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...
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 arraylike 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 arraylike 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 arraylike 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 ⛓
Generate and return the natural logarithm of the normalization factor of the Probability Density Func...
This module contains classes and procedures for computing various statistical quantities related to t...
 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.
 See also
 getGammaLogPDF
setGammaLogPDF
Example usage ⛓
12 integer(IK) ,
parameter :: NP
= 1000_IK
13 real ,
allocatable :: invSigma(:), logPDFNF(:), Kappa(:)
15 type(display_type) :: disp
20 allocate(logPDFNF,
mold = Kappa)
23 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
24 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
25 call disp%show(
"! Compute the natural logarithm of the normalization factor of the Gamma distribution PDF.")
26 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
27 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
31 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
32 call disp%show(
"! Compute the PDF at an input scalar real value.")
33 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
37 call disp%show(
"[Kappa(1), invSigma(1)]")
38 call disp%show( [Kappa(
1), invSigma(
1)] )
39 call disp%show(
"logPDFNF(1) = getGammaLogPDFNF(kappa = Kappa(1), invSigma = invSigma(1))")
46 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
47 call disp%show(
"! Compute the PDF at at a mix of scalar and vector input values.")
48 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
52 call disp%show(
"[Kappa(1), invSigma(1)]")
53 call disp%show( [Kappa(
1), invSigma(
1)] )
54 call disp%show(
"logPDFNF(1) = getGammaLogPDFNF(kappa = Kappa(1), invSigma = invSigma(1))")
63 call disp%show(
"logPDFNF(1:NP:NP/5) = getGammaLogPDFNF(kappa = Kappa(1:NP:NP/5), invSigma = 1.)")
64 logPDFNF(
1:NP:NP
/5)
= getGammaLogPDFNF(kappa
= Kappa(
1:NP:NP
/5), invSigma
= 1.)
65 call disp%show(
"logPDFNF(1:NP:NP/5)")
66 call disp%show( logPDFNF(
1:NP:NP
/5) )
72 call disp%show(
"logPDFNF(1:NP:NP/5) = getGammaLogPDFNF(kappa = 1., invSigma = invSigma(1:NP:NP/5))")
73 logPDFNF(
1:NP:NP
/5)
= getGammaLogPDFNF(kappa
= 1., invSigma
= invSigma(
1:NP:NP
/5))
74 call disp%show(
"logPDFNF(1:NP:NP/5)")
75 call disp%show( logPDFNF(
1:NP:NP
/5) )
81 call disp%show(
"logPDFNF(1:NP:NP/5) = getGammaLogPDFNF(kappa = Kappa(1:NP:NP/5), invSigma = invSigma(1:NP:NP/5))")
82 logPDFNF(
1:NP:NP
/5)
= getGammaLogPDFNF(kappa
= Kappa(
1:NP:NP
/5), invSigma
= invSigma(
1:NP:NP
/5))
83 call disp%show(
"logPDFNF(1:NP:NP/5)")
84 call disp%show( logPDFNF(
1:NP:NP
/5) )
92 integer :: fileUnit, i
94 open(newunit
= fileUnit, file
= "getGammaLogPDFNF.RK.txt")
95 write(fileUnit,
"(2(g0,:,' '))") (Kappa(i),
exp(logPDFNF(i)), i
= 1,
size(logPDFNF))
Generate count evenly spaced points over the interval [x1, x2] if x1 < x2, or [x2,...
Return the linSpace output argument with size(linSpace) elements of evenlyspaced values over the int...
This is a generic method of the derived type display_type with pass attribute.
This is a generic method of the derived type display_type with pass attribute.
This module contains procedures and generic interfaces for generating arrays with linear or logarithm...
This module contains classes and procedures for input/output (IO) or generic display operations on st...
type(display_type) disp
This is a scalar module variable an object of type display_type for general display.
