Getting Started with NNS: Partial Moments

Fred Viole

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

## [1] 0.09040591
## [1] 0.09040591

Variance

## [1] 0.8332328
## [1] 0.8249005
## [1] 0.8332328
## [1] 0.8332328

Standard Deviation

## [1] 0.9128159
## [1] 0.9128159

Covariance

## [1] -0.04372107
## [1] -0.04372107

Covariance Elements and Covariance Matrix

## $clpm
##           x         y
## x 0.4033078 0.1559295
## y 0.1559295 0.3939005
## 
## $cupm
##           x         y
## x 0.4299250 0.1033601
## y 0.1033601 0.5411626
## 
## $dlpm
##           x         y
## x 0.0000000 0.1469182
## y 0.1560924 0.0000000
## 
## $dupm
##           x         y
## x 0.0000000 0.1560924
## y 0.1469182 0.0000000
## 
## $matrix
##             x           y
## x  0.83323283 -0.04372107
## y -0.04372107  0.93506310

Pearson Correlation

## [1] -0.04953215
## [1] -0.04953215

CDFs (Discrete and Continuous)

## [1] 0.48
## [1] 0.83
## [1] 0.48
## [1] 0.83
## [1] 0.48 0.83

## [1] 0.28
## [1] 0.28 0.73

PDFs

##      Intervals          PDF
##   1: -2.309169 0.0001908930
##   2: -2.264204 0.0003890644
##   3: -2.219239 0.0004040456
##   4: -2.174274 0.0004199092
##   5: -2.129309 0.0004367259
##  ---                       
##  97:  2.007473 0.0014780900
##  98:  2.052438 0.0011915263
##  99:  2.097403 0.0009296535
## 100:  2.142368 0.0008033076
## 101:  2.187333 0.0003484623

Numerical Integration

Partial moments are asymptotic area approximations of \(f(x)\) akin to the familiar Trapezoidal and Simpson’s rules. More observations, more accuracy…

\[[UPM(1,0,f(x))-LPM(1,0,f(x))]\asymp\frac{[F(b)-F(a)]}{[b-a]}\]

## [1] 0.3335

\[0.3333=\frac{\int_{0}^{1} x^2 dx}{1-0}\] 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: