pm_distPiwiPoweto::getPiwiPowetoCDF Interface Reference

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)).
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Detailed Description

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)).

See the documentation of pm_distPiwiPoweto for more information on the (Truncated) PiwiPoweto distribution.

Parameters

[in]	logx	: The input scalar of type real of kind any supported by the processor (e.g., RK, RK32, RK64, or RK128), containing the natural logarithm of the x value within the support of the distribution at which the CDF must be computed.
[in]	alpha	: The input vector of the same type and kind as logx, of the same size n as the number of the power-law components of the distribution, containing the shape parameter of the distribution (i.e., the exponents of the power-law components of the distribution).
[in]	logLimX	: The input vector of the same type and kind as alpha, of size size(alpha) + 1 containing the natural logarithm of the scale parameters (i.e., the break points, or the limits) of the n power-law components of the distribution. The scale parameter(s) must be in ascending order, such that logLimX(1) <= x <= logLimX(size(logLimX)). Setting logLimX(1) <= -log(huge(logLimX)) effectively implies a left-opened semi-infinite support for the distribution. Setting logLimX(size(logLimX)) >= log(huge(logLimX)) effectively implies a right-opened semi-infinite support for the distribution.
[in]	logPDFNF	: The input vector of the same type, kind, and size as alpha, containing the natural logarithm of the normalization factors ( \(\eta\)) of power-law components of the distribution of the (Truncated) PiwiPoweto distribution. Specifying this argument when calling this procedure repeatedly with fixed \((\alpha, x_\mathrm{lim})\) parameters significantly improves the runtime performance. (optional. It must be present if and only if cumSumArea is also present. Default = getPiwiPowetoLogPDFNF(alpha, logLimX))
[in]	cumSumArea	: The output vector of the same type, kind, and size as logLimX, each element of which corresponds to cumulative area underneath the distribution from the minimum of the support exp(logLimX(1)) to the corresponding element of exp(logLimX). By definition, cumSumArea(1) == 0. and cumSumArea(size(cumSumArea)) == 1., and isAscending(cumSumArea) hold. This output vector is a side-product of the computation of the normalization factors. It is also required for random number generation from the (Truncated) PiwiPoweto distribution. Precomputing and supplying this vector to the random number generator routines significantly improves the runtime performance. (optional. It must be present if and only if logPDFNF is also present. The default is set by getPiwiPowetoLogPDFNF(alpha, logLimX, cumSumArea).)

Returns

cdf : The output scalar of the same type and kind the input argument logx, containing the Cumulative Distribution Function (CDF) at the specified point within the support of the distribution.

Possible calling interfaces ⛓

use pm_distPiwiPoweto, only: getPiwiPowetoCDF
 
cdf = getPiwiPowetoCDF(logx, alpha(1:n), logLimX(1:n+1))
cdf = getPiwiPowetoCDF(logx, alpha(1:n), logLimX(1:n+1), logPDFNF(1:n), cumSumArea(1:n+1))

Warning

The condition size(alpha) > 0 must hold for the corresponding input arguments.
The condition size(logLimX) == size(alpha) + 1 must hold for the corresponding input arguments.
The condition size(cumSumArea) == size(alpha) + 1 must hold for the corresponding input arguments.
The condition size(logPDFNF) == size(alpha) must hold for the corresponding input arguments.
The conditions logLimX(1) <= logx .and. logx <= logLimX(size(logLimX)) must hold for the corresponding input arguments.
These conditions are verified only if the library is built with the preprocessor macro CHECK_ENABLED=1.

Remarks

The procedures under discussion are impure.

See also

setPiwiPowetoCDF

Example usage ⛓

program example
 
    use pm_kind, only: SK, IK, LK
    use pm_distPiwiPoweto, only: getPiwiPowetoLogPDFNF
    use pm_distPiwiPoweto, only: getPiwiPowetoCDF
    use pm_arraySpace, only: setLinSpace
    use pm_io, only: display_type
 
    implicit none
 
    integer(IK), parameter  :: NP = 999_IK
    real                    :: logx(NP), CDF(NP), cumSumArea(NP)
 
    type(display_type)      :: disp
    disp = display_type(file = "main.out.F90")
 
    call disp%skip()
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("! Compute the Cumulative Distribution Function (CDF) of the (truncated) PiwiPoweto distribution.")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%skip()
 
