ParaMonte Fortran 2.0.0
Parallel Monte Carlo and Machine Learning Library
See the latest version documentation. |
This module contains classes and procedures for computing various statistical quantities related to the Kolmogorov distribution. More...
Data Types | |
type | distKolm_type |
This is the derived type for signifying distributions that are of type Kolmogorov as defined in the description of pm_distKolm. More... | |
interface | getKolmCDF |
Generate and return the Cumulative Distribution Function (CDF) of the Kolmogorov distribution for an input x within the support of the distribution \(X \in [0, +\infty)\). More... | |
interface | getKolmPDF |
Generate and return the Probability Density Function (PDF) of the Kolmogorov distribution for an input x within the support of the distribution \(X \in [0, +\infty)\). More... | |
interface | getKolmQuan |
Generate and return a scalar (or array of arbitrary rank) of the quantile corresponding to the specified CDF of Kolmogorov distribution. More... | |
interface | getKolmRand |
Generate and return a scalar (or array of arbitrary rank) of the random value(s) from the Kolmogorov distribution. More... | |
interface | setKolmCDF |
Return the Cumulative Distribution Function (CDF) of the Kolmogorov distribution for an input x within the support of the distribution \(X \in [0, +\infty)\). More... | |
interface | setKolmPDF |
Return the Probability Density Function (PDF) of the Kolmogorov distribution for an input x within the support of the distribution \(X \in [0, +\infty)\). More... | |
interface | setKolmQuan |
Return a scalar (or array of arbitrary rank) of the quantile corresponding to the specified CDF of Kolmogorov distribution. More... | |
interface | setKolmRand |
Return a scalar (or array of arbitrary rank) of the random value(s) from the Kolmogorov distribution. More... | |
Variables | |
character(*, SK), parameter | MODULE_NAME = "@pm_distKolm" |
This module contains classes and procedures for computing various statistical quantities related to the Kolmogorov distribution.
Specifically, this module contains routines for computing the following quantities of the Kolmogorov distribution:
The Kolmogorov distribution is the distribution of the random variable
\begin{equation} \large X = \sup_{t\in [0,1]}|B(t)| ~, \end{equation}
where \(B(t)\) is the Brownian bridge.
The cumulative distribution function (CDF) of \(K\) over the non-negative support \(X \in [0, +\infty)\) is given by,
\begin{equation} \large \ms{CDF}(X\leq x) = 1 - 2\sum_{k=1}^{\infty}(-1)^{k-1}e^{-2k^{2}x^{2}} = {\frac{\sqrt{2\pi}}{x}}\sum_{k=1}^{\infty}e^{-(2k-1)^{2}\pi ^{2}/(8x^{2})} ~, \end{equation}
which can also be expressed by the Jacobi theta function \(\vartheta_{01}(z=0;\tau =2ix^{2}/\pi)\).
The distribution is named after Andrey Kolmogorov.
The corresponding PDF \(\pi(\cdot)\) of the Kolmogorov distribution can be obtained by taking the derivative of the CDF of the distribution with respect to \(x\),
\begin{eqnarray} \large \pi(x) &=& 8x \sum_{k=1}^{\infty}(-1)^{k-1} k^2 e^{-2k^{2}x^{2}} \nonumber \\ &=& {\frac{\sqrt{2\pi}}{x^4}}\sum_{k=1}^{\infty} \left[(2k - 1)^2\pi^2/4 - x^2\right] e^{-(2k-1)^{2}\pi ^{2}/(8x^{2})} \nonumber \\ &=& {\frac{2\sqrt{2\pi}}{x^2}}\sum_{k=1}^{\infty} \left[\frac{(2k - 1)^2\pi^2}{8x^2} - \frac{1}{2}\right] e^{-(2k-1)^{2}\pi ^{2}/(8x^{2})} ~, \end{eqnarray}
where the symbol \(\pi\) on the righthand side represents the mathematical number \(\pi = 3.1415\ldots\).
There is no close form expression for the inverse CDF of the Kolmogorov distribution.
However, root finding methods can be used to refine an initial guess toward an acceptable answer.
In the most naive scenario, the quantile function can be used for random number generation.
Final Remarks ⛓
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For details on the naming conventions, see this page.
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character(*, SK), parameter pm_distKolm::MODULE_NAME = "@pm_distKolm" |
Definition at line 90 of file pm_distKolm.F90.