Generate and 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

Generate and 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

[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.`)
[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.`)

Returns: 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.

Possible calling interfaces ⛓

: use pm_distGamma, only: getGammaCDF

cdf = getGammaCDF(x, kappa = kappa, invSigma = invSigma)

pm_distGamma::getGammaCDF
Generate and return the Cumulative Distribution Function (CDF) of the Gamma distribution for an input...
Definition: pm_distGamma.F90:781

pm_distGamma
This module contains classes and procedures for computing various statistical quantities related to t...
Definition: pm_distGamma.F90:75

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.; The choice of invSigma vs. sigma as the input argument to the procedures of this generic interface may seem rather odd.
This choice was made deliberately to increase the similarity and consistency of the functional interface here with the subroutine interface setGammaCDF.
Note that the primary reason for preferring invSigma vs. sigma as the choice of input argument is the enhanced computational efficiency of the routines.
Additionally, invSigma matches the alternative rate interpretation of the second parameter of the Gamma distribution.

See also: setGammaCDF

Example usage ⛓

: 1program example

2

3 use pm_kind, only: SK, IK, LK

4 use pm_distGamma, only: getGammaCDF

5 use pm_arraySpace, only: setLogSpace

6 use pm_arraySpace, only: getLinSpace

7 use pm_io, only: display_type

8

9 implicit none

10

11 integer(IK), parameter :: NP = 999_IK

12 real :: Point(NP), CDF(NP)

13

14 type(display_type) :: disp

15 disp = display_type(file = "main.out.F90")

16

17 call setLogSpace(Point, logx1 = -5., logx2 = 5.)

18

19 call disp%skip()

20 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")

21 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")

22 call disp%show("! Compute the Cumulative Distribution Function (CDF) of the Gamma distribution.")

23 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")

24 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")

25 call disp%skip()

26

27 call disp%skip()

28 call disp%show("Point(1)")

29 call disp%show( Point(1) )

30 call disp%show("CDF(1) = getGammaCDF(Point(1), 2., 2.)")

31 CDF(1) = getGammaCDF(Point(1), 2., 2.)

32 call disp%show("CDF(1)")

33 call disp%show( CDF(1) )

34 call disp%skip()

35

36 call disp%skip()

37 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")

38 call disp%show("! Accelerate the runtime performance for repeated calls when `kappa` and `invSigma` are fixed.")

39 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")

40 call disp%skip()

41

42 call disp%skip()

43 call disp%show("Point(1:NP:NP/4)")

44 call disp%show( Point(1:NP:NP/4) )

45 call disp%show("CDF(1:NP:NP/4) = getGammaCDF(Point(1:NP:NP/4), 2., 2.)")

46 CDF(1:NP:NP/4) = getGammaCDF(Point(1:NP:NP/4), 2., 2.)

47 call disp%show("CDF(1:NP:NP/4)")

48 call disp%show( CDF(1:NP:NP/4) )

49 call disp%skip()

50

51 call disp%skip()

52 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")

53 call disp%show("! A vector of CDF at different points with the same CDF parameters.")

54 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")

55 call disp%skip()

56

57 call disp%skip()

58 call disp%show("Point(1:NP:NP/4)")

59 call disp%show( Point(1:NP:NP/4) )

60 call disp%show("CDF(1:NP:NP/4) = getGammaCDF(Point(1:NP:NP/4), kappa = 0.5, invSigma = 5.)")

61 CDF(1:NP:NP/4) = getGammaCDF(Point(1:NP:NP/4), kappa = 0.5, invSigma = 5.)

62 call disp%show("CDF(1:NP:NP/4)")

63 call disp%show( CDF(1:NP:NP/4) )

64 call disp%skip()

65

66 call disp%skip()

67 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")

68 call disp%show("! A vector of CDF at the same point but with different CDF parameters.")

