Return the square of the Mahalanobis distance of a (set of npnt) point(s) from a single (or a set of nsam independent) sample(s) characterized by a (set of) Multivariate Normal (MVN) distribution(s) in ndim dimensions. More...

Detailed Description

Return the square of the Mahalanobis distance of a (set of npnt) point(s) from a single (or a set of nsam independent) sample(s) characterized by a (set of) Multivariate Normal (MVN) distribution(s) in ndim dimensions.

See pm_distanceMahal for the mathematical definition of the Mahalanobis distance.

Parameters

[out]	mahalSq	: The output of the same type and kind as the input `point` representing the square of the Mahalanobis distance of `point` from the MVN distribution(s) specified by the arguments `center` and `invCov`. On output, `mahalSq` is, a scalar if, the input `point` is a scalar. the input `point` is of shape `(1:ndim)` and `invCov` is of shape `(1:ndim, 1:ndim)`. a vector of size `(1:nsam)` if the input `point` is of shape `(1:ndim)` and `invCov` is of shape `(1:ndim, 1:ndim, 1:nsam)`. a vector of size `(1:npnt)` if the input `point` is of shape `(1:ndim, 1:npnt)` and `invCov` is of shape `(1:ndim, 1:ndim)`. a matrix of size `(1:nsam, 1:npnt)` if the input `point` is of shape `(1:ndim, 1:npnt)` and `invCov` is of shape `(1:ndim, 1:ndim, 1:nsam)`.
[in]	point	: The input scalar, or vector of size `ndim`, or array of shape `(ndim, npnt)` of either type `complex` of kind any supported by the processor (e.g., CK, CK32, CK64, or CK128), or type `real` of kind any supported by the processor (e.g., RK, RK32, RK64, or RK128), representing the `ndim`-dimensional coordinates of (`npnt`) point(s) whose distance(s) from the specified MVN distribution(s) should be computed.
[in]	invCov	: The input scalar or matrix of shape `(ndim, ndim)` or array of shape `(ndim, ndim, nsam)` of the same type and kind as the input `point`, representing the inverse covariance matrix(es) of the MVN distribution(s). (optional, default = `1` if and only if `point` is a scalar, otherwise, the Identity matrix of shape `(ndim, ndim)`).
[in]	center	: The input scalar or vector size `ndim` or matrix of shape `(ndim, nsam)` of the same type and kind as the input `point`, representing the center of the MVN distribution(s). (optional, default = `0`).

Possible calling interfaces ⛓

: use pm_distanceMahal, only: setDisMahalSq

call setDisMahalSq(mahalSq, point, invCov) ! elemental

call setDisMahalSq(mahalSq, point, invCov, mean) ! elemental

call setDisMahalSq(mahalSq, point(1:ndim), invCov(1:ndim, 1:ndim))

call setDisMahalSq(mahalSq, point(1:ndim), invCov(1:ndim, 1:ndim), center(1:ndim))

call setDisMahalSq(mahalSq(1:npnt), point(1:ndim, 1:npnt), invCov(1:ndim, 1:ndim))

call setDisMahalSq(mahalSq(1:npnt), point(1:ndim, 1:npnt), invCov(1:ndim, 1:ndim), center(1:ndim))

call setDisMahalSq(mahalSq(1:nsam), point(1:ndim), invCov(1:ndim, 1:ndim, 1:nsam))

call setDisMahalSq(mahalSq(1:nsam), point(1:ndim), invCov(1:ndim, 1:ndim, 1:nsam), center(1:ndim, 1:nsam))

call setDisMahalSq(mahalSq(1:nsam, 1:npnt), point(1:ndim, 1:npnt), invCov(1:ndim, 1:ndim, 1:nsam))

call setDisMahalSq(mahalSq(1:nsam, 1:npnt), point(1:ndim, 1:npnt), invCov(1:ndim, 1:ndim, 1:nsam), center(1:ndim, 1:nsam))

pm_distanceMahal::setDisMahalSq
Return the square of the Mahalanobis distance of a (set of npnt) point(s) from a single (or a set of ...
Definition: pm_distanceMahal.F90:1533

pm_distanceMahal
This module contains classes and procedures for computing the Mahalanobis statistical distance.
Definition: pm_distanceMahal.F90:82