    call disp%skip()
    call disp%show(∞"cdf = getPiwiPowetoCDF(logx = log(3.), alpha = [-2.], logLimX = [log(2.), log(huge(0.))]) ! left-truncated PiwiPoweto distribution at [2., +)")
                    cdf = getPiwiPowetoCDF(logx = log(3.), alpha = [-2.], logLimX = [log(2.), log(huge(0.))])
    call disp%show("cdf")
    call disp%show( cdf )
    call disp%skip()
 
    call disp%skip()
    call disp%show("cdf = getPiwiPowetoCDF(logx = log(2.5), alpha = [1., 0.], logLimX = [0.5, 1., 1.5]) ! doubly-truncated PiwiPoweto distribution at `[0.5, 1.5]`.")
                    cdf = getPiwiPowetoCDF(logx = log(2.5), alpha = [1., 0.], logLimX = [0.5, 1., 1.5]) ! doubly-truncated PiwiPoweto distribution at `[0.5, 1.5]`.
    call disp%show("cdf")
    call disp%show( cdf )
    call disp%skip()
 
    call disp%skip()
    call disp%show("cdf = getPiwiPowetoCDF(logx = log(2.5), alpha = [1., 0., -1.5], logLimX = [0.5, 1., 1.5, log(huge(0.))]) ! (truncated) PiwiPoweto with Semi-Infinite support.")
                    cdf = getPiwiPowetoCDF(logx = log(2.5), alpha = [1., 0., -1.5], logLimX = [0.5, 1., 1.5, log(huge(0.))]) ! (truncated) PiwiPoweto with Semi-Infinite support.
    call disp%show("cdf")
    call disp%show( cdf )
    call disp%skip()
 
    call disp%skip()
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("! Expedite repeated PDF computations by precomputing the normalization factors.")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%skip()
 
    call disp%skip()
    call disp%show("cdf = getPiwiPowetoCDF(logx = log(2.5), alpha = [1., 0., -1.5], logLimX = [0.5, 1., 1.5, log(huge(0.))], logPDFNF = getPiwiPowetoLogPDFNF(alpha = [1., 0., -1.5], logLimX = [0.5, 1., 1.5, log(huge(0.))], cumSumArea = cumSumArea(1:4)), cumSumArea = cumSumArea(1:4))")
                    cdf = getPiwiPowetoCDF(logx = log(2.5), alpha = [1., 0., -1.5], logLimX = [0.5, 1., 1.5, log(huge(0.))], logPDFNF = getPiwiPowetoLogPDFNF(alpha = [1., 0., -1.5], logLimX = [0.5, 1., 1.5, log(huge(0.))], cumSumArea = cumSumArea(1:4)), cumSumArea = cumSumArea(1:4))
    call disp%show("cdf")
    call disp%show( cdf )
    call disp%skip()
 