69 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")

70 call disp%skip()

71

72 call disp%skip()

73 call disp%show("Point(NP/4)")

74 call disp%show( Point(NP/4) )

75 call disp%show("CDF(1:NP:NP/4) = getGammaCDF(Point(NP/4), kappa = getLinSpace(0.5, 5., 5), invSigma = getLinSpace(5., .5, 5))")

76 CDF(1:NP:NP/4) = getGammaCDF(Point(NP/4), kappa = getLinSpace(0.5, 5., 5), invSigma = getLinSpace(5., .5, 5))

77 call disp%show("CDF(1:NP:NP/4)")

78 call disp%show( CDF(1:NP:NP/4) )

79 call disp%skip()

80

81 call disp%skip()

82 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")

83 call disp%show("! A vector of CDF at different points with different CDF parameters.")

84 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")

85 call disp%skip()

86

87 call disp%skip()

88 call disp%show("Point(1:NP:NP/4)")

89 call disp%show( Point(1:NP:NP/4) )

90 call disp%show("CDF(1:NP:NP/4) = getGammaCDF(Point(1:NP:NP/4), kappa = getLinSpace(0.5, 5., 5), invSigma = getLinSpace(5., .5, 5))")

91 CDF(1:NP:NP/4) = getGammaCDF(Point(1:NP:NP/4), kappa = getLinSpace(0.5, 5., 5), invSigma = getLinSpace(5., .5, 5))

92 call disp%show("CDF(1:NP:NP/4)")

93 call disp%show( CDF(1:NP:NP/4) )

94 call disp%skip()

95

96 !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

97 ! Output an example array for visualization.

98 !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

99

100 block

101 integer(IK) :: fileUnit, i

102 open(newunit = fileUnit, file = "getGammaCDF.RK.txt")

103 write(fileUnit,"(5(g0,:,' '))") (Point(i), getGammaCDF(Point(i), kappa = [0.5, 1.0, 2.0, 7.5], invSigma = [1.0, 0.5, 0.5, 1.0]), i = 1, NP)

104 close(fileUnit)

105 end block

106

107end program example

pm_arraySpace::getLinSpace
Generate count evenly spaced points over the interval [x1, x2] if x1 < x2, or [x2,...
Definition: pm_arraySpace.F90:108

pm_arraySpace::setLogSpace
Return the logSpace output argument with size(logSpace) elements of logarithmically-evenly-spaced val...
Definition: pm_arraySpace.F90:765

pm_io::show
This is a generic method of the derived type display_type with pass attribute.
Definition: pm_io.F90:11726

pm_io::skip
This is a generic method of the derived type display_type with pass attribute.
Definition: pm_io.F90:11508

pm_arraySpace
This module contains procedures and generic interfaces for generating arrays with linear or logarithm...
Definition: pm_arraySpace.F90:33

pm_io
This module contains classes and procedures for input/output (IO) or generic display operations on st...
Definition: pm_io.F90:252

pm_io::disp
type(display_type) disp
This is a scalar module variable an object of type display_type for general display.
Definition: pm_io.F90:11393

pm_kind
This module defines the relevant Fortran kind type-parameters frequently used in the ParaMonte librar...
Definition: pm_kind.F90:268

pm_kind::LK
integer, parameter LK
The default logical kind in the ParaMonte library: kind(.true.) in Fortran, kind(....
Definition: pm_kind.F90:541

pm_kind::IK
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

pm_kind::SK
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

pm_io::display_type
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 Cumulative Distribution Function (CDF) of the Gamma distribution.

5!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

6!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

7

8

9Point(1)

10+0.673794653E-2

11CDF(1) = getGammaCDF(Point(1), 2., 2.)

12CDF(1)

13+0.899882580E-4

14

15

16!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

17! Accelerate the runtime performance for repeated calls when `kappa` and `invSigma` are fixed.

18!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

19

20

21Point(1:NP:NP/4)

22+0.673794653E-2, +0.816747844E-1, +0.990029752, +12.0007629, +145.468491

23CDF(1:NP:NP/4) = getGammaCDF(Point(1:NP:NP/4), 2., 2.)

24CDF(1:NP:NP/4)

25+0.899882580E-4, +0.119738970E-1, +0.588569999, +1.00000000, +1.00000000

26

27

28!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

29! A vector of CDF at different points with the same CDF parameters.

