pm_distPower Module Reference

This module contains classes and procedures for computing various statistical quantities related to the (Truncated) Power distribution. More...

Data Types
type	distPower_type
	This is the derived type for signifying distributions that are of type Power as defined in the description of pm_distPower. More...
interface	getPowerLogCDF
	Generate and return the natural logarithm of the Cumulative Distribution Function (CDF) of the (Truncated) Power distribution for an input log(x) within the support of the distribution \(x \in [0 < x_\mathrm{min}, x_\mathrm{max} < +\infty]\). More...
interface	getPowerLogCDFNF
	Generate and return the natural logarithm of the normalization factor of the CDF of the (Truncated) Power distribution for an input parameter set \((\alpha, x_\mathrm{min}, x_\mathrm{max})\). More...
interface	getPowerLogPDF
	Generate and return the natural logarithm of the Probability Density Function (PDF) of the (Truncated) Power distribution for an input log(x) within the support of the distribution \(x \in [0 < x_\mathrm{min}, x_\mathrm{max} < +\infty]\). More...
interface	getPowerLogPDFNF
	Generate and return the natural logarithm of the normalization factor of the PDF of the (Truncated) Power distribution for an input parameter set \((\alpha, x_\mathrm{min}, x_\mathrm{max})\). More...
interface	getPowerLogQuan
	Generate and return a scalar (or array of arbitrary rank) of the natural logarithm(s) of quantile corresponding to the specified CDF of (Truncated) Power distribution with parameters \((\alpha, x_\mathrm{min}, x_\mathrm{max})\). More...
interface	getPowerLogRand
	Generate and return a scalar (or array of arbitrary rank) of the natural logarithm(s) of random value(s) from the (Truncated) Power distribution with parameters \((\alpha, x_\mathrm{min}, x_\mathrm{max})\). More...
interface	setPowerLogCDF
	Return the natural logarithm of the Cumulative Distribution Function (CDF) of the (Truncated) Power distribution for an input log(x) within the support of the distribution \(x \in [0 < x_\mathrm{min}, x_\mathrm{max} < +\infty]\). More...
interface	setPowerLogPDF
	Return the natural logarithm of the Probability Density Function (PDF) of the (Truncated) Power distribution for an input log(x) within the support of the distribution \(x \in [0 < x_\mathrm{min}, x_\mathrm{max} < +\infty]\). More...
interface	setPowerLogQuan
	Return a scalar (or array of arbitrary rank) of the natural logarithm(s) of quantile corresponding to the specified CDF of (Truncated) Power distribution with parameters \((\alpha, x_\mathrm{min}, x_\mathrm{max})\). More...
interface	setPowerLogRand
	Return a scalar (or array of arbitrary rank) of the natural logarithm(s) of random value(s) from the (Truncated) Power distribution with parameters \((\alpha, x_\mathrm{min}, x_\mathrm{max})\). More...

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

Detailed Description

This module contains classes and procedures for computing various statistical quantities related to the (Truncated) Power distribution.

Specifically, this module contains routines for computing the following quantities of the (Truncated) Power distribution:

1. the Probability Density Function (PDF)
2. the Cumulative Distribution Function (CDF)
3. the Random Number Generation from the distribution (RNG)
4. the Inverse Cumulative Distribution Function (ICDF) or the Quantile Function

The PDF of the (Truncated) Power distribution over a strictly-positive support \(x \in [x_\mathrm{min}, x_\mathrm{max}]\) is defined with the three (shape, scale, scale) parameters \((\alpha, x_\mathrm{min}, x_\mathrm{max})\) as,

\begin{equation} \large \pi(x | \alpha, x_\mathrm{min}, x_\mathrm{max}) = \eta(\alpha, x_\mathrm{min}, x_\mathrm{max}) ~ x^{\alpha - 1} ~, \end{equation}

where \(\mathbf{0 < \alpha < +\infty}\) and \(\mathbf{0 < x_\mathrm{min} \leq x \leq x_\mathrm{max} < +\infty}\) hold, and \(\eta(\alpha, x_\mathrm{min}, x_\mathrm{max})\) is the normalization factor of the PDF,

\begin{equation} \large \eta(\alpha, x_\mathrm{min}, x_\mathrm{max}) = \frac{\alpha}{x_\mathrm{max}^\alpha - x_\mathrm{min}^\alpha} ~, \end{equation}

When \(x_\mathrm{min} \rightarrow 0\), the Truncated Power distribution simplifies to the Power Distribution with PDF.

\begin{equation} \large \lim_{x_\mathrm{min} \rightarrow 0} \pi(x | \alpha, x_\mathrm{min}, x_\mathrm{max}) = \frac{\alpha}{x_\mathrm{max}^\alpha} x^{\alpha - 1} ~, \end{equation}

The equation for \(\eta(\cdot)\) for the Power distribution simplifies to,

\begin{equation} \large \lim_{x_\mathrm{min} \rightarrow 0} \eta(\alpha, x_\mathrm{min}, x_\mathrm{max}) = \alpha x_\mathrm{max}^{-\alpha} ~,~ 0 < \alpha < +\infty. \end{equation}

