pm_distPareto::getParetoLogPDFNF Interface Reference

Generate and return the natural logarithm of the normalization factor of the PDF of the (Truncated) Pareto distribution for an input parameter set \((\alpha, x_\mathrm{min}, x_\mathrm{max})\). More...

Detailed Description

Generate and return the natural logarithm of the normalization factor of the PDF of the (Truncated) Pareto distribution for an input parameter set \((\alpha, x_\mathrm{min}, x_\mathrm{max})\).

The normalization factor \(\eta(\alpha, x_\mathrm{min}, x_\mathrm{max})\) of the PDF of the Pareto distribution is,

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

When \(x_\mathrm{max} \rightarrow +\infty\), the Truncated Pareto distribution simplifies to the Pareto Distribution.
The equation for \(\eta(\cdot)\) for the Pareto distribution simplifies to,

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

See the documentation of pm_distPareto for more information on the (Truncated) Pareto distribution.

The primary use of this interface is to compute the normalization factor of the PDF of the (Truncated) Pareto distribution for a fixed set of parameters and use it in subsequent repeated calculations of the properties of the (Truncated) Pareto distribution to improve the runtime performance by eliminating redundant calculations.

Parameters

[in]	alpha	: The input scalar or array of the same shape as other array-like arguments, of type real of kind any supported by the processor (e.g., RK, RK32, RK64, or RK128), containing the shape parameter of the distribution.
[in]	logMinX	: The input scalar or array of the same shape as other array-like arguments, of the same type and kind as alpha, containing the natural logarithm of the first scale parameter of the distribution, representing the minimum of the support of the distribution.
[in]	logMaxX	: The input scalar or array of the same shape as other array-like arguments, of the same type and kind as alpha, containing the natural logarithm of the second scale parameter of the distribution, representing the maximum of the support of the distribution. (optional, default = \(+\infty\))

Returns

logPDFNF : The output scalar or array of the same shape as any input array-like argument, of the same type and kind as the input argument alpha, containing the natural logarithm of the normalization factor of the PDF of (Truncated) Pareto distribution.

Possible calling interfaces ⛓

use pm_distPareto, only: getParetoLogPDFNF
 
logPDFNF = getParetoLogPDFNF(alpha, logMinX) ! Pareto distribution.
logPDFNF = getParetoLogPDFNF(alpha, logMinX, logMaxX) ! Truncated Pareto distribution.
!

Warning

The condition alpha < 0 must hold for the corresponding input arguments.
The condition logMinX < logMaxX must hold for the corresponding input arguments.
These conditions are verified only if the library is built with the preprocessor macro CHECK_ENABLED=1.

The pure procedure(s) documented herein become impure when the ParaMonte library is compiled with preprocessor macro CHECK_ENABLED=1.
By default, these procedures are pure in release build and impure in debug and testing builds.

Remarks

The procedures under discussion are elemental.

See also

getParetoLogPDF
setParetoLogPDF

Example usage ⛓

program example
 
    use pm_kind, only: IK
    use pm_kind, only: SK
    use pm_io, only: display_type
    use pm_arraySpace, only: getLogSpace
    use pm_distPareto, only: getParetoLogPDFNF
 
    implicit none
 
    real :: logPDFNF(3)
 
    type(display_type) :: disp
    disp = display_type(file = "main.out.F90")
 
    call disp%skip()
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("! Compute the natural logarithm of the normalization factor of the (Truncated) Pareto distribution PDF.")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%skip()
 
    call disp%skip()
    call disp%show("logPDFNF(1) = getParetoLogPDFNF(alpha = -1., logMinX = 1.)")
                    logPDFNF(1) = getParetoLogPDFNF(alpha = -1., logMinX = 1.)
    call disp%show("logPDFNF(1)")
    call disp%show( logPDFNF(1) )
    call disp%skip()
 
    call disp%skip()
    call disp%show("logPDFNF(1:3) = getParetoLogPDFNF(alpha = -2., logMinX = log([1., 2., 3.]))")
                    logPDFNF(1:3) = getParetoLogPDFNF(alpha = -2., logMinX = log([1., 2., 3.]))
    call disp%show("logPDFNF(1:3)")
    call disp%show( logPDFNF(1:3) )
    call disp%skip()
 
    call disp%skip()
    call disp%show("logPDFNF(1:3) = getParetoLogPDFNF(alpha = -[+2., +3., +4.], logMinX = log([1., 2., 3.]))")
                    logPDFNF(1:3) = getParetoLogPDFNF(alpha = -[+2., +3., +4.], logMinX = log([1., 2., 3.]))
    call disp%show("logPDFNF(1:3)")
    call disp%show( logPDFNF(1:3) )
    call disp%skip()
 
    call disp%skip()
    call disp%show("logPDFNF(1:3) = getParetoLogPDFNF(alpha = -[+2., +3., +4.], logMinX = log([1., 2., 3.]), logMaxX = log(20.))")
                    logPDFNF(1:3) = getParetoLogPDFNF(alpha = -[+2., +3., +4.], logMinX = log([1., 2., 3.]), logMaxX = log(20.))
    call disp%show("logPDFNF(1:3)")
    call disp%show( logPDFNF(1:3) )
    call disp%skip()
 
