Generate and return the Probability Density Function (PDF) of the Beta distribution for an input x within the support of the distribution $x \in [0,1]$. More...

Detailed Description

Generate and return the Probability Density Function (PDF) of the Beta distribution for an input x within the support of the distribution $x \in [0,1]$.

See the documentation of pm_distBeta for more information on the Beta distribution.

Contrary to the higher-performance procedure interfaces getBetaLogPDF and setBetaLogPDF that only accept $x \in (0,1)$, the procedures under this generic interface also accept the boundary values of the support: $x = 0$, and $x = 1$.

This interface therefore has two primary use cases:

When the PDF of the Beta distribution (as opposed to the natural logarithm of PDF) is needed.
When the coverage of the full closed interval $[0,1]$ for $x$ is desired.

Note, however, the output of the procedures of this generic interface are prone to overflow at $(x \sim 0, \alpha < 1)$ or $(x \sim 1, \beta < 1)$.
The occurrence of the overflow depends on the specified values for $(x, \alpha, \beta)$ and the requested real precision (kind) for the output PDF value.

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 PDF must be computed.
[in]	alpha	: The input scalar or array of the same shape as other array-like arguments, of the same type and kind as `x`, containing the first shape parameter of the distribution.
[in]	beta	: The input scalar or array of the same shape as other array-like arguments, of the same type and kind as `x`, containing the second shape parameter of the distribution.

Returns: pdf : 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 x, containing the PDF of the distribution.

Possible calling interfaces ⛓

: use pm_distBeta, only: getBetaPDF

pdf = getBetaPDF(x, alpha, beta)

pm_distBeta::getBetaPDF
Generate and return the Probability Density Function (PDF) of the Beta distribution for an input x wi...
Definition: pm_distBeta.F90:227

pm_distBeta
This module contains classes and procedures for computing various statistical quantities related to t...
Definition: pm_distBeta.F90:99

Warning: The condition $x \in (0,1)$, $\alpha \geq 0$, and $\beta \geq 0$ must hold for the triple input values (x, alpha, beta).
These conditions are verified only if the library is built with the preprocessor macro CHECK_ENABLED=1.; The procedures of this generic interface are prone to overflow at extreme values of x, alpha, beta.
Use getBetaLogPDF or setBetaLogPDF which more tolerant of extreme input values.; 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: getBetaLogPDF
setBetaLogPDF

Example usage ⛓

: 1program example

2

3 use pm_kind, only: SK, IK

4 use pm_distBeta, only: getBetaPDF

5 use pm_arraySpace, only: setLinSpace

6 use pm_arraySpace, only: getLinSpace

7 use pm_mathBeta, only: getLogBeta

8 use pm_io, only: display_type

9

10 implicit none

11

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

13 real :: Point(NP), PDF(NP)

14

15 type(display_type) :: disp

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

17

18 call setLinSpace(Point, x1 = 0.001, x2 = .999)

19

20 call disp%skip()

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

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

23 call disp%show("! Compute the Probability Density Function (PDF) of the Beta distribution.")

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

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

26 call disp%skip()

27

28 call disp%skip()

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

30 call disp%show( Point(1) )

31 call disp%show("PDF(1) = getBetaPDF(Point(1), 2., 2.)")

32 PDF(1) = getBetaPDF(Point(1), 2., 2.)

33 call disp%show("PDF(1)")

34 call disp%show( PDF(1) )

35 call disp%skip()

36

37 call disp%skip()

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

39 call disp%show("! Accelerate the runtime performance for repeated calls when `alpha` and `beta` are fixed (i.e., the PDF normalization constant is fixed).")

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

41 call disp%skip()

42

43 call disp%skip()

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

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

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

47 call disp%skip()

48

49 call disp%skip()

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

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

52 call disp%show("PDF(1:NP:NP/4) = getBetaPDF(Point(1:NP:NP/4), alpha = 0.5, beta = 5.)")

