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.
## [1] 0.09040591
## [1] 0.09040591
## [1] 0.8332328
## [1] 0.8249005
## [1] 0.8332328
# 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))
## [1] 0.8332328
## [1] 0.9128159
## [1] 0.9128159
## [1] -0.04372107
## [1] -0.04372107
## $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
## [1] -0.04953215
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)
## [1] -0.04953215
## [1] 0.48
## [1] 0.83
## [1] 0.48
## [1] 0.83
## [1] 0.48 0.83
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")
## [1] 0.28
## [1] 0.28 0.73
# Continuous CDF:
plot(sort(x), LPM.ratio(1, sort(x), x), type = "l", col = "blue", lwd = 3, xlab = "x")
## 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
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: \[[UPM(1,0,f(x))+LPM(1,0,f(x))]\asymp\left\lvert{\int_{a}^{b} f(x)dx}\right\rvert\]
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: