Generate and return the Bhattacharyya distance of two univariate (discrete or continuous) distributions. More...

Detailed Description

Generate and return the Bhattacharyya distance of two univariate (discrete or continuous) distributions.

See pm_distanceBhat for the mathematical definition of the Bhattacharyya distance.

Parameters

[in]	p	: The vector of the same size as the non-zero size of the input argument `q`, of the same type and kind as the output `bhat`, representing the Probability Mass Function (PMF) of the first distribution in the computation of the squared Bhattacharyya distance. Alternatively, the input `p` can be a non-elemental function that takes a scalar of the same type and kind as the output `bhat` and returns the output `pdf` of the same type and kind as `x`, representing the Probability Density Function (PDF) of the first distribution in the computation of the squared Bhattacharyya distance. The following illustrates the generic interface of `p`, function p(x) result(pdf) real(RKG), intent(in) :: x real(RKG) :: pdf end function where `RKG` is the kind type parameter of the output `bhat`. The input arguments `p` and `q` must be of the same type and kind (and size if they are vectors).
[in]	q	: The vector of the same size as the non-zero size of the input argument `p`, of the same type and kind as the output `bhat`, representing the Probability Mass Function (PMF) of the second distribution in the computation of the squared Bhattacharyya distance. Alternatively, the input `p` can be a non-elemental function that takes a scalar of the same type and kind as the output `bhat` and returns the output `pdf` of the same type and kind as `x`, representing the Probability Density Function (PDF) of the second distribution in the computation of the squared Bhattacharyya distance. The following illustrates the generic interface of `p`, function p(x) result(pdf) real(RKG), intent(in) :: x real(RKG) :: pdf end function where `RKG` is the kind type parameter of the output `bhat`. The input arguments `p` and `q` must be of the same type and kind (and size if they are vectors).
[in]	lb	: The input scalar of type `real` of the same kind as `integral`, representing the lower limit of integration. Set `lb = -huge(lb)` or to the IEEE-compliant negative infinity (`lb =`getInfNeg(lb)) to imply \(-\infty\) as the lower bound of integration. See also the corresponding argument of isFailedQuad(). (optional, default = getInfNeg(lb). It can be present only if the input arguments `p` and `q` are procedures.)
[in]	ub	: The input scalar of type `real` of the same kind as `integral`, representing the upper limit of integration. Set `ub = huge(ub)` or to the IEEE-compliant positive infinity (`ub =`getInfPos(lb)) to imply \(+\infty\) as the upper bound of integration. See also the corresponding argument of isFailedQuad(). (optional, default = getInfPos(lb). It can be present only if the input arguments `p` and `q` are procedures.)
[out]	failed	: The output scalar of type `logical` of default kind LK, that is set to `.true.` if and only if the integration Bhattacharyya fails to converge within the tolerances. Otherwise, it is set `.false.` if the integration succeeds with no errors. See also the corresponding output argument of isFailedQuad(). See also the description of the output argument `err` of getQuadErr for information on the kinds of integration failures that can happen. (optional. If missing and the integration fails, the procedure will halt the program by calling `error stop`.)
[out]	msg	: The output scalar argument of type `character` of default kind SK of arbitrary length type parameter that is set to a diagnostic message if the integration fails to converge. A length type parameter of `127` is sufficient to capture all error messages. If `msg` has shorter length parameter, the output message will be trimmed from the end, otherwise padded with blanks as necessary. See also the corresponding argument of isFailedQuad(). (optional. If missing, no diagnostic message will be returned.)

Returns

bhat : The output scalar of

type real of kind any supported by the processor (e.g., RK, RK32, RK64, or RK128),

representing the Bhattacharyya distance of the discrete or continuous distributions represented by p and q.

Possible calling interfaces ⛓

: use pm_distanceBhat, only: getDisBhat

! discrete distributions.

bhat = getDisBhat(p(:), q(:))

! continuous univariate distributions.

bhat = getDisBhat(p, q, lb = lb, ub = ub)

pm_distanceBhat::getDisBhat
Generate and return the Bhattacharyya distance of two univariate (discrete or continuous) distributio...
Definition: pm_distanceBhat.F90:249

pm_distanceBhat
This module contains classes and procedures for computing the Bhattacharyya statistical distance betw...
Definition: pm_distanceBhat.F90:127

Warning: The condition sum(p) == 1 must hold for the corresponding input arguments.
The condition sum(q) == 1 must hold for the corresponding input arguments.
The condition all(0 <= p) .and. all(p <= 1) must hold for the corresponding input arguments.
The condition all(0 <= q) .and. all(q <= 1) must hold for the corresponding input arguments.
The condition size(p) == size(q) must hold for the corresponding input arguments.
This condition is 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. The procedures under this generic interface are always impure when the input arguments p and q are procedures.

