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 (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" |
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:
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}
Final Remarks ⛓
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character(*, SK), parameter pm_distPower::MODULE_NAME = "@pm_distPower" |
Definition at line 144 of file pm_distPower.F90.