The corresponding CDF of the (Truncated) Power distribution is given by,

\begin{equation} \large \mathrm{CDF}(x | \alpha, x_\mathrm{min}, x_\mathrm{max}) = \zeta(\alpha, x_\mathrm{min}, x_\mathrm{max}) \bigg[\bigg(\frac{x}{x_\mathrm{min}}\bigg)^\alpha - 1\bigg] ~, \end{equation}

where \(\mathbf{0 < \alpha < +\infty}\) and \(\mathbf{0 < x_\mathrm{min} \leq x \leq x_\mathrm{max} < +\infty}\) hold, and \(\zeta(\alpha, x_\mathrm{min}, x_\mathrm{max})\) is the normalization factor of the CDF,

\begin{eqnarray} \large \zeta(\alpha, x_\mathrm{min}, x_\mathrm{max}) &=& \frac{x_\mathrm{min}^\alpha}{x_\mathrm{max}^\alpha - x_\mathrm{min}^\alpha} ~, \\ &=& \frac{x_\mathrm{min}^\alpha}{\alpha} \eta(\alpha, x_\mathrm{min}, x_\mathrm{max}) ~, \end{eqnarray}

where \(\eta(\cdot)\) is the normalization factor of the PDF.

When \(x_\mathrm{min} \rightarrow 0\), the Truncated Power distribution simplifies to the Power Distribution with CDF,

\begin{equation} \large \lim_{x_\mathrm{min} \rightarrow 0} \mathrm{CDF}(x | \alpha, x_\mathrm{min}, x_\mathrm{max}) = \bigg(\frac{x}{x_\mathrm{max}}\bigg)^\alpha ~, \end{equation}

with,

\begin{equation} \large \lim_{x_\mathrm{min} \rightarrow 0} \zeta(\alpha, x_\mathrm{min}, x_\mathrm{max}) = x_\mathrm{max}^{-\alpha} ~. \end{equation}

The corresponding Inverse CDF of the (Truncated) Power distribution is given by,

\begin{equation} \large Q(\mathrm{CDF}(x); \alpha, x_\mathrm{min}, x_\mathrm{max}) \equiv x = x_\mathrm{min} \bigg(1 + \frac{\mathrm{CDF}(x)}{\zeta(\alpha, x_\mathrm{min}, x_\mathrm{max})}\bigg)^{\frac{1}{\alpha}} ~, \end{equation}

where \(\mathbf{0 < \alpha < +\infty}\) and \(\mathbf{0 < x_\mathrm{min} \leq Q(\mathrm{CDF}(x); \alpha, x_\mathrm{min}, x_\mathrm{max}) \leq x_\mathrm{max} < +\infty}\) hold, and \(\zeta(\alpha, x_\mathrm{min}, x_\mathrm{max})\) is the normalization factor of the CDF,

\begin{equation} \large \zeta(\alpha, x_\mathrm{min}, x_\mathrm{max}) = \frac{x_\mathrm{min}^\alpha}{x_\mathrm{max}^\alpha - x_\mathrm{min}^\alpha} ~. \end{equation}

When \(x_\mathrm{max} \rightarrow +\infty\), the Truncated Power distribution simplifies to the Power Distribution, with,

\begin{equation} \large Q(\mathrm{CDF}(x); \alpha, x_\mathrm{min}, x_\mathrm{max}) \equiv x = x_\mathrm{max} \big(\mathrm{CDF}(x)\big)^{\frac{1}{\alpha}} = \bigg(\frac{\mathrm{CDF}(x)}{\zeta(\alpha, x_\mathrm{max})}\bigg)^{\frac{1}{\alpha}} ~, \end{equation}

Random Number Generation

Assuming that \(U \in [0, 1)\) is a uniformly-distributed random variate, the transformed random variable,

\begin{equation} \large x = x_\mathrm{min} \bigg(1 + \frac{U}{\zeta(\alpha, x_\mathrm{min}, x_\mathrm{max})}\bigg)^{\frac{1}{\alpha}} ~, \end{equation}

follows a Truncated Power distribution with parameters \((\alpha, x_\mathrm{min}, x_\mathrm{max}\)) where \(\mathbf{0 < \alpha < +\infty}\) and \(\mathbf{0 < x_\mathrm{min} \leq x \leq x_\mathrm{max} < +\infty}\) hold, and \(\zeta(\alpha, x_\mathrm{min}, x_\mathrm{max})\) is the normalization factor of the CDF,

\begin{equation} \large \zeta(\alpha, x_\mathrm{min}, x_\mathrm{max}) = \frac{x_\mathrm{min}^\alpha}{x_\mathrm{max}^\alpha - x_\mathrm{min}^\alpha} ~. \end{equation}

When \(x_\mathrm{max} \rightarrow +\infty\), the Truncated Power distribution simplifies to the Power Distribution, with,

\begin{equation} \large x = x_\mathrm{max} U^{\frac{1}{\alpha}} = \bigg(\frac{U}{\zeta(\alpha, x_\mathrm{max})}\bigg)^{\frac{1}{\alpha}} ~, \end{equation}

Remarks

See pm_distPareto for the case of \(-\infty < \alpha < 0\).
See pm_distPoweto for the case of \(\alpha = 0\).

See also

pm_distPower
pm_distPareto
pm_distPoweto

Test:

test_pm_distPower

Final Remarks ⛓

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Copyright

Computational Data Science Lab

Author:

Amir Shahmoradi, Oct 16, 2009, 11:14 AM, Michigan

Variable Documentation

MODULE_NAME

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

Definition at line 144 of file pm_distPower.F90.