This module defines the relevant Fortran kind typeparameters frequently used in the ParaMonte librar...
integer, parameter IK
The default integer kind in the ParaMonte library: int32 in Fortran, c_int32_t in CFortran Interoper...
integer, parameter SK
The default character kind in the ParaMonte library: kind("a") in Fortran, c_char in CFortran Intero...
Generate and return an object of type display_type.
Example Unix compile command via Intel ifort
compiler ⛓
3ifort fpp standardsemantics O3 Wl,rpath,../../../lib I../../../inc main.F90 ../../../lib/libparamonte* o main.exe
Example Windows Batch compile command via Intel ifort
compiler ⛓
2set PATH=..\..\..\lib;%PATH%
3ifort /fpp /standardsemantics /O3 /I:..\..\..\include main.F90 ..\..\..\lib\libparamonte*.lib /exe:main.exe
Example Unix / MinGW compile command via GNU gfortran
compiler ⛓
3gfortran cpp ffreelinelengthnone O3 Wl,rpath,../../../lib I../../../inc main.F90 ../../../lib/libparamonte* o main.exe
Example output ⛓
14[Kappa(
1), invSigma(
1)]
15+0.999999978E2,
+0.999999978E2
26[Kappa(
1), invSigma(
1)]
27+0.999999978E2,
+0.999999978E2
34+0.999999978E2,
+2.00999999,
+4.01000023,
+6.01000023,
+8.01000023
35logPDFNF(
1:NP:NP
/5)
= getGammaLogPDFNF(kappa
= Kappa(
1:NP:NP
/5), invSigma
= 1.)
374.59948015,
0.426001893E2,
1.80433512,
4.80456209,
8.54532528
41+0.999999978E2,
+2.00999999,
+4.01000023,
+6.01000023,
+8.01000023
42logPDFNF(
1:NP:NP
/5)
= getGammaLogPDFNF(kappa
= 1., invSigma
= invSigma(
1:NP:NP
/5))
444.60517025,
+0.698134720,
+1.38879132,
+1.79342484,
+2.08069086
48+0.999999978E2,
+2.00999999,
+4.01000023,
+6.01000023,
+8.01000023
49logPDFNF(
1:NP:NP
/5)
= getGammaLogPDFNF(kappa
= Kappa(
1:NP:NP
/5), invSigma
= invSigma(
1:NP:NP
/5))
519.20465088,
+0.693874717,
0.415543795,
3.01113725,
6.46463442
Postprocessing of the example output ⛓
3import matplotlib.pyplot
as plt
15xlab = {
"CK" :
r"shape, Rate parameters: $a, b$ ( real/imaginary )"
16 ,
"IK" :
r"shape, Rate parameters: $a, b$ ( integervalued )"
17 ,
"RK" :
r"shape, Rate parameters: $a, b$ ( realvalued )"
20for kind
in [
"IK",
"CK",
"RK"]:
22 pattern =
"*." + kind +
".txt"
23 fileList = glob.glob(pattern)
24 if len(fileList) == 1:
26 df = pd.read_csv(fileList[0], delimiter =
" ")
28 fig = plt.figure(figsize = 1.25 * np.array([6.4, 4.8]), dpi = 200)
32 plt.plot( df.values[:, 0]
37 plt.plot( df.values[:, 1]
43 plt.plot( df.values[:, 0]
49 plt.xticks(fontsize = fontsize  2)
50 plt.yticks(fontsize = fontsize  2)
51 ax.set_xlabel(xlab[kind], fontsize = fontsize)
52 ax.set_ylabel(
"Normalization Factor of the PDF", fontsize = fontsize)
54 plt.grid(visible =
True, which =
"both", axis =
"both", color =
"0.85", linestyle =
"")
55 ax.tick_params(axis =
"y", which =
"minor")
56 ax.tick_params(axis =
"x", which =
"minor")
59 plt.savefig(fileList[0].replace(
".txt",
".png"))
61 elif len(fileList) > 1:
63 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.

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.

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.