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) PieceWise Power/Pareto distribution (hence the name PiwiPoweto). More...
Data Types | |
interface | getPiwiPowetoCDF |
Generate and return the Cumulative Distribution Function (CDF) of the (Truncated) PiwiPoweto distribution for an input logx within the support of the distribution logLimX(1) <= logx <= logLimX(size(logLimX)) .More... | |
interface | getPiwiPowetoLogPDF |
Generate and return the natural logarithm of the Probability Density Function (PDF) of the (Truncated) PiwiPoweto distribution for an input logx within the support of the distribution logLimX(1) <= logx <= logLimX(n+1) .More... | |
interface | getPiwiPowetoLogPDFNF |
Generate and return the natural logarithm of the normalization factors of the components of the Probability Density Function (PDF) of the (Truncated) PiwiPoweto distribution for the input parameter vectors \((\alpha, x_\mathrm{lim})\). More... | |
interface | setPiwiPowetoCDF |
Return the Cumulative Distribution Function (CDF) of the (Truncated) PiwiPoweto distribution for an input logx within the support of the distribution logLimX(1) <= logx <= logLimX(size(logLimX)) .More... | |
interface | setPiwiPowetoLogPDF |
Return the natural logarithm of the Probability Density Function (PDF) of the (Truncated) PiwiPoweto distribution for an input logx within the support of the distribution logLimX(1) <= logx <= logLimX(n+1) . More... | |
Variables | |
character(*, SK), parameter | MODULE_NAME = "@pm_distPiwiPoweto" |
This module contains classes and procedures for computing various statistical quantities related to the (Truncated) PieceWise Power/Pareto distribution (hence the name PiwiPoweto).
Specifically, this module contains routines for computing the following quantities of the (Truncated) PieceWise Power/Pareto distribution:
The (Truncated) PiwiPoweto distribution is better known by its special case: the Broken Power Law distribution.
The PDF of an \(n\)-piece (Truncated) PiwiPoweto distribution (with \(n > 0\)) over a strictly-positive support \(x \in [0 < x_\mathrm{lim,1}, x_\mathrm{lim,n+1} \leq +\infty)\) is defined with the ascending-ordered scale vector \(x_\mathrm{lim} = \{ x_\mathrm{lim,i}, i = 1 : n + 1\}\) and shape vector \(\alpha = \{\alpha_i, i = 1 : n \}\) as,
\begin{equation} \large \pi(x | \alpha, x_\mathrm{lim}) = \begin{cases} \eta_1 x^{\alpha_1 - 1} &, ~ 0 \leq x_\mathrm{lim,1} \leq x < x_\mathrm{lim,2} ~, \\ \ldots \\ \eta_i x^{\alpha_i - 1} &, ~ x_\mathrm{lim,i} \leq x < x_\mathrm{lim,i+1} ~, \\ \ldots \\ \eta_n x^{\alpha_n - 1} &, ~ x_\mathrm{lim,n} \leq x < x_\mathrm{lim,n+1} \leq +\infty ~, \end{cases} \end{equation}
where the conditions \(\{\alpha_i \in \mathbb{R}, i = 1 : n\}\) and \(\alpha_1 > 0 \lor x_\mathrm{lim,1} > 0\) and \(\alpha_n < 0 \lor x_\mathrm{lim,n+1} < +\infty\) must hold.
These conditions must be met for the PDF to be normalizable.
The component normalization factors \(\eta_i\) are computed according to the following relationship,
\begin{equation} \large \eta_i = \eta_1 \prod_{j = 2}^i ~ x_\mathrm{lim,j}^{(\alpha_{j - 1} - \alpha_j)} ~, ~i = 2:n \end{equation}
where \(\eta_1\) is a normalization factor that properly normalizes the integral of the PDF over its support to unity.
The corresponding Cumulative Distribution Function (CDF) of the (Truncated) PiwiPoweto is,
\begin{equation} \large \mathrm{CDF}(x | \alpha, x_\mathrm{lim}) = \sum_{i = 1}^{n : x_\mathrm{min,n} < x} ~ S_i ~, \end{equation}
where \(S_i ~:~ 1 \leq i \leq n\) is expressed via the following,
\begin{equation} \large S_i = \begin{cases} \frac{\eta_i}{\alpha_i} \log\big( \frac{\min(x, x_{i+1})}{x_i} \big) &, ~\alpha_i = 0 ~, \\ \frac{\eta_i}{\alpha_i} \big( \min(x, x_{i+1})^{\alpha_i} - x_{i}^{\alpha_i} \big) &, ~\alpha_i \neq 0 ~, \end{cases} \end{equation}
where \(\eta_i\) are the normalization factors of the Power-Law components of the distribution.
Final Remarks ⛓
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For details on the naming abbreviations, see this page.
For details on the naming conventions, see this page.
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character(*, SK), parameter pm_distPiwiPoweto::MODULE_NAME = "@pm_distPiwiPoweto" |
Definition at line 100 of file pm_distPiwiPoweto.F90.