Warning: The condition size(point, 1) == size(center, 1) must hold for the corresponding input arguments.
The condition all(size(point, 1) == [size(invCov, 1), size(invCov, 2)]) must hold for the corresponding input arguments.
The condition size(center, rank(center)) == size(invCov, rank(invCov)) must hold for the corresponding input arguments.
The condition isMatClass(invCov, posdefmat) must hold for the corresponding input arguments.
The size and shape of the input mahalSq must be consistent with other 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 this generic interface are elemental only when all input arguments are scalars or arrays of the same rank as other array-like arguments.

Note: The computation of the Mahalanobis distance for complex arguments follows the normal intrinsic Fortran rules for complex arithmetic.

Example usage ⛓

: 1program example

2

3 use pm_kind, only: SK, IK

4 use pm_arrayFill, only: getFilled

5 use pm_distUnif, only: getUnifRand

6 use pm_arrayResize, only: setResized

7 use pm_distCov, only: getCovRand

8 use pm_distanceMahal, only: setDisMahalSq

9 use pm_sampleShift, only: getShifted

10 use pm_matrixInit, only: getMatInit

11 use pm_matrixInit, only: uppLowDia

12 use pm_io, only: display_type

13

14 implicit none

15

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

17

18 type(display_type) :: disp

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

20

21 call disp%skip()

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

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

24 call disp%show("! Compute the Mahalanobis distance squared for real-valued arguments.")

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

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

27 call disp%skip()

28

29 block

30

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

32 real(RKG), allocatable :: point(:,:), mahalSq(:,:), mean(:,:), invCov(:,:,:)

33

34 call disp%show("ndim = 3; npnt = 5; nsam = 2")

35 ndim = 3; npnt = 5; nsam = 2

36 call disp%show("point = getUnifRand(0., 1., ndim, npnt)")

37 point = getUnifRand(0., 1., ndim, npnt)

38 call disp%show("point")

39 call disp%show( point )

40 call disp%show("call setResized(mean, [ndim, nsam])")

41 call setResized(mean, [ndim, nsam])

42 call disp%show("call setResized(invCov, [ndim, ndim, nsam])")

43 call setResized(invCov, [ndim, ndim, nsam])

44 call disp%show("do isam = 1, nsam")

45 do isam = 1, nsam

46 call disp%show("invCov(:,:,isam) = getCovRand(mold = 1., ndim = ndim)")

47 invCov(:,:,isam) = getCovRand(mold = 1., ndim = ndim)

48 call disp%show("mean(:,isam) = getFilled(isam - 1, ndim)")

49 mean(:,isam) = getFilled(isam - 1, ndim)

50 call disp%show("end do")

51 end do

52 call disp%show("invCov")

53 call disp%show( invCov )

54 call disp%show("mean")

55 call disp%show( mean )

56 call disp%show("call setResized(mahalSq, [nsam, npnt])")

57 call setResized(mahalSq, [nsam, npnt])

58

59

60 call disp%show("call setDisMahalSq(mahalSq, point, invCov) ! for a set of ndim-dimensional points against nsam independent samples centered at origin.")

61 call setDisMahalSq(mahalSq, point, invCov)

62 call disp%show("mahalSq")

63 call disp%show( mahalSq )

64 call disp%show("call setDisMahalSq(mahalSq, point, invCov, mean) ! for a set of ndim-dimensional points against nsam independent samples centered at mean.")

65 call setDisMahalSq(mahalSq, point, invCov, mean)

66 call disp%show("mahalSq")

67 call disp%show( mahalSq )

68

69 call disp%show("call setDisMahalSq(mahalSq(1:nsam, 1), point(1:ndim, 1), invCov) ! for a single ndim-dimensional point against nsam independent samples centered at origin.")

