pm_distGamma::setGammaCDF Interface Reference

Return the Cumulative Distribution Function (CDF) of the Gamma distribution for an input x within the support of the distribution \(x \in (0,+\infty)\). More...

Detailed Description

Return the Cumulative Distribution Function (CDF) of the Gamma distribution for an input x within the support of the distribution \(x \in (0,+\infty)\).

See the documentation of pm_distGamma for more information on the Gamma CDF.

Parameters

[out]	cdf	: The output scalar or array of the same shape as any input array-like argument, of the same type and kind the input argument x, containing the CDF of the distribution at the specified x.
[in]	x	: 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 values at which the CDF must be computed.
[in]	kappa	: The input scalar or array of the same shape as other array-like arguments, of the same type and kind as x, containing the shape parameter of the distribution. (optional, default = 1.. It must be present if invSigma is also present.)
[in]	invSigma	: The input scalar or array of the same shape as other array-like arguments, of the same type and kind as x, containing the rate (inverse scale) parameter of the distribution. (optional, default = 1.. It can be present only if all previous arguments are also present.)
[out]	info	: The output scalar of type integer of default kind IK. On output, it is set to positive the number of iterations taken for the series representation of the Gamma function to converge. If the algorithm fails to converge, then info is set to the negative of the number of iterations taken by the algorithm. An negative output value signifies the lack of convergence and failure to compute the CDF. This is likely to happen if the input value for kappa is too large.

Possible calling interfaces ⛓

use pm_distGamma, only: setGammaCDF
 
call setGammaCDF(cdf, x, info)
call setGammaCDF(cdf, x, kappa, invSigma, info)

Warning

The condition 0 < x must hold for the corresponding input arguments.
The condition 0 < kappa must hold for the corresponding input arguments.
The condition 0 < invSigma 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

getGammaCDF

Example usage ⛓

program example
 
    use pm_kind, only: SK, IK, LK
    use pm_distGamma, only: setGammaCDF
    use pm_arraySpace, only: setLogSpace
    use pm_arraySpace, only: getLinSpace
    use pm_io, only: display_type
 
    implicit none
 
    integer(IK), parameter  :: NP = 999_IK
    real                    :: point(NP), CDF(NP)
    integer(IK)             :: info(NP)
 
    type(display_type)      :: disp
    disp = display_type(file = "main.out.F90")
 
    call setLogSpace(point, logx1 = -5., logx2 = 5.)
 
    call disp%skip()
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("! Compute the Cumulative Distribution Function (CDF) of the Gamma distribution.")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%skip()
 
    call disp%skip()
    call disp%show("point(1)")
    call disp%show( point(1) )
    call disp%show("call setGammaCDF(CDF(1), point(1), info(1))")
                    call setGammaCDF(CDF(1), point(1), info(1))
    call disp%show("if (info(1) < 0) error stop 'setGammaCDF() failed.'")
                    if (info(1) < 0) error stop 'setGammaCDF() failed.'
    call disp%show("CDF(1)")
    call disp%show( CDF(1) )
    call disp%skip()
 
    call disp%skip()
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("! Accelerate the runtime performance for repeated calls when `kappa` and `invSigma` are fixed.")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%skip()
 
    call disp%skip()
    call disp%show("point(1:NP:NP/4)")
    call disp%show( point(1:NP:NP/4) )
    call disp%show("call setGammaCDF(CDF(1:NP:NP/4), point(1:NP:NP/4), kappa = 2., invSigma = 2., info = info(1:NP:NP/4))")
                    call setGammaCDF(CDF(1:NP:NP/4), point(1:NP:NP/4), kappa = 2., invSigma = 2., info = info(1:NP:NP/4))
    call disp%show("if (any(info(1:NP:NP/4) < 0)) error stop 'setGammaCDF() failed.'")
                    if (any(info(1:NP:NP/4) < 0)) error stop 'setGammaCDF() failed.'
    call disp%show("CDF(1:NP:NP/4)")
    call disp%show( CDF(1:NP:NP/4) )
    call disp%skip()
 
    call disp%skip()
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("! A vector of CDF at different points with the same CDF parameters.")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%skip()
 
    call disp%skip()
    call disp%show("point(1:NP:NP/4)")
    call disp%show( point(1:NP:NP/4) )
    call disp%show("call setGammaCDF(CDF(1:NP:NP/4), point(1:NP:NP/4), kappa = 0.5, invSigma = 5., info = info(1:NP:NP/4))")
                    call setGammaCDF(CDF(1:NP:NP/4), point(1:NP:NP/4), kappa = 0.5, invSigma = 5., info = info(1:NP:NP/4))
    call disp%show("if (any(info(1:NP:NP/4) < 0)) error stop 'setGammaCDF() failed.'")
                    if (any(info(1:NP:NP/4) < 0)) error stop 'setGammaCDF() failed.'
    call disp%show("CDF(1:NP:NP/4)")
    call disp%show( CDF(1:NP:NP/4) )
    call disp%skip()
 
    call disp%skip()
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("! A vector of CDF at the same point but with different CDF parameters.")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%skip()
 
    call disp%skip()
    call disp%show("point(NP/4)")
    call disp%show( point(NP/4) )
    call disp%show("call setGammaCDF(CDF(1:NP:NP/4), point(NP/4), kappa = getLinSpace(0.5, 5., 5), invSigma = getLinSpace(5., .5, 5), info = info(1:NP:NP/4))")
                    call setGammaCDF(CDF(1:NP:NP/4), point(NP/4), kappa = getLinSpace(0.5, 5., 5), invSigma = getLinSpace(5., .5, 5), info = info(1:NP:NP/4))
    call disp%show("if (any(info(1:NP:NP/4) < 0)) error stop 'setGammaCDF() failed.'")
                    if (any(info(1:NP:NP/4) < 0)) error stop 'setGammaCDF() failed.'
    call disp%show("CDF(1:NP:NP/4)")
    call disp%show( CDF(1:NP:NP/4) )
    call disp%skip()
 
