pm_distanceKolm Module Reference

This module contains classes and procedures for computing the Kolmogorov statistical distance. More...

Data Types
interface	getDisKolm
	Generate and return the Kolmogorov distance of a sample1 of size nsam1 from another sample sample2 of size nsam2 or the CDF of the Uniform or a custom reference distribution. More...
interface	setDisKolm
	Return the Kolmogorov distance of a sample1 of size nsam1 from another sample sample2 of size nsam2 or the CDF of the Uniform or a custom reference distribution. More...

Variables
character(*, SK), parameter	MODULE_NAME = "@pm_distanceKolm"

Detailed Description

This module contains classes and procedures for computing the Kolmogorov statistical distance.

The Kolmogorov distance of a univariate observational sample from another univariate observational sample is the largest separation between the Empirical Distribution Functions of the two samples.
Formally, the empirical distribution function \(F_n\) for \(n\) independent and identically distributed (i.i.d.) ordered observations \(X_i\) is defined as,

\begin{equation} F_{n}(x) = {\frac{{\text{number of (elements in the sample}} \leq x)}{n}} = {\frac{1}{n}} \sum_{i=1}^{n}1_{(-\infty ,x]}(X_{i}) ~, \end{equation}

where \(1_{(-\infty ,x]}(X_{i})\) is the indicator function, equal to \(1\) if \(X_{i}\leq x\) and equal to \(0\) otherwise.
The Kolmogorov–Smirnov distance (or statistic) for a given cumulative distribution function \(F(x)\) is,

\begin{equation} D_{n} = \sup_{x}|F_{n}(x) - F(x)| ~, \end{equation}

where \(\sup_x\) is the supremum of the set of distances.
Intuitively, the statistic takes the largest absolute difference between the two distribution functions across all \(x\) values.
By the Glivenko–Cantelli theorem, if the sample comes from distribution \(F(x)\), then \(D_n\) converges to \(0\) almost surely in the limit when \(n\) goes to infinity.
Kolmogorov strengthened this result, by effectively providing the rate of this convergence through the definition of the Kolmogorov distribution.

See also

pm_distKolm

Test:

test_pm_distanceKolm

Final Remarks ⛓

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Copyright

Computational Data Science Lab

Author:

Amir Shahmoradi, March 22, 2012, 2:21 PM, National Institute for Fusion Studies, The University of Texas at Austin

Variable Documentation

MODULE_NAME

character(*, SK), parameter pm_distanceKolm::MODULE_NAME = "@pm_distanceKolm"

Definition at line 61 of file pm_distanceKolm.F90.