# Getting Started with NNS: Partial Moments" In NNS: Nonlinear Nonparametric Statistics

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.