30!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

31

32

33Point(1:NP:NP/4)

34+0.673794653E-2, +0.816747844E-1, +0.990029752, +12.0007629, +145.468491

35CDF(1:NP:NP/4) = getGammaCDF(Point(1:NP:NP/4), kappa = 0.5, invSigma = 5.)

36CDF(1:NP:NP/4)

37+0.204808801, +0.633867264, +0.998347461, +1.00000000, +1.00000000

38

39

40!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

41! A vector of CDF at the same point but with different CDF parameters.

42!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

43

44

45Point(NP/4)

46+0.808604807E-1

47CDF(1:NP:NP/4) = getGammaCDF(Point(NP/4), kappa = getLinSpace(0.5, 5., 5), invSigma = getLinSpace(5., .5, 5))

48CDF(1:NP:NP/4)

49+0.631466985, +0.860468596E-1, +0.307858689E-2, +0.173809440E-4, +0.870413130E-9

50

51

52!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

53! A vector of CDF at different points with different CDF parameters.

54!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

55

56

57Point(1:NP:NP/4)

58+0.673794653E-2, +0.816747844E-1, +0.990029752, +12.0007629, +145.468491

59CDF(1:NP:NP/4) = getGammaCDF(Point(1:NP:NP/4), kappa = getLinSpace(0.5, 5., 5), invSigma = getLinSpace(5., .5, 5))

60CDF(1:NP:NP/4)

61+0.204808801, +0.872944593E-1, +0.573785663, +0.999996066, +1.00000000

62

63

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

9linewidth = 2

10fontsize = 17

11

12marker ={ "CK" : "-"

13 , "IK" : "."

14 , "RK" : "-"

15 }

16xlab = { "CK" : "X ( real/imaginary components )"

17 , "IK" : "X ( integer-valued )"

18 , "RK" : "X ( real-valued )"

19 }

20legends = [ r"$\kappa = 0.5,~\sigma = 1.0$"

21 , r"$\kappa = 1.0,~\sigma = 2.0$"

22 , r"$\kappa = 2.0,~\sigma = 2.0$"

23 , r"$\kappa = 7.5,~\sigma = 1.0$"

24 ]

25

26for kind in ["IK", "CK", "RK"]:

27

28 pattern = "*." + kind + ".txt"

29 fileList = glob.glob(pattern)

30 if len(fileList) == 1:

31

32 df = pd.read_csv(fileList[0], delimiter = " ")

33

34 fig = plt.figure(figsize = 1.25 * np.array([6.4, 4.8]), dpi = 200)

35 ax = plt.subplot()

36

37 if kind == "CK":

38 plt.plot( df.values[:, 0]

39 , df.values[:,1:5]

40 , marker[kind]

41 , linewidth = linewidth

42 #, color = "r"

43 )

44 plt.plot( df.values[:, 1]

45 , df.values[:,1:5]

46 , marker[kind]

47 , linewidth = linewidth

48 #, color = "blue"

49 )

50 else:

51 plt.plot( df.values[:, 0]

52 , df.values[:,1:5]

53 , marker[kind]

54 , linewidth = linewidth

55 #, color = "r"

56 )

57 ax.legend ( legends

58 , fontsize = fontsize

59 )

60

61 plt.xticks(fontsize = fontsize - 2)

62 plt.yticks(fontsize = fontsize - 2)

63 ax.set_xlabel(xlab[kind], fontsize = 17)

64 ax.set_ylabel("Cumulative Distribution Function (CDF)", fontsize = 17)

65 ax.set_xlim([0, 20.])

66

67 plt.grid(visible = True, which = "both", axis = "both", color = "0.85", linestyle = "-")

68 ax.tick_params(axis = "y", which = "minor")

69 ax.tick_params(axis = "x", which = "minor")

70

71 plt.savefig(fileList[0].replace(".txt",".png"))

72

73 elif len(fileList) > 1:

74

75 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 781 of file pm_distGamma.F90.

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

src/fortran/main/pm_distGamma.F90