ParaMonte Fortran 2.0.0
Parallel Monte Carlo and Machine Learning Library
See the latest version documentation.
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

logPDFNF = getGammaLogPDFNF(kappa)
logPDFNF = getGammaLogPDFNF(kappa, invSigma)
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.
Remarks
The procedures under discussion are elemental.
See also
getGammaLogPDF
setGammaLogPDF


Example usage

1program example
2
3 use pm_kind, only: IK
4 use pm_kind, only: SK
5 use pm_io, only: display_type
9
10 implicit none
11
12 integer(IK) , parameter :: NP = 1000_IK
13 real , allocatable :: invSigma(:), logPDFNF(:), Kappa(:)
14
15 type(display_type) :: disp
16 disp = display_type(file = "main.out.F90")
17
18 Kappa = getLinSpace(0.01, 10., count = NP)
19 invSigma = getLinSpace(0.01, 10., count = NP)
20 allocate(logPDFNF, mold = Kappa)
21
22 call disp%skip()
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("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
28 call disp%skip()
29
30 call disp%skip()
31 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
32 call disp%show("! Compute the PDF at an input scalar real value.")
33 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
34 call disp%skip()
35
36 call disp%skip()
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))")
40 logPDFNF(1) = getGammaLogPDFNF(kappa = Kappa(1), invSigma = invSigma(1))
41 call disp%show("logPDFNF(1)")
42 call disp%show( logPDFNF(1) )
43 call disp%skip()
44
45 call disp%skip()
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("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
49 call disp%skip()
50
51 call disp%skip()
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))")
55 logPDFNF(1) = getGammaLogPDFNF(kappa = Kappa(1), invSigma = invSigma(1))
56 call disp%show("logPDFNF(1)")
57 call disp%show( logPDFNF(1) )
58 call disp%skip()
59
60 call disp%skip()
61 call disp%show("Kappa(1:NP:NP/5)")
62 call disp%show( Kappa(1:NP:NP/5) )
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) )
67 call disp%skip()
68
69 call disp%skip()
70 call disp%show("Kappa(1:NP:NP/5)")
71 call disp%show( Kappa(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) )
76 call disp%skip()
77
78 call disp%skip()
79 call disp%show("Kappa(1:NP:NP/5)")
80 call disp%show( Kappa(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) )
85 call disp%skip()
86
87 !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
88 ! Output an example array for visualization.
89 !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
90
91 block
92 integer :: fileUnit, i
93 logPDFNF = getGammaLogPDFNF(Kappa, invSigma)
94 open(newunit = fileUnit, file = "getGammaLogPDFNF.RK.txt")
95 write(fileUnit,"(2(g0,:,' '))") (Kappa(i), exp(logPDFNF(i)), i = 1, size(logPDFNF))
96 close(fileUnit)
97 end block
98
99end program example
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 evenly-spaced values over the int...
This is a generic method of the derived type display_type with pass attribute.
Definition: pm_io.F90:11726
This is a generic method of the derived type display_type with pass attribute.
Definition: pm_io.F90:11508
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...
Definition: pm_io.F90:252
type(display_type) disp
This is a scalar module variable an object of type display_type for general display.
Definition: pm_io.F90:11393
This module defines the relevant Fortran kind type-parameters frequently used in the ParaMonte librar...
Definition: pm_kind.F90:268
integer, parameter IK
The default integer kind in the ParaMonte library: int32 in Fortran, c_int32_t in C-Fortran Interoper...
Definition: pm_kind.F90:540
integer, parameter SK
The default character kind in the ParaMonte library: kind("a") in Fortran, c_char in C-Fortran Intero...
Definition: pm_kind.F90:539
Generate and return an object of type display_type.
Definition: pm_io.F90:10282

Example Unix compile command via Intel ifort compiler
1#!/usr/bin/env sh
2rm main.exe
3ifort -fpp -standard-semantics -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
4./main.exe

Example Windows Batch compile command via Intel ifort compiler
1del main.exe
2set PATH=..\..\..\lib;%PATH%
3ifort /fpp /standard-semantics /O3 /I:..\..\..\include main.F90 ..\..\..\lib\libparamonte*.lib /exe:main.exe
4main.exe

Example Unix / MinGW compile command via GNU gfortran compiler
1#!/usr/bin/env sh
2rm main.exe
3gfortran -cpp -ffree-line-length-none -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
4./main.exe

Example output
1
2!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
3!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
4! Compute the natural logarithm of the normalization factor of the Gamma distribution PDF.
5!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
6!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
7
8
9!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
10! Compute the PDF at an input scalar real value.
11!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
12
13
14[Kappa(1), invSigma(1)]
15+0.999999978E-2, +0.999999978E-2
16logPDFNF(1) = getGammaLogPDFNF(kappa = Kappa(1), invSigma = invSigma(1))
17logPDFNF(1)
18-9.20465088
19
20
21!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
22! Compute the PDF at at a mix of scalar and vector input values.
23!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
24
25
26[Kappa(1), invSigma(1)]
27+0.999999978E-2, +0.999999978E-2
28logPDFNF(1) = getGammaLogPDFNF(kappa = Kappa(1), invSigma = invSigma(1))
29logPDFNF(1)
30-9.20465088
31
32
33Kappa(1:NP:NP/5)
34+0.999999978E-2, +2.00999999, +4.01000023, +6.01000023, +8.01000023
35logPDFNF(1:NP:NP/5) = getGammaLogPDFNF(kappa = Kappa(1:NP:NP/5), invSigma = 1.)
36logPDFNF(1:NP:NP/5)
37-4.59948015, -0.426001893E-2, -1.80433512, -4.80456209, -8.54532528
38
39
40Kappa(1:NP:NP/5)
41+0.999999978E-2, +2.00999999, +4.01000023, +6.01000023, +8.01000023
42logPDFNF(1:NP:NP/5) = getGammaLogPDFNF(kappa = 1., invSigma = invSigma(1:NP:NP/5))
43logPDFNF(1:NP:NP/5)
44-4.60517025, +0.698134720, +1.38879132, +1.79342484, +2.08069086
45
46
47Kappa(1:NP:NP/5)
48+0.999999978E-2, +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))
50logPDFNF(1:NP:NP/5)
51-9.20465088, +0.693874717, -0.415543795, -3.01113725, -6.46463442
52
53

Postprocessing of the example output
1#!/usr/bin/env python
2
3import matplotlib.pyplot as plt
4import pandas as pd
5import numpy as np
6import glob
7import sys
8
9fontsize = 17
10
11marker ={ "CK" : "-"
12 , "IK" : "."
13 , "RK" : "-"
14 }
15xlab = { "CK" : r"shape, Rate parameters: $a, b$ ( real/imaginary )"
16 , "IK" : r"shape, Rate parameters: $a, b$ ( integer-valued )"
17 , "RK" : r"shape, Rate parameters: $a, b$ ( real-valued )"
18 }
19
20for kind in ["IK", "CK", "RK"]:
21
22 pattern = "*." + kind + ".txt"
23 fileList = glob.glob(pattern)
24 if len(fileList) == 1:
25
26 df = pd.read_csv(fileList[0], delimiter = " ")
27
28 fig = plt.figure(figsize = 1.25 * np.array([6.4, 4.8]), dpi = 200)
29 ax = plt.subplot()
30
31 if kind == "CK":
32 plt.plot( df.values[:, 0]
33 , df.values[:,2]
34 , marker[kind]
35 , color = "r"
36 )
37 plt.plot( df.values[:, 1]
38 , df.values[:,3]
39 , marker[kind]
40 , color = "blue"
41 )
42 else:
43 plt.plot( df.values[:, 0]
44 , df.values[:, 1]
45 , marker[kind]
46 , color = "r"
47 )
48
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)
53
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")
57
58 plt.tight_layout()
59 plt.savefig(fileList[0].replace(".txt",".png"))
60
61 elif len(fileList) > 1:
62
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.

  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.

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: