pm_distGamma::getGammaLogPDFNF Interface Reference

Generate and return the natural logarithm of the normalization factor of the Probability Density Function (PDF) of the Gamma distribution for an input parameter set \((\kappa,\sigma)\). More...

Detailed Description

Generate and return the natural logarithm of the normalization factor of the Probability Density Function (PDF) of the Gamma distribution for an input parameter set \((\kappa,\sigma)\).

The natural logarithm of the normalization factor of the Gamma distribution is given by,

\begin{equation} \large \ms{logPDFNF} \equiv \log(\eta) = \log\bigg( \frac {1} {\sigma\Gamma(\kappa)} \bigg) ~, \end{equation}

where \(\Gamma(\kappa)\) is the Gamma function whose natural logarithm is returned by the Fortran intrinsic log_gamma().
See the documentation of pm_distGamma for more information on the Gamma distribution.

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

Parameters

[in]	kappa	: 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]	invSigma	: The input scalar or array of the same shape as other array-like arguments, of the same type and kind as kappa, containing the rate (inverse scale) parameter of the distribution. (optional, default = 1.)

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 kappa, containing the natural logarithm of the normalization factor of the PDF of Gamma distribution.

Possible calling interfaces ⛓

use pm_distGamma, only: getGammaLogPDFNF
 
logPDFNF = getGammaLogPDFNF(kappa)
logPDFNF = getGammaLogPDFNF(kappa, invSigma)

Warning

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

getGammaLogPDF
setGammaLogPDF

Example usage ⛓

program example
 
    use pm_kind, only: IK
    use pm_kind, only: SK
    use pm_io, only: display_type
    use pm_arraySpace, only: getLinSpace
    use pm_arraySpace, only: setLinSpace
    use pm_distGamma, only: getGammaLogPDFNF
 
    implicit none
 
    integer(IK) , parameter :: NP = 1000_IK
    real        , allocatable :: invSigma(:), logPDFNF(:), Kappa(:)
 
    type(display_type) :: disp
    disp = display_type(file = "main.out.F90")
 
    Kappa = getLinSpace(0.01, 10., count = NP)
    invSigma = getLinSpace(0.01, 10., count = NP)
    allocate(logPDFNF, mold = Kappa)
 
    call disp%skip()
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("! Compute the natural logarithm of the normalization factor of the Gamma distribution PDF.")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%skip()
 
    call disp%skip()
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("! Compute the PDF at an input scalar real value.")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%skip()
 
    call disp%skip()
    call disp%show("[Kappa(1), invSigma(1)]")
    call disp%show( [Kappa(1), invSigma(1)] )
    call disp%show("logPDFNF(1) = getGammaLogPDFNF(kappa = Kappa(1), invSigma = invSigma(1))")
                    logPDFNF(1) = getGammaLogPDFNF(kappa = Kappa(1), invSigma = invSigma(1))
    call disp%show("logPDFNF(1)")
    call disp%show( logPDFNF(1) )
    call disp%skip()
 
    call disp%skip()
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("! Compute the PDF at at a mix of scalar and vector input values.")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%skip()
 
    call disp%skip()
    call disp%show("[Kappa(1), invSigma(1)]")
    call disp%show( [Kappa(1), invSigma(1)] )
    call disp%show("logPDFNF(1) = getGammaLogPDFNF(kappa = Kappa(1), invSigma = invSigma(1))")
                    logPDFNF(1) = getGammaLogPDFNF(kappa = Kappa(1), invSigma = invSigma(1))
    call disp%show("logPDFNF(1)")
    call disp%show( logPDFNF(1) )
    call disp%skip()
 
    call disp%skip()
    call disp%show("Kappa(1:NP:NP/5)")
    call disp%show( Kappa(1:NP:NP/5) )
    call disp%show("logPDFNF(1:NP:NP/5) = getGammaLogPDFNF(kappa = Kappa(1:NP:NP/5), invSigma = 1.)")
                    logPDFNF(1:NP:NP/5) = getGammaLogPDFNF(kappa = Kappa(1:NP:NP/5), invSigma = 1.)
    call disp%show("logPDFNF(1:NP:NP/5)")
    call disp%show( logPDFNF(1:NP:NP/5) )
    call disp%skip()
 