53 PDF(1:NP:NP/4) = getBetaPDF(Point(1:NP:NP/4), alpha = 0.5, beta = 5.)

54 call disp%show("PDF(1:NP:NP/4)")

55 call disp%show( PDF(1:NP:NP/4) )

56 call disp%skip()

57

58 call disp%skip()

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

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

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

62 call disp%skip()

63

64 call disp%skip()

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

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

67 call disp%show("PDF(1:NP:NP/4) = getBetaPDF(Point(NP/4), alpha = getLinSpace(0.5, 5., 5), beta = getLinSpace(5., .5, 5))")

68 PDF(1:NP:NP/4) = getBetaPDF(Point(NP/4), alpha = getLinSpace(0.5, 5., 5), beta = getLinSpace(5., .5, 5))

69 call disp%show("PDF(1:NP:NP/4)")

70 call disp%show( PDF(1:NP:NP/4) )

71 call disp%skip()

72

73 call disp%skip()

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

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

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

77 call disp%skip()

78

79 call disp%skip()

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

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

82 call disp%show("PDF(1:NP:NP/4) = getBetaPDF(Point(1:NP:NP/4), alpha = getLinSpace(0.5, 5., 5), beta = getLinSpace(5., .5, 5))")

83 PDF(1:NP:NP/4) = getBetaPDF(Point(1:NP:NP/4), alpha = getLinSpace(0.5, 5., 5), beta = getLinSpace(5., .5, 5))

84 call disp%show("PDF(1:NP:NP/4)")

85 call disp%show( PDF(1:NP:NP/4) )

86 call disp%skip()

87

88 !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

89 ! Output an example array for visualization.

90 !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

91

92 block

93 integer(IK) :: fileUnit, i

94 open(newunit = fileUnit, file = "getBetaPDF.RK.txt")

95 write(fileUnit,"(5(g0,:,' '))") ( Point(i) &

96 , getBetaPDF(Point(i), alpha = 0.5, beta = 0.5) &

97 , getBetaPDF(Point(i), alpha = 2.0, beta = 2.0) &

98 , getBetaPDF(Point(i), alpha = 2.0, beta = 5.0) &

99 , getBetaPDF(Point(i), alpha = 5.0, beta = 2.0) &

100 , i = 1, NP &

101 )

102 close(fileUnit)

103 end block

104

105end 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::setLinSpace
Return the linSpace output argument with size(linSpace) elements of evenly-spaced values over the int...
Definition: pm_arraySpace.F90:324

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_mathBeta::getLogBeta
Generate and return the natural logarithm of the Beta Function as defined in the details section of ...
Definition: pm_mathBeta.F90:144

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::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_mathBeta
This module contains classes and procedures for computing the mathematical Beta Function and its inve...
Definition: pm_mathBeta.F90:84

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 Probability Density Function (PDF) of the Beta distribution.

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

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

7

8

9Point(1)

10+0.100000005E-2

11PDF(1) = getBetaPDF(Point(1), 2., 2.)

12PDF(1)

13+0.599400001E-2

14

15

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

17! Accelerate the runtime performance for repeated calls when `alpha` and `beta` are fixed (i.e., the PDF normalization constant is fixed).

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

19

20

21!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

22! A vector of PDF at different points with the same PDF parameters.

23!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

24

25

26Point(1:NP:NP/4)

27+0.100000005E-2, +0.250000000, +0.499000013, +0.748000026, +0.997000039

28PDF(1:NP:NP/4) = getBetaPDF(Point(1:NP:NP/4), alpha = 0.5, beta = 5.)

29PDF(1:NP:NP/4)

30+38.7554321, +0.778655887, +0.109741561, +0.573748583E-2, +0.998127483E-10

31

32

33!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

34! A vector of PDF at the same point but with different PDF parameters.

35!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

36

37

38Point(NP/4)

39+0.249000013

40PDF(1:NP:NP/4) = getBetaPDF(Point(NP/4), alpha = getLinSpace(0.5, 5., 5), beta = getLinSpace(5., .5, 5))

41PDF(1:NP:NP/4)

42+0.784387290, +2.09134030, +1.07599258, +0.174454987, +0.545818219E-2

43

44

45!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

46! A vector of PDF at different points with different PDF parameters.

47!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

48

49

50Point(1:NP:NP/4)

51+0.100000005E-2, +0.250000000, +0.499000013, +0.748000026, +0.997000039

52PDF(1:NP:NP/4) = getBetaPDF(Point(1:NP:NP/4), alpha = getLinSpace(0.5, 5., 5), beta = getLinSpace(5., .5, 5))

53PDF(1:NP:NP/4)

54+38.7554321, +2.08856964, +1.78847313, +2.08294487, +22.1969547

55

56

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"$\alpha = 0.5,~\beta = 0.5$"

21 , r"$\alpha = 2.0,~\beta = 2.0$"

22 , r"$\alpha = 2.0,~\beta = 5.0$"

23 , r"$\alpha = 5.0,~\beta = 2.0$"

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("Probability Density Function (PDF)", fontsize = 17)

65

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

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

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

69 plt.tight_layout()

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_distBeta

Todo:: Normal 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 227 of file pm_distBeta.F90.

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

src/fortran/main/pm_distBeta.F90