    !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    ! Output an example CDF array for visualization.
    !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
    block
        real, parameter :: LOG_HUGE = log(huge(0.))
        real, allocatable :: cumSumArea(:), alpha(:), logLimX(:)
        integer(IK) :: fileUnit, i
        real :: CDF(4)
        alpha = [3., 1., -1., -5.]
        logLimX = log([2., 5., 10., 15.])
        call setLinSpace(logx, x1 = log(0.001), x2 = log(20.), fopen = .true._LK, lopen = .true._LK)
        if (allocated(cumSumArea)) deallocate(cumSumArea); allocate(cumSumArea(size(logLimX)+2))
        open(newunit = fileUnit, file = "getPiwiPowetoCDF.RK.txt")
        do i = 1, NP
            CDF(1) = getPiwiPowetoCDF(logx(i), [alpha(1:2), 0., alpha(3:4)], [-LOG_HUGE, logLimX(1:4), LOG_HUGE], getPiwiPowetoLogPDFNF([alpha(1:2), 0., alpha(3:4)], [-LOG_HUGE, logLimX(1:4), LOG_HUGE], cumSumArea(1:6)), cumSumArea(1:6)) ! PiwiPoweto
            if (logx(i) > logLimX(1)) then
                CDF(2) = getPiwiPowetoCDF(logx(i), alpha(2:4), [logLimX(1:3), LOG_HUGE], getPiwiPowetoLogPDFNF(alpha(2:4), [logLimX(1:3), LOG_HUGE], cumSumArea(1:4)), cumSumArea(1:4)) ! left-truncated PiwiPoweto
            else
                CDF(2) = 0.
            end if
            if (logx(i) < logLimX(4)) then
                CDF(3) = getPiwiPowetoCDF(logx(i), alpha(2:4), [-LOG_HUGE, logLimX(2:4)], getPiwiPowetoLogPDFNF(alpha(2:4), [-LOG_HUGE, logLimX(2:4)], cumSumArea(1:4)), cumSumArea(1:4)) ! right-truncated PiwiPoweto
            else
                CDF(3) = 1.
            end if
            if (logx(i) > logLimX(1) .and. logx(i) < logLimX(4)) then
                CDF(4) = getPiwiPowetoCDF(logx(i), alpha(1:3), logLimX(1:4), getPiwiPowetoLogPDFNF(alpha(1:3), logLimX(1:4), cumSumArea(1:4)), cumSumArea(1:4)) ! doubly-truncated PiwiPoweto
            elseif (logx(i) <= logLimX(1)) then
                CDF(4) = 0.
            else
                CDF(4) = 1.
            end if
            write(fileUnit,"(*(g0,:,', '))") exp(logx(i)), CDF
        end do
        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 Cumulative Distribution Function (CDF) of the (truncated) PiwiPoweto distribution.
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
cdf = getPiwiPowetoCDF(logx = log(3.), alpha = [-2.], logLimX = [log(2.), log(huge(0.))]) ! left-truncated PiwiPoweto distribution at [2., +∞)
cdf
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cdf = getPiwiPowetoCDF(logx = log(2.5), alpha = [1., 0.], logLimX = [0.5, 1., 1.5]) ! doubly-truncated PiwiPoweto distribution at `[0.5, 1.5]`.
cdf
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cdf = getPiwiPowetoCDF(logx = log(2.5), alpha = [1., 0., -1.5], logLimX = [0.5, 1., 1.5, log(huge(0.))]) ! (truncated) PiwiPoweto with Semi-Infinite support.
cdf
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!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! Expedite repeated PDF computations by precomputing the normalization factors.
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
cdf = getPiwiPowetoCDF(logx = log(2.5), alpha = [1., 0., -1.5], logLimX = [0.5, 1., 1.5, log(huge(0.))], logPDFNF = getPiwiPowetoLogPDFNF(alpha = [1., 0., -1.5], logLimX = [0.5, 1., 1.5, log(huge(0.))], cumSumArea = cumSumArea(1:4)), cumSumArea = cumSumArea(1:4))
cdf
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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
 
linewidth = 2
fontsize = 17
 
marker ={ "CK" : "-"
        , "IK" : "."
        , "RK" : "-"
        }
xlab =  { "CK" : "X ( real/imaginary components )"
        , "IK" : "X ( integer-valued )"
        , "RK" : "X ( real-valued )"
        }
legends =   [ r"5-piece Poweto"
            , r"3-piece left-truncated Poweto"
            , r"4-piece right-truncated Poweto"
            , r"4-piece doubly-truncated Poweto"
            ]
 
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[:,[2,4]]
                    , marker[kind]
                    , linewidth = linewidth
                   #, color = "r"
                    )
            plt.plot( df.values[:, 1]
                    , df.values[:,[3,5]]
                    , marker[kind]
                    , linewidth = linewidth
                   #, color = "blue"
                    )
        else:
            plt.plot( df.values[:, 0]
                    , df.values[:,1:]
                    , marker[kind]
                    , linewidth = linewidth
                   #, color = "r"
                    )
            ax.legend   ( legends
                        , fontsize = fontsize
                        )
 
        plt.xticks(fontsize = fontsize - 2)
        plt.yticks(fontsize = fontsize - 2)
        ax.set_xlabel(xlab[kind], fontsize = 17)
        ax.set_ylabel("Cumulative Distribution Function (CDF)", fontsize = 17)
        #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_distPiwiPoweto

Todo:

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

Final Remarks ⛓

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Copyright

Computational Data Science Lab

Author:

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

Definition at line 742 of file pm_distPiwiPoweto.F90.

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

• src/fortran/main/pm_distPiwiPoweto.F90