    call disp%skip()
    call disp%show("Kappa(1:NP:NP/5)")
    call disp%show( Kappa(1:NP:NP/5) )
    call disp%show("logPDFNF(1:NP:NP/5) = getGammaLogPDFNF(kappa = 1., invSigma = invSigma(1:NP:NP/5))")
                    logPDFNF(1:NP:NP/5) = getGammaLogPDFNF(kappa = 1., invSigma = invSigma(1:NP:NP/5))
    call disp%show("logPDFNF(1:NP:NP/5)")
    call disp%show( logPDFNF(1:NP:NP/5) )
    call disp%skip()
 
    call disp%skip()
    call disp%show("Kappa(1:NP:NP/5)")
    call disp%show( Kappa(1:NP:NP/5) )
    call disp%show("logPDFNF(1:NP:NP/5) = getGammaLogPDFNF(kappa = Kappa(1:NP:NP/5), invSigma = invSigma(1:NP:NP/5))")
                    logPDFNF(1:NP:NP/5) = getGammaLogPDFNF(kappa = Kappa(1:NP:NP/5), invSigma = invSigma(1:NP:NP/5))
    call disp%show("logPDFNF(1:NP:NP/5)")
    call disp%show( logPDFNF(1:NP:NP/5) )
    call disp%skip()
 
    !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    ! Output an example array for visualization.
    !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
    block
        integer :: fileUnit, i
        logPDFNF = getGammaLogPDFNF(Kappa, invSigma)
        open(newunit = fileUnit, file = "getGammaLogPDFNF.RK.txt")
        write(fileUnit,"(2(g0,:,' '))") (Kappa(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 Gamma distribution PDF.
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! Compute the PDF at an input scalar real value.
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
[Kappa(1), invSigma(1)]
+0.999999978E-2, +0.999999978E-2
logPDFNF(1) = getGammaLogPDFNF(kappa = Kappa(1), invSigma = invSigma(1))
logPDFNF(1)
-9.20465088
 
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! Compute the PDF at at a mix of scalar and vector input values.
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
[Kappa(1), invSigma(1)]
+0.999999978E-2, +0.999999978E-2
logPDFNF(1) = getGammaLogPDFNF(kappa = Kappa(1), invSigma = invSigma(1))
logPDFNF(1)
-9.20465088
 
 
Kappa(1:NP:NP/5)
+0.999999978E-2, +2.00999999, +4.01000023, +6.01000023, +8.01000023
logPDFNF(1:NP:NP/5) = getGammaLogPDFNF(kappa = Kappa(1:NP:NP/5), invSigma = 1.)
logPDFNF(1:NP:NP/5)
-4.59948015, -0.426001893E-2, -1.80433512, -4.80456209, -8.54532528
 
 
Kappa(1:NP:NP/5)
+0.999999978E-2, +2.00999999, +4.01000023, +6.01000023, +8.01000023
logPDFNF(1:NP:NP/5) = getGammaLogPDFNF(kappa = 1., invSigma = invSigma(1:NP:NP/5))
logPDFNF(1:NP:NP/5)
-4.60517025, +0.698134720, +1.38879132, +1.79342484, +2.08069086
 
 
Kappa(1:NP:NP/5)
+0.999999978E-2, +2.00999999, +4.01000023, +6.01000023, +8.01000023
logPDFNF(1:NP:NP/5) = getGammaLogPDFNF(kappa = Kappa(1:NP:NP/5), invSigma = invSigma(1:NP:NP/5))
logPDFNF(1:NP:NP/5)
-9.20465088, +0.693874717, -0.415543795, -3.01113725, -6.46463442
 
 

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"shape, Rate parameters: $a, b$ ( real/imaginary )"
        , "IK" : r"shape, Rate parameters: $a, b$ ( integer-valued )"
        , "RK" : r"shape, Rate parameters: $a, b$ ( 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)
 
        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_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 198 of file pm_distGamma.F90.

---

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

• src/fortran/main/pm_distGamma.F90