examples/Partial Moments Equivalences.md

In OVVO-Financial/NNS: Nonlinear Nonparametric Statistics

Partial Moments Equivalences

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

Why is this relevant?

The additional information generated from partial moments permits a level of analysis simply not possible with traditional summary statistics. There is further introductory material on partial moments and their extension into nonlinear analysis & behavioral finance applications available at:

https://www.linkedin.com/pulse/elements-variance-fred-viole

Installation

require(devtools); install_github('OVVO-Financial/NNS', ref = "NNS-Beta-Version")

Mean

A difference between the upside area and the downside area of f(x).

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

> mean(x)
[1] 0.09040591
> UPM(1,0,x)-LPM(1,0,x)
[1] 0.09040591

Variance

A sum of the squared upside area and the squared downside area.

> var(x)
[1] 0.8332328
# Sample Variance:
> UPM(2,mean(x),x)+LPM(2,mean(x),x)
[1] 0.8249005
# Population Variance:
> (UPM(2,mean(x),x)+LPM(2,mean(x),x))*(length(x)/(length(x)-1))
[1] 0.8332328
# 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))
[1] 0.8332328

The first 4 moments are returned with the function NNS.moments. For sample statistics, set population = FALSE.

> NNS.moments(x)
$mean
[1] 0.09040591

$variance
[1] 0.8332328

$skewness
[1] 0.06049948

$kurtosis
[1] -0.161053

> NNS.moments(x, population = FALSE)
$mean
[1] 0.09040591

$variance
[1] 0.8249005

$skewness
[1] 0.06235774

$kurtosis
[1] -0.1069186

Standard Deviation

> sd(x)
[1] 0.9128159
> ((UPM(2,mean(x),x)+LPM(2,mean(x),x))*(length(x)/(length(x)-1)))^.5
[1] 0.9128159

Covariance

> cov(x,y)
[1] -0.04372107
> (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))
[1] -0.04372107

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.

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

> cov.mtx = PM.matrix(LPM_degree = 1, UPM_degree = 1, target = 'mean', variable = cbind(x,y), pop_adj = TRUE)
> cov.mtx
$cupm
          x         y
x 0.4299250 0.1033601
y 0.1033601 0.5411626

$dupm
          x         y
x 0.0000000 0.1469182
y 0.1560924 0.0000000

$dlpm
          x         y
x 0.0000000 0.1560924
y 0.1469182 0.0000000

$clpm
          x         y
x 0.4033078 0.1559295
y 0.1559295 0.3939005

$cov.matrix
            x           y
x  0.83323283 -0.04372107
y -0.04372107  0.93506310


# Reassembled Covariance Matrix
> cov.mtx$cupm + cov.mtx$clpm - cov.mtx$dupm - cov.mtx$dlpm
            x           y
x  0.83323283 -0.04372107
y -0.04372107  0.93506310


# Standard Covariance Matrix
> cov(cbind(x,y))
            x           y
x  0.83323283 -0.04372107
y -0.04372107  0.93506310

Pearson Correlation

> cor(x,y)
[1] -0.04953215
> 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)
[1] -0.04953215

Skewness

A normalized difference between upside area and downside area.

> library(PerformanceAnalytics)
> skewness(x)
[1] 0.06049948
> ((UPM(3,mean(x),x)-LPM(3,mean(x),x))/(UPM(2,mean(x),x)+LPM(2,mean(x),x))^(3/2))
[1] 0.06049948

UPM/LPM - a more intuitive measure of skewness. (Upside area / Downside area)

> UPM(2,mean(x),x)/LPM(2,mean(x),x)
[1] 1.065997

Kurtosis

A normalized sum of upside area and downside area.

> library(PerformanceAnalytics)
> kurtosis(x)
[1] -0.161053
> ((UPM(4,mean(x),x)+LPM(4,mean(x),x))/(UPM(2,mean(x),x)+LPM(2,mean(x),x))^2)-3
[1] -0.161053

CDFs

> P = ecdf(x)
> P(0); P(1)
[1] 0.48
[1] 0.83
> LPM(0,0,x); LPM(0,1,x)
[1] 0.48
[1] 0.83
# Vectorized targets:
> LPM(0,c(0,1),x)
[1] 0.48 0.83
# Joint CDF:
> Co.LPM(0,x,y,0,0)
[1] 0.28
# Vectorized targets:
> Co.LPM(0,x,y,c(0,1),c(0,1))
[1] 0.28 0.73

# Alternatively via NNS.CDF()
> NNS.CDF(x)

Copulas

# 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)
[1] 0.26

Numerical Integration - [UPM(1,0,f(x))-LPM(1,0,f(x))]=[F(b)-F(a)]/[b-a]

# x is uniform sample over interval [a,b]; y = f(x)
> x = seq(0,1,.001); y = x^2
# [F(b)-F(a)] = [UPM(1,0,f(x))-LPM(1,0,f(x))] * [b-a] 
> (UPM(1,0,y)-LPM(1,0,y)) * (1-0)
[1] 0.3335

Bayes' Theorem

See the following example explaining Bayes' Theorem and partial moments: 

https://github.com/OVVO-Financial/NNS/blob/NNS-Beta-Version/examples/Bayes'%20Theorem%20From%20Partial%20Moments.pdf

OVVO-Financial/NNS documentation built on April 22, 2024, 10:26 p.m.