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

knitr::opts_chunk$set(echo = TRUE) require(NNS)  # 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 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(1,1,x,x,mean(x),mean(x))+Co.UPM(1,1,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  ## Covariance cov(x,y) (Co.LPM(1,1,x,y,mean(x),mean(y))+Co.UPM(1,1,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 PM.matrix(LPM.degree = 1,UPM.degree = 1,target = 'mean', variable = cbind(x,y), pop.adj = TRUE)  ## Pearson Correlation cor(x,y) cov.xy=(Co.LPM(1,1,x,y,mean(x),mean(y))+Co.UPM(1,1,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(0,0,x,y,0,0) # Vectorized targets: Co.LPM(0,0,x,y,c(0,1),c(0,1)) # Continuous CDF: plot(sort(x),LPM.ratio(1,sort(x),x),type = 'l',col='blue',lwd=3,xlab="x")  ## PDFs NNS.PDF(degree=1,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... $$[UPM(1,0,f(x))-LPM(1,0,f(x))]\asymp\frac{[F(b)-F(a)]}{[b-a]}$$ x=seq(0,1,.001);y=x^2 UPM(1,0,y)-LPM(1,0,y)  $$0.3333=\frac{\int_{0}^{1} x^2 dx}{1-0}$$ For the total area, not just the definite integral, simply sum the partial moments: $$[UPM(1,0,f(x))+LPM(1,0,f(x))]\asymp\left\lvert{\int_{a}^{b} f(x)dx}\right\rvert$$ ## 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 Dec. 8, 2017, 5:03 p.m.