    call disp%skip()
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("! A vector of CDF at different points with different CDF parameters.")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%skip()
 
    call disp%skip()
    call disp%show("point(1:NP:NP/4)")
    call disp%show( point(1:NP:NP/4) )
    call disp%show("call setGammaCDF(CDF(1:NP:NP/4), point(1:NP:NP/4), kappa = getLinSpace(0.5, 5., 5), invSigma = getLinSpace(5., .5, 5), info = info(1:NP:NP/4))")
                    call setGammaCDF(CDF(1:NP:NP/4), point(1:NP:NP/4), kappa = getLinSpace(0.5, 5., 5), invSigma = getLinSpace(5., .5, 5), info = info(1:NP:NP/4))
    call disp%show("if (any(info(1:NP:NP/4) < 0)) error stop 'setGammaCDF() failed.'")
                    if (any(info(1:NP:NP/4) < 0)) error stop 'setGammaCDF() failed.'
    call disp%show("CDF(1:NP:NP/4)")
    call disp%show( CDF(1:NP:NP/4) )
    call disp%skip()
 
    !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    ! Output an example array for visualization.
    !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
    block
        integer(IK) :: fileUnit, i
        real, parameter :: kappa(*) = [0.5, 1.0, 2.0, 7.5]
        real, parameter :: invSigma(*) = [1.0, 0.5, 0.5, 1.0]
        open(newunit = fileUnit, file = "setGammaCDF.RK.txt")
        do i = 1, NP
            call setGammaCDF(CDF(1:4), point(i), kappa, invSigma, info(1:4))
            if (any(info(1:4) < 0)) error stop 'setGammaCDF() failed.'
            write(fileUnit,"(5(g0,:,' '))") point(i), CDF(1:4)
        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 Gamma distribution.
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
point(1)
+0.673794653E-2
call setGammaCDF(CDF(1), point(1), info(1))
if (info(1) < 0) error stop 'setGammaCDF() failed.'
CDF(1)
+0.671529910E-2
 
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! Accelerate the runtime performance for repeated calls when `kappa` and `invSigma` are fixed.
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
point(1:NP:NP/4)
+0.673794653E-2, +0.816747844E-1, +0.990029752, +12.0007629, +145.468491
call setGammaCDF(CDF(1:NP:NP/4), point(1:NP:NP/4), kappa = 2., invSigma = 2., info = info(1:NP:NP/4))
if (any(info(1:NP:NP/4) < 0)) error stop 'setGammaCDF() failed.'
CDF(1:NP:NP/4)
+0.899882580E-4, +0.119738970E-1, +0.588569999, +1.00000000, +1.00000000
 
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! A vector of CDF at different points with the same CDF parameters.
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
point(1:NP:NP/4)
+0.673794653E-2, +0.816747844E-1, +0.990029752, +12.0007629, +145.468491
call setGammaCDF(CDF(1:NP:NP/4), point(1:NP:NP/4), kappa = 0.5, invSigma = 5., info = info(1:NP:NP/4))
if (any(info(1:NP:NP/4) < 0)) error stop 'setGammaCDF() failed.'
CDF(1:NP:NP/4)
+0.204808801, +0.633867264, +0.998347461, +1.00000000, +1.00000000
 
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! A vector of CDF at the same point but with different CDF parameters.
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
point(NP/4)
+0.808604807E-1
call setGammaCDF(CDF(1:NP:NP/4), point(NP/4), kappa = getLinSpace(0.5, 5., 5), invSigma = getLinSpace(5., .5, 5), info = info(1:NP:NP/4))
if (any(info(1:NP:NP/4) < 0)) error stop 'setGammaCDF() failed.'
CDF(1:NP:NP/4)
+0.631466985, +0.860468596E-1, +0.307858689E-2, +0.173809440E-4, +0.870413130E-9
 
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! A vector of CDF at different points with different CDF parameters.
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
point(1:NP:NP/4)
+0.673794653E-2, +0.816747844E-1, +0.990029752, +12.0007629, +145.468491
call setGammaCDF(CDF(1:NP:NP/4), point(1:NP:NP/4), kappa = getLinSpace(0.5, 5., 5), invSigma = getLinSpace(5., .5, 5), info = info(1:NP:NP/4))
if (any(info(1:NP:NP/4) < 0)) error stop 'setGammaCDF() failed.'
CDF(1:NP:NP/4)
+0.204808801, +0.872944593E-1, +0.573785663, +0.999996066, +1.00000000
 
 

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"$\kappa = 0.5,~\sigma = 1.0$"
            , r"$\kappa = 1.0,~\sigma = 2.0$"
            , r"$\kappa = 2.0,~\sigma = 2.0$"
            , r"$\kappa = 7.5,~\sigma = 1.0$"
            ]
 
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[:,1:5]
                    , marker[kind]
                    , linewidth = linewidth
                   #, color = "r"
                    )
            plt.plot( df.values[:, 1]
                    , df.values[:,1:5]
                    , marker[kind]
                    , linewidth = linewidth
                   #, color = "blue"
                    )
        else:
            plt.plot( df.values[:, 0]
                    , df.values[:,1:5]
                    , 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_xlim([0, 20.])
 
        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.savefig(fileList[0].replace(".txt",".png"))
 
    elif len(fileList) > 1:
 
        sys.exit("Ambiguous file list exists.")

Visualization of the example output ⛓

Test:

test_pm_distGamma

Todo:

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 917 of file pm_distGamma.F90.

---

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

• src/fortran/main/pm_distGamma.F90