Getting Started with NNS: Partial Moments"

knitr::opts_chunk$set(echo = TRUE)

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

library(NNS)
set.seed(123) ; x = rnorm(100) ; y = rnorm(100)

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

Variance

var(x)

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

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

# 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))) * (length(x) / (length(x) - 1))

Standard Deviation

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.

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.

# Continuous
NNS.mode(x)

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

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. $$ \Sigma = CLPM + CUPM - DLPM - DUPM $$

PM.matrix(LPM_degree = 1, UPM_degree = 1,target = 'mean', variable = cbind(x, y), pop_adj = TRUE)

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

Pearson Correlation

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)

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))

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

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

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

PDFs

NNS.PDF(x)

Numerical Integration

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

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

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_x, degree_y, 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.

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

References

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



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NNS documentation built on Jan. 8, 2023, 1:08 a.m.