note the partition identifications are limited to 1's and 2's (left and right of the partition respectively), not the 4 values per the $x$ and $y$ partitioning.
```{r res2, echo=FALSE}
NNS.part(x,y,order = 4, type = "XONLY")
```

## Clusters Used in Regression
The right column of plots shows the corresponding regression for the order of `NNS` partitioning.
```{r depreg},results='hide'}
for(i in 1 : 3){NNS.part(x, y, order = i, obs.req = 0, Voronoi = TRUE) ; NNS.reg(x, y, order = i, ncores = 1)}
```


# NNS Regression `NNS.reg()`
**`NNS.reg`** can fit any $f(x)$, for both uni- and multivariate cases.  **`NNS.reg`** returns a self-evident list of values provided below.

## Univariate:
```{r nonlinear,fig.width=5,fig.height=3,fig.align = "center"}
NNS.reg(x, y, ncores = 1)
```

## Multivariate:
Multivariate regressions return a plot of $y$ and $\hat{y}$, as well as the regression points (`$RPM`) and partitions (`$rhs.partitions`) for each regressor.
```{r nonlinear multi,fig.width=5,fig.height=3,fig.align = "center"}
f = function(x, y) x ^ 3 + 3 * y - y ^ 3 - 3 * x
y = x ; z <- expand.grid(x, y)
g = f(z[ , 1], z[ , 2])
NNS.reg(z, g, order = "max", plot = FALSE, ncores = 1)
```

## Inter/Extrapolation
`NNS.reg` can inter- or extrapolate any point of interest.  The **`NNS.reg(x, y, point.est = ...)`** parameter permits any sized data of similar dimensions to $x$ and called specifically with **`NNS.reg(...)$Point.est`**.


## NNS Dimension Reduction Regression
**`NNS.reg`** also provides a dimension reduction regression by including a parameter **`NNS.reg(x, y, dim.red.method = "cor", ...)`**.  Reducing all regressors to a single dimension using the returned equation **`NNS.reg(..., dim.red.method = "cor", ...)$equation`**.
```{r nonlinear_class,fig.width=5,fig.height=3,fig.align = "center", message = FALSE}
NNS.reg(iris[ , 1 : 4], iris[ , 5], dim.red.method = "cor", location = "topleft", ncores = 1)$equation
```

```{r nonlinear_class2,fig.width=5,fig.height=3,fig.align = "center", message = FALSE, echo=FALSE}
a = NNS.reg(iris[ , 1 : 4], iris[ , 5], dim.red.method = "cor", location = "topleft", ncores = 1, plot = FALSE)$equation
```
Thus, our model for this regression would be:
$$Species = \frac{`r round(a$Coefficient[1],3)`*Sepal.Length `r round(a$Coefficient[2],3)`*Sepal.Width +`r round(a$Coefficient[3],3)`*Petal.Length +`r round(a$Coefficient[4],3)`*Petal.Width}{4} $$


### Threshold
**`NNS.reg(x, y, dim.red.method = "cor", threshold = ...)`** offers a method of reducing regressors further by controlling the absolute value of required correlation.
```{r nonlinear class threshold,fig.width=5,fig.height=3,fig.align = "center"}
NNS.reg(iris[ , 1 : 4], iris[ , 5], dim.red.method = "cor", threshold = .75, location = "topleft", ncores = 1)$equation
```

```{r nonlinear class threshold 2,fig.width=5,fig.height=3,fig.align = "center", echo=FALSE}
a = NNS.reg(iris[ , 1 : 4], iris[ , 5], dim.red.method = "cor", threshold = .75, location = "topleft", ncores = 1, plot = FALSE)$equation
```

Thus, our model for this further reduced dimension regression would be:
$$Species = \frac{\: `r round(a$Coefficient[1],3)`*Sepal.Length + `r round(a$Coefficient[2],3)`*Sepal.Width +`r round(a$Coefficient[3],3)`*Petal.Length +`r round(a$Coefficient[4],3)`*Petal.Width}{3} $$

and the `point.est = (...)` operates in the same manner as the full regression above, again called with **`NNS.reg(...)$Point.est`**.
```{r final,fig.width=5,fig.height=3,fig.align = "center"}
NNS.reg(iris[ , 1 : 4], iris[ , 5], dim.red.method = "cor", threshold = .75, point.est = iris[1 : 10, 1 : 4], location = "topleft", ncores = 1)$Point.est
```


