knitr::opts_chunk$set(echo = TRUE) library(NNS) library(data.table) data.table::setDTthreads(2L) options(mc.cores = 1) Sys.setenv("OMP_THREAD_LIMIT" = 2)
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.
library(NNS) set.seed(123) ; x = rnorm(100) ; y = rnorm(100) mean(x) UPM(1, 0, x) - LPM(1, 0, x)
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))
sd(x) ((UPM(2, mean(x), x) + LPM(2, mean(x), x)) * (length(x) / (length(x) - 1))) ^ .5
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)
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)
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))
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 # Reassembled Covariance Matrix cov.mtx$clpm + cov.mtx$cupm - cov.mtx$dlpm - cov.mtx$dupm # Standard Covariance Matrix cov(cbind(x, y))
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)
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")
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 $[b - a]$: $$[UPM(1,0,f(x))+LPM(1,0,f(x))] *[b-a]\asymp\left\lvert{\int_{a}^{b} f(x)dx}\right\rvert$$
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
.
$$P(A|B)=\frac{Co.UPM(0,0,A,B,0,0)}{UPM(0,0,B)}$$
If the user is so motivated, detailed arguments and proofs are provided within the following:
Sys.setenv("OMP_THREAD_LIMIT" = "")
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