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

```{r setup, include=FALSE, message = FALSE}
knitr::opts_chunk$set(echo = TRUE)
library(NNS)
library(data.table)
data.table::setDTthreads(1L)
options(mc.cores = 1)
RcppParallel::setThreadOptions(numThreads = 1)
Sys.setenv("OMP_THREAD_LIMIT" = 1)
```

# Partial Moments

Why is it necessary to parse the variance with partial moments?  The additional information generated from partial moments permits a level of analysis simply not possible with traditional summary statistics.

Below are some basic equivalences demonstrating partial moments role as the elements of variance.

## Mean
```{r mean, message=FALSE}
library(NNS)
set.seed(123) ; x = rnorm(100) ; y = rnorm(100)

mean(x)
UPM(1, 0, x) - LPM(1, 0, x)
```

## Variance
```{r variance}
# Sample Variance (base R):
var(x)

# Sample Variance:
(UPM(2, mean(x), x) + LPM(2, mean(x), x)) * (length(x) / (length(x) - 1))


# Population Adjustment of Sample Variance (base R):
var(x) * ((length(x) - 1) / length(x))

# Population Variance:
UPM(2, mean(x), x) + LPM(2, mean(x), x)


# Variance is also the co-variance of itself:
(Co.LPM(1, x, x, mean(x), mean(x)) + Co.UPM(1, x, x, mean(x), mean(x)) - D.LPM(1, 1, x, x, mean(x), mean(x)) - D.UPM(1, 1, x, x, mean(x), mean(x)))
```


## Standard Deviation
```{r stdev}
sd(x)
((UPM(2, mean(x), x) + LPM(2, mean(x), x)) * (length(x) / (length(x) - 1))) ^ .5
```


## First 4 Moments
The first 4 moments are returned with the function `NNS.moments`.  For sample statistics, set `population = FALSE`.
```{r moments}
NNS.moments(x)

NNS.moments(x, population = FALSE)
```


## Statistical Mode of a Continuous Distribution
`NNS.mode` offers support for discrete valued distributions as well as recognizing multiple modes.

```{r mode}
# Continuous
NNS.mode(x)

# Discrete and multiple modes
NNS.mode(c(1, 2, 2, 3, 3, 4, 4, 5), discrete = TRUE, multi = TRUE)
```


## Covariance
```{r covariance}
cov(x, y)
(Co.LPM(1, x, y, mean(x), mean(y)) + Co.UPM(1, x, y, mean(x), mean(y)) - D.LPM(1, 1, x, y, mean(x), mean(y)) - D.UPM(1, 1, x, y, mean(x), mean(y))) * (length(x) / (length(x) - 1))
```

## Covariance Elements and Covariance Matrix
The covariance matrix $(\Sigma)$ is equal to the sum of the co-partial moments matrices less the divergent partial moments matrices.
$$ \Sigma = CLPM + CUPM - DLPM - DUPM $$

```{r cov_dec, warning=FALSE}
cov.mtx = PM.matrix(LPM_degree = 1, UPM_degree = 1,target = 'mean', variable = cbind(x, y), pop_adj = TRUE)
cov.mtx

# Reassembled Covariance Matrix
cov.mtx$clpm + cov.mtx$cupm - cov.mtx$dlpm - cov.mtx$dupm


# Standard Covariance Matrix
cov(cbind(x, y))
```

## Pearson Correlation
```{r pearson}
cor(x, y)
cov.xy = (Co.LPM(1, x, y, mean(x), mean(y)) + Co.UPM(1, x, y, mean(x), mean(y)) - D.LPM(1, 1, x, y, mean(x), mean(y)) - D.UPM(1, 1, x, y, mean(x), mean(y))) * (length(x) / (length(x) - 1))
sd.x = ((UPM(2, mean(x), x) + LPM(2, mean(x), x)) * (length(x) / (length(x) - 1))) ^ .5
sd.y = ((UPM(2, mean(y), y) + LPM(2, mean(y) , y)) * (length(y) / (length(y) - 1))) ^ .5
cov.xy / (sd.x * sd.y)
```

## CDFs (Discrete and Continuous)
```{r cdfs,fig.align="center",fig.width=5,fig.height=3, results='hide'}
P = ecdf(x)
P(0) ; P(1)
LPM(0, 0, x) ; LPM(0, 1, x)

# Vectorized targets:
LPM(0, c(0, 1), x)

plot(ecdf(x))
points(sort(x), LPM(0, sort(x), x), col = "red")
legend("left", legend = c("ecdf", "LPM.CDF"), fill = c("black", "red"), border = NA, bty = "n")

# Joint CDF:
Co.LPM(0, x, y, 0, 0)

# Vectorized targets:
Co.LPM(0, x, y, c(0, 1), c(0, 1))

# Copula
# Transform x and y so that they are uniform
u_x = LPM.ratio(0, x, x)
u_y = LPM.ratio(0, y, y)

# Value of copula at c(.5, .5)
Co.LPM(0, u_x, u_y, .5, .5)

# Continuous CDF:
NNS.CDF(x, 1)

# CDF with target:
NNS.CDF(x, 1, target = mean(x))

# Survival Function:
NNS.CDF(x, 1, type = "survival")
```



## Numerical Integration
Partial moments are asymptotic area approximations of $f(x)$ akin to the familiar Trapezoidal and Simpson's rules.  More observations, more accuracy...

$$[UPM(1,0,f(x))-LPM(1,0,f(x))]\asymp\frac{[F(b)-F(a)]}{[b-a]}$$
$$[UPM(1,0,f(x))-LPM(1,0,f(x))] *[b-a] \asymp[F(b)-F(a)]$$

```{r numerical integration}
x = seq(0, 1, .001) ; y = x ^ 2
(UPM(1, 0, y) - LPM(1, 0, y)) * (1 - 0)
```

$$0.3333 * [1-0] = \int_{0}^{1} x^2 dx$$
For the total area, not just the definite integral, simply sum the partial moments and multiply by $[b - a]$:
$$[UPM(1,0,f(x))+LPM(1,0,f(x))] *[b-a]\asymp\left\lvert{\int_{a}^{b} f(x)dx}\right\rvert$$

## Bayes' Theorem
For example, when ascertaining the probability of an increase in $A$ given an increase in $B$, the `Co.UPM(degree_upm, x, y, target_x, target_y)` target parameters are set to `target_x = 0` and `target_y = 0` and the `UPM(degree, target, variable)` target parameter is also set to `target = 0`.

$$P(A|B)=\frac{Co.UPM(0,A,B,0,0)}{UPM(0,0,B)}$$

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

* [Partial Moments as a Unifying Primitive: Distributional Structure, Benchmark-Relative Utility, Adaptive Estimation, and Learned Neural Nonlinearities](https://doi.org/10.2139/ssrn.6249658)

* [Nonlinear Nonparametric Statistics: Using Partial Moments](https://github.com/OVVO-Financial/NNS/blob/NNS-Beta-Version/examples/index.md)

* [Cumulative Distribution Functions and UPM/LPM Analysis](https://doi.org/10.2139/ssrn.2148482)

* [Continuous CDFs and ANOVA with NNS](https://doi.org/10.2139/ssrn.3007373)

* [f(Newton)](https://doi.org/10.2139/ssrn.2186471)

* [Bayes' Theorem From Partial Moments](https://doi.org/10.2139/ssrn.3457377)

