The Elements of Variance
What are partial moments? Partial moments can be viewed as the elements of variance. They represent each of the possible interactions between variables. The aggregate of these interactions is commonly represented by variance and covariance measures. However, looking at things individually reveals much more information than possible at the aggregate level.
Univariate
The variance of a variable is simply the sum of the degree 2 partial moments taken from the mean of the distribution, shown in the code below.
Lower Partial Moment in red, Upper Partial Moment in blue:
#R-commands:
> require(NNS)
> set.seed(123); x=rnorm(100); y=rnorm(100)
#Mean:
> mean(x)
[1] 0.09040591
> UPM(1,0,x)-LPM(1,0,x)
[1] 0.09040591
#Variance:
> var(x)
[1] 0.8332328
#Sample Variance:
> UPM(2,mean(x),x)+LPM(2,mean(x),x)
[1] 0.8249005
#Population Variance:
> (UPM(2,mean(x),x)+LPM(2,mean(x),x))*(length(x)/(length(x)-1))
[1] 0.8332328
Multivariate
Co-Lower Partial Moment highlighted:
> cov(x,y)
[1] -0.04372107
> (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))
[1] -0.04372107
# Full Covariance Matrix Deconstruction
> cov.mtx = PM.matrix(LPM_degree = 1, UPM_degree = 1, target = 'mean', variable = cbind(x,y), pop_adj = TRUE)
> cov.mtx
$cupm
x y
x 0.4299250 0.1033601
y 0.1033601 0.5411626
$dupm
x y
x 0.0000000 0.1469182
y 0.1560924 0.0000000
$dlpm
x y
x 0.0000000 0.1560924
y 0.1469182 0.0000000
$clpm
x y
x 0.4033078 0.1559295
y 0.1559295 0.3939005
$cov.matrix
x y
x 0.83323283 -0.04372107
y -0.04372107 0.93506310
# Reassembled Covariance Matrix
> cov.mtx$cupm + cov.mtx$clpm - cov.mtx$dupm - cov.mtx$dlpm
x y
x 0.83323283 -0.04372107
y -0.04372107 0.93506310
# Standard Covariance Matrix
> cov(cbind(x,y))
x y
x 0.83323283 -0.04372107
y -0.04372107 0.93506310
If you feel like reading, the derivation and proof of all the matrices is available here: Cumulative Distribution Functions and UPM/LPM Analysis
Partial Moments in R
Here is a link to some more R-commands demonstrating the basic variance & covariance equivalences above and other classical statistics:
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Why Bother?
Why are partial moments relevant? From a behavioral finance perspective, partial moments are critical in representing the different views an individual has for an above target observation and a below target observation (note the term "target" instead of "mean", yet another benefit versus the variance based statistics). Partial moments also avoid the philosophical inconsistency when a single observation is used as both a reward and a risk in their respective descriptive statistics, mean and variance for instance.
From a statistical standpoint, partial moments remove the underlying reliance upon lines & geometry when trying to determine the relationship between variables. Focusing on the interactions discloses more relationship information than revealed from classic linear based techniques such as Pearson's correlation coefficient, linear regression, and PCA. Partial moments also allow us to do this all nonparametrically, such that we do not have to fit a distribution to the data.
Please review this presentation illustrating the influence of partial moments in nonlinear nonparametric statistics as well as their natural extension into behavioral finance in my research. Article descriptions as well as links are provided. If you would like to learn more, feel free to reach out.
Further Applications
Please check out this follow up post explaining the use of partial moments in various statistics and links to R code for each function.
And please check out this follow up post presenting a visual introduction of how partial moments can capture individual preferences.
Finally, for a finance specific application, see the following post for an explanation of expected partial moments.