# Classification
For a classification problem, we simply set **`NNS.reg(x, y, type = "CLASS", ...)`**.

**NOTE: Base category of response variable should be 1, not 0 for classification problems.**

```{r class,fig.width=5,fig.height=3,fig.align = "center", message=FALSE}
NNS.reg(iris[ , 1 : 4], iris[ , 5], type = "CLASS", point.est = iris[1 : 10, 1 : 4], location = "topleft", ncores = 1)$Point.est
```


# Cross-Validation `NNS.stack()`
The **`NNS.stack`** routine cross-validates for a given objective function the `n.best` parameter in the multivariate **`NNS.reg`** function as well as the `threshold` parameter in the dimension reduction **`NNS.reg`** version.  **`NNS.stack`** can be used for classification:

**`NNS.stack(..., type = "CLASS", ...)`** 

or continuous dependent variables:

**`NNS.stack(..., type = NULL, ...)`**.

Any objective function `obj.fn` can be called using `expression()` with the terms `predicted` and `actual`.


```{r stack,fig.width=5,fig.height=3,fig.align = "center", message=FALSE, eval=FALSE}
NNS.stack(IVs.train = iris[ , 1 : 4], 
          DV.train = iris[ , 5], 
          IVs.test = iris[1 : 10, 1 : 4],
          dim.red.method = "cor",
          obj.fn = expression( mean(round(predicted) == actual) ),
          objective = "max", type = "CLASS", 
          folds = 1, ncores = 1)
```

```{r stackevalres, eval = FALSE}
Folds Remaining = 0 
Current NNS.reg(... , threshold = 0.79 ) MAX Iterations Remaining = 1 
Current NNS.reg(... , threshold = 0.44 ) MAX Iterations Remaining = 0 
Current NNS.reg(... , n.best = 1 ) MAX Iterations Remaining = 12 
Current NNS.reg(... , n.best = 2 ) MAX Iterations Remaining = 11 
Current NNS.reg(... , n.best = 3 ) MAX Iterations Remaining = 10 
Current NNS.reg(... , n.best = 4 ) MAX Iterations Remaining = 9 
$OBJfn.reg
[1] 1

$NNS.reg.n.best
[1] 1

$probability.threshold
[1] 0.445

$OBJfn.dim.red
[1] 0.9133333

$NNS.dim.red.threshold
[1] 0.79

$reg
[1] 1 1 1 1 1 1 1 1 1 1

$reg.pred.int
NULL

$dim.red
[1] 1 1 1 1 1 1 1 1 1 1

$dim.red.pred.int
NULL

$stack
[1] 1 1 1 1 1 1 1 1 1 1

$pred.int
NULL
```

## Increasing Dimensions
Given multicollinearity is not an issue for nonparametric regressions as it is for OLS, in the case of an ill-fit univariate model a better option may be to increase the dimensionality of regressors with a copy of itself and cross-validate the number of clusters `n.best` via:

**`NNS.stack(IVs.train = cbind(x, x), DV.train = y, method = 1, ...)`**.

```{r stack2, message = FALSE,fig.width=5,fig.height=3,fig.align = "center",results='hide', eval = FALSE}
set.seed(123)
x = rnorm(100); y = rnorm(100)

nns.params = NNS.stack(IVs.train = cbind(x, x),
                        DV.train = y,
                        method = 1, ncores = 1)
```

```{r stack2optim, echo = FALSE}
set.seed(123)
x = rnorm(100); y = rnorm(100)

nns.params = list()
nns.params$NNS.reg.n.best = 9
```

```{r stack2res, fig.width=5,fig.height=3,fig.align = "center",results='hide'}
NNS.reg(cbind(x, x), y, 
        n.best = nns.params$NNS.reg.n.best,
        point.est = cbind(x, x), 
        residual.plot = TRUE,  
        ncores = 1, confidence.interval = .95)
```


# References
If the user is so motivated, detailed arguments further examples are provided within the following:

* [Nonlinear Nonparametric Statistics: Using Partial Moments](https://www.amazon.com/dp/1490523995/ref=cm_sw_su_dp)

* [Deriving Nonlinear Correlation Coefficients from Partial Moments](https://www.ssrn.com/abstract=2148522)

* [Nonparametric Regression Using Clusters](https://link.springer.com/article/10.1007/s10614-017-9713-5)

* [Clustering and Curve Fitting by Line Segments](https://www.ssrn.com/abstract=2861339)

* [Classification Using NNS Clustering Analysis](https://www.ssrn.com/abstract=2864711)