Example usage ⛓

: 1program example

2

3 use pm_kind, only: SK, IK

4 use pm_arrayRange, only: getRange

5 use pm_arrayResize, only: setResized

6 use pm_distanceBhat, only: getDisBhat

7 use pm_distGeomCyclic, only: getGeomCyclicLogPMF

8 use pm_distNorm, only: getNormLogPDF

9 use pm_distCov, only: getCovRand

10 use pm_io, only: display_type

11

12 implicit none

13

14 integer(IK) :: ndim, npnt, nsam, isam

15

16 type(display_type) :: disp

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

18

19 call disp%skip()

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

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

22 call disp%show("! Compute the Bhattacharyya distance squared for real-valued PMFs.")

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

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

25 call disp%skip()

26

27 block

28

29 use pm_kind, only: RKG => RKS ! all other real kinds are also supported.

30 real(RKG), allocatable :: p(:), q(:), stepSuccess(:)

31 integer(IK) :: period

32 real(RKG) :: bhat

33

34 call disp%skip()

35 call disp%show("period = 10;")

36 period = 10;

37 call disp%show("stepSuccess = getRange(1, period)")

38 stepSuccess = getRange(1, period)

39 call disp%show("stepSuccess")

40 call disp%show( stepSuccess )

41 call disp%show("p = exp(getGeomCyclicLogPMF(stepSuccess, probSuccess = .1_RKG, period = period))")

42 p = exp(getGeomCyclicLogPMF(stepSuccess, probSuccess = .1_RKG, period = period))

43 call disp%show("p")

44 call disp%show( p )

45 call disp%show("q = p")

46 q = p

47 call disp%show("q")

48 call disp%show( q )

49 call disp%show("bhat = getDisBhat(p, q)")

50 bhat = getDisBhat(p, q)

51 call disp%show("bhat")

52 call disp%show( bhat )

53 call disp%skip()

54

55 call disp%skip()

56 call disp%show("period = 10;")

57 period = 10;

58 call disp%show("stepSuccess = getRange(1, period)")

59 stepSuccess = getRange(1, period)

60 call disp%show("stepSuccess")

61 call disp%show( stepSuccess )

62 call disp%show("p = exp(getGeomCyclicLogPMF(stepSuccess, probSuccess = .1_RKG, period = period))")

63 p = exp(getGeomCyclicLogPMF(stepSuccess, probSuccess = .1_RKG, period = period))

64 call disp%show("p")

65 call disp%show( p )

66 call disp%show("q = exp(getGeomCyclicLogPMF(stepSuccess, probSuccess = .9_RKG, period = period))")

67 q = exp(getGeomCyclicLogPMF(stepSuccess, probSuccess = .9_RKG, period = period))

68 call disp%show("q")

69 call disp%show( q )

70 call disp%show("bhat = getDisBhat(p, q)")

71 bhat = getDisBhat(p, q)

72 call disp%show("bhat")

73 call disp%show( bhat )

74 call disp%skip()

75

76 end block

77

78 call disp%skip()

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

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

81 call disp%show("! Compute the Bhattacharyya distance squared for real-valued PDFs.")

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

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

84 call disp%skip()

85

86 block

87

88 use pm_kind, only: RKG => RKD ! all other real kinds are also supported.

89 real(RKG) :: bhat

90

91 call disp%skip()

92 call disp%show("bhat = getDisBhat(getp, getq)")

93 bhat = getDisBhat(getp, getq)

94 call disp%show("bhat")

95 call disp%show( bhat )

96 call disp%skip()

97 call disp%show("bhat = getDisBhat(getp, getq, lb = -huge(0._RKG), ub = +huge(0._RKG))")

98 bhat = getDisBhat(getp, getq, lb = -huge(0._RKG), ub = +huge(0._RKG))

99 call disp%show("bhat")

100 call disp%show( bhat )

101 call disp%skip()

102

103 end block

104

105contains

106

107 function getp(x) result(pdf)

108 use pm_kind, only: RKG => RKD ! all other real kinds are also supported.

109 real(RKG), intent(in) :: x

110 real(RKG) :: pdf

111 pdf = exp(getNormLogPDF(x))

112 end function

113

114 function getq(x) result(pdf)

115 use pm_kind, only: RKG => RKD ! all other real kinds are also supported.