    !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    ! Output an example array for visualization.
    !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
    block
        integer :: fileUnit, i
        real, allocatable :: alpha(:), logPDFNF(:)
        alpha = -getLogSpace(+10., -10., count = 500_IK)
        logPDFNF = getParetoLogPDFNF(alpha, logMinX = -1., logMaxX = +1.)
        open(newunit = fileUnit, file = "getParetoLogPDFNF.RK.txt")
        write(fileUnit,"(2(g0,:,' '))") (-alpha(i), exp(logPDFNF(i)), i = 1, size(logPDFNF))
        close(fileUnit)
    end block
 
end program example

Example Unix compile command via Intel ifort compiler ⛓

#!/usr/bin/env sh
rm main.exe
ifort -fpp -standard-semantics -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
./main.exe

Example Windows Batch compile command via Intel ifort compiler ⛓

del main.exe
set PATH=..\..\..\lib;%PATH%
ifort /fpp /standard-semantics /O3 /I:..\..\..\include main.F90 ..\..\..\lib\libparamonte*.lib /exe:main.exe
main.exe

Example Unix / MinGW compile command via GNU gfortran compiler ⛓

#!/usr/bin/env sh
rm main.exe
gfortran -cpp -ffree-line-length-none -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
./main.exe

Example output ⛓

 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! Compute the natural logarithm of the normalization factor of the (Truncated) Pareto distribution PDF.
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
logPDFNF(1) = getParetoLogPDFNF(alpha = -1., logMinX = 1.)
logPDFNF(1)
+1.00000000
 
 
logPDFNF(1:3) = getParetoLogPDFNF(alpha = -2., logMinX = log([1., 2., 3.]))
logPDFNF(1:3)
+0.693147182, +2.07944155, +2.89037180
 
 
logPDFNF(1:3) = getParetoLogPDFNF(alpha = -[+2., +3., +4.], logMinX = log([1., 2., 3.]))
logPDFNF(1:3)
+0.693147182, +3.17805386, +5.78074360
 
 
logPDFNF(1:3) = getParetoLogPDFNF(alpha = -[+2., +3., +4.], logMinX = log([1., 2., 3.]), logMaxX = log(20.))
logPDFNF(1:3)
+0.695650339, +3.17905426, +5.78125000
 
 

Postprocessing of the example output ⛓

#!/usr/bin/env python
 
import matplotlib.pyplot as plt
import pandas as pd
import numpy as np
import glob
import sys
 
fontsize = 17
 
marker ={ "CK" : "-"
        , "IK" : "."
        , "RK" : "-"
        }
xlab =  { "CK" : r"$-\alpha$ ( real/imaginary )"
        , "IK" : r"$-\alpha$ ( integer-valued )"
        , "RK" : r"$-\alpha$ ( real-valued )"
        }
 
for kind in ["IK", "CK", "RK"]:
 
    pattern = "*." + kind + ".txt"
    fileList = glob.glob(pattern)
    if len(fileList) == 1:
 
        df = pd.read_csv(fileList[0], delimiter = " ")
 
        fig = plt.figure(figsize = 1.25 * np.array([6.4, 4.8]), dpi = 200)
        ax = plt.subplot()
 
        if kind == "CK":
            plt.plot( df.values[:, 0]
                    , df.values[:,2]
                    , marker[kind]
                    , color = "r"
                    )
            plt.plot( df.values[:,1]
                    , df.values[:,3]
                    , marker[kind]
                    , color = "blue"
                    )
        else:
            plt.plot( df.values[:, 0]
                    , df.values[:,1]
                    , marker[kind]
                    , color = "r"
                    )
 
        plt.xticks(fontsize = fontsize - 2)
        plt.yticks(fontsize = fontsize - 2)
        ax.set_xlabel(xlab[kind], fontsize = fontsize)
        ax.set_ylabel("Normalization Factor of the PDF", fontsize = fontsize)
        ax.set_xscale("log")
        ax.set_yscale("log")
 
        plt.grid(visible = True, which = "both", axis = "both", color = "0.85", linestyle = "-")
        ax.tick_params(axis = "y", which = "minor")
        ax.tick_params(axis = "x", which = "minor")
 
        plt.tight_layout()
        plt.savefig(fileList[0].replace(".txt",".png"))
 
    elif len(fileList) > 1:
 
        sys.exit("Ambiguous file list exists.")

Visualization of the example output ⛓

Test:

test_pm_distPareto

Todo:

Very Low Priority: This generic interface can be extended to complex arguments.

Final Remarks ⛓

---

If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.
For details on the naming abbreviations, see this page.
For details on the naming conventions, see this page.
This software is distributed under the MIT license with additional terms outlined below.

1. If you use any parts or concepts from this library to any extent, please acknowledge the usage by citing the relevant publications of the ParaMonte library.
   
2. If you regenerate any parts/ideas from this library in a programming environment other than those currently supported by this ParaMonte library (i.e., other than C, C++, Fortran, MATLAB, Python, R), please also ask the end users to cite this original ParaMonte library.
   

This software is available to the public under a highly permissive license.
Help us justify its continued development and maintenance by acknowledging its benefit to society, distributing it, and contributing to it.

Copyright

Computational Data Science Lab

Author:

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

Definition at line 272 of file pm_distPareto.F90.

---

The documentation for this interface was generated from the following file:

• src/fortran/main/pm_distPareto.F90