pm_distPiwiPoweto Module Reference

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"

Detailed Description

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:

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 (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.

See also

pm_distPower
pm_distPareto

Test:

test_pm_distPiwiPoweto

Todo:

Generic interfaces for computing the logarithm of CDF robustly (without numerical rounding) must be added in the future.

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_distPiwiPoweto::MODULE_NAME = "@pm_distPiwiPoweto"

Definition at line 100 of file pm_distPiwiPoweto.F90.