116 real(RKG), intent(in) :: x

117 real(RKG) :: pdf

118 pdf = exp(getNormLogPDF(x, mu = 1._RKG))

119 end function

120

121end program example

pm_arrayRange::getRange
Generate minimally-spaced character, integer, real sequences or sequences at fixed intervals of size ...
Definition: pm_arrayRange.F90:135

pm_arrayResize::setResized
Allocate or resize (shrink or expand) an input allocatable scalar string or array of rank 1....
Definition: pm_arrayResize.F90:249

pm_distCov::getCovRand
Generate and return a random positive-definite (correlation or covariance) matrix using the Gram meth...
Definition: pm_distCov.F90:394

pm_distGeomCyclic::getGeomCyclicLogPMF
Generate and return the natural logarithm of the Probability Mass Function (PMF) of the Cyclic Geomet...
Definition: pm_distGeomCyclic.F90:198

pm_distNorm::getNormLogPDF
Generate the natural logarithm of probability density function (PDF) of the univariate Normal distrib...
Definition: pm_distNorm.F90:294

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_arrayRange
This module contains procedures and generic interfaces for generating ranges of discrete character,...
Definition: pm_arrayRange.F90:36

pm_arrayResize
This module contains procedures and generic interfaces for resizing allocatable arrays of various typ...
Definition: pm_arrayResize.F90:81

pm_distCov
This module contains classes and procedures for generating random matrices distributed on the space o...
Definition: pm_distCov.F90:72

pm_distGeomCyclic
This module contains classes and procedures for computing various statistical quantities related to t...
Definition: pm_distGeomCyclic.F90:80

pm_distNorm
This module contains classes and procedures for computing various statistical quantities related to t...
Definition: pm_distNorm.F90:179

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::RKD
integer, parameter RKD
The double precision real kind in Fortran mode. On most platforms, this is an 64-bit real kind.
Definition: pm_kind.F90:568

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_kind::RKS
integer, parameter RKS
The single-precision real kind in Fortran mode. On most platforms, this is an 32-bit real kind.
Definition: pm_kind.F90:567

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 Bhattacharyya distance squared for real-valued PMFs.

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

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

7

8

9period = 10;

10stepSuccess = getRange(1, period)

11stepSuccess

12+1.00000000, +2.00000000, +3.00000000, +4.00000000, +5.00000000, +6.00000000, +7.00000000, +8.00000000, +9.00000000, +10.0000000

13p = exp(getGeomCyclicLogPMF(stepSuccess, probSuccess = .1_RKG, period = period))

14p

15+0.153533965, +0.138180569, +0.124362521, +0.111926265, +0.100733615, +0.906602591E-1, +0.815942287E-1, +0.734348074E-1, +0.660913289E-1, +0.594821833E-1

16q = p

17q

18+0.153533965, +0.138180569, +0.124362521, +0.111926265, +0.100733615, +0.906602591E-1, +0.815942287E-1, +0.734348074E-1, +0.660913289E-1, +0.594821833E-1

19bhat = getDisBhat(p, q)

20bhat

21+0.178813949E-6

22

23

24period = 10;

25stepSuccess = getRange(1, period)

26stepSuccess

27+1.00000000, +2.00000000, +3.00000000, +4.00000000, +5.00000000, +6.00000000, +7.00000000, +8.00000000, +9.00000000, +10.0000000

28p = exp(getGeomCyclicLogPMF(stepSuccess, probSuccess = .1_RKG, period = period))

29p

30+0.153533965, +0.138180569, +0.124362521, +0.111926265, +0.100733615, +0.906602591E-1, +0.815942287E-1, +0.734348074E-1, +0.660913289E-1, +0.594821833E-1

31q = exp(getGeomCyclicLogPMF(stepSuccess, probSuccess = .9_RKG, period = period))

32q

33+0.899999976, +0.900000185E-1, +0.900000334E-2, +0.900000334E-3, +0.900000305E-4, +0.900000759E-5, +0.900000259E-6, +0.900002348E-7, +0.900001940E-8, +0.900001462E-9

34bhat = getDisBhat(p, q)

35bhat

36+0.632928014

37

38

39!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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

41! Compute the Bhattacharyya distance squared for real-valued PDFs.

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

43!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

44

45

46bhat = getDisBhat(getp, getq)

47bhat

48+0.12500000000000006

49

50bhat = getDisBhat(getp, getq, lb = -huge(0._RKG), ub = +huge(0._RKG))

51bhat

52+0.12500000000000006

53

54

Test:: test_pm_distanceBhat

Todo:: Critical Priority: The runtime checks for the complex input invCov must be implemented.

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, Monday March 6, 2017, 3:22 pm, Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin.

Definition at line 249 of file pm_distanceBhat.F90.

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

src/fortran/main/pm_distanceBhat.F90