NNS examples: Partial Moments Equivalences

Why is it necessary to parse 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.

Installation

install.packages("NNS")

Setup

library(NNS)
set.seed(123)
x <- rnorm(100)
y <- rnorm(100)

Mean

mean(x)
## [1] 0.09040591

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

Variance

# Sample Variance (base R):
var(x)
## [1] 0.8332328

# Sample Variance from partial moments:
(UPM(2, mean(x), x) + LPM(2, mean(x), x)) * (length(x) / (length(x) - 1))
## [1] 0.8332328

# Population Adjustment of Sample Variance (base R):
var(x) * ((length(x) - 1) / length(x))
## [1] 0.8249005

# Population Variance:
UPM(2, mean(x), x) + LPM(2, mean(x), x)
## [1] 0.8249005

# Variance is also the covariance 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)))
## [1] 0.8249005

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

sqrt(NNS.moments(x, population = FALSE)$variance)
## [1] 0.9128159

First 4 Moments

The first four moments are returned with NNS.moments(). For sample statistics, set population = FALSE.

NNS.moments(x)
## $mean
## [1] 0.09040591
##
## $variance
## [1] 0.8249005
##
## $skewness
## [1] 0.06049948
##
## $kurtosis
## [1] -0.161053

NNS.moments(x, population = FALSE)
## $mean
## [1] 0.09040591
##
## $variance
## [1] 0.8332328
##
## $skewness
## [1] 0.06235774
##
## $kurtosis
## [1] -0.1069186

Statistical Mode of a Continuous Distribution

NNS.mode offers support for discrete valued distributions as well as recognizing multiple modes.

# Continuous
NNS.mode(x)
## [1] -0.4132834

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

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 (Σ) is equal to the sum of the co-partial moments matrices less the divergent partial moments matrices.

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$clpm + cov.mtx$cupm - cov.mtx$dlpm - cov.mtx$dupm
##             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

PerformanceAnalytics::skewness(x)

(UPM(3, mean(x), x) - LPM(3, mean(x), x)) /
  (UPM(2, mean(x), x) + LPM(2, mean(x), x)) ^ (3 / 2)

NNS.moments(x)$skewness

UPM / LPM Ratio

UPM(2, mean(x), x) / LPM(2, mean(x), x)

Kurtosis

PerformanceAnalytics::kurtosis(x)

((UPM(4, mean(x), x) + LPM(4, mean(x), x)) /
  (UPM(2, mean(x), x) + LPM(2, mean(x), x)) ^ 2) - 3

NNS.moments(x)$kurtosis

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 CDF vs LPM:
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")

Copulas

# Transform x and y to uniform marginals
u_x <- LPM.ratio(0, x, x)
u_y <- LPM.ratio(0, y, y)

# Value of copula at c(0.5, 0.5)
Co.LPM(0, u_x, u_y, 0.5, 0.5)

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) ≍ F(b) − F(a)

x <- seq(0, 1, .001)
y <- x ^ 2

(UPM(1, 0, y) - LPM(1, 0, y)) * (1 - 0)
## [1] 0.3335

For the total area, not just the definite integral, sum the partial moments and multiply by (b − a):

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.

Partial Moments Equivalences