70 call setDisMahalSq(mahalSq(1:nsam, 1), point(1:ndim, 1), invCov)

71 call disp%show("mahalSq(1:nsam, 1)")

72 call disp%show( mahalSq(1:nsam, 1) )

73 call disp%show("call setDisMahalSq(mahalSq(1:nsam, 1), point(1:ndim, 1), invCov, mean) ! for a single ndim-dimensional point against nsam independent samples centered at mean.")

74 call setDisMahalSq(mahalSq(1:nsam, 1), point(1:ndim, 1), invCov, mean)

75 call disp%show("mahalSq(1:nsam, 1)")

76 call disp%show( mahalSq(1:nsam, 1) )

77

78

79 call disp%show("call setDisMahalSq(mahalSq(1, 1:npnt), point, invCov(:,:,1)) ! for a set of ndim-dimensional points against one sample centered at origin.")

80 call setDisMahalSq(mahalSq(1, 1:npnt), point, invCov(:,:,1))

81 call disp%show("mahalSq(1, 1:npnt)")

82 call disp%show( mahalSq(1, 1:npnt) )

83 call disp%show("call setDisMahalSq(mahalSq(1, 1:npnt), point, invCov(:,:,1), mean(:,1)) ! for a set of ndim-dimensional points against one sample centered at mean.")

84 call setDisMahalSq(mahalSq(1, 1:npnt), point, invCov(:,:,1), mean(:,1))

85 call disp%show("mahalSq(1, 1:npnt)")

86 call disp%show( mahalSq(1, 1:npnt) )

87

88 call disp%show("call setDisMahalSq(mahalSq(1, 1), point(1:ndim, 1), invCov(:,:,1)) ! for a single ndim-dimensional point one sample centered at origin.")

89 call setDisMahalSq(mahalSq(1, 1), point(1:ndim, 1), invCov(:,:,1))

90 call disp%show("mahalSq(1, 1)")

91 call disp%show( mahalSq(1, 1) )

92 call disp%show("call setDisMahalSq(mahalSq(1, 1), point(1:ndim, 1), invCov(:,:,1), mean(:,1)) ! for a single ndim-dimensional point one sample centered at mean.")

93 call setDisMahalSq(mahalSq(1, 1), point(1:ndim, 1), invCov(:,:,1), mean(:,1))

94 call disp%show("mahalSq(1, 1)")

95 call disp%show( mahalSq(1, 1) )

96

97

98 call disp%show("call setDisMahalSq(mahalSq(1, 1:npnt), point(1, 1:npnt), invCov(1,1,1)) ! for a set of single-dimensional points against one sample centered at origin.")

99 call setDisMahalSq(mahalSq(1, 1:npnt), point(1, 1:npnt), invCov(1,1,1))

100 call disp%show("mahalSq(1, 1:npnt)")

101 call disp%show( mahalSq(1, 1:npnt) )

102 call disp%show("call setDisMahalSq(mahalSq(1, 1:npnt), point(1, 1:npnt), invCov(1,1,1), mean(1,1)) ! for a set of single-dimensional points against one sample centered at mean.")

103 call setDisMahalSq(mahalSq(1, 1:npnt), point(1, 1:npnt), invCov(1,1,1), mean(1,1))

104 call disp%show("mahalSq(1, 1:npnt)")

105 call disp%show( mahalSq(1, 1:npnt) )

106

107 call disp%show("call setDisMahalSq(mahalSq(1, 1), point(1, 1), invCov(1,1,1)) ! for a single single-dimensional point one sample centered at origin.")

108 call setDisMahalSq(mahalSq(1, 1), point(1, 1), invCov(1,1,1))

109 call disp%show("mahalSq(1, 1)")

110 call disp%show( mahalSq(1, 1) )

111 call disp%show("call setDisMahalSq(mahalSq(1, 1), point(1, 1), invCov(1,1,1), mean(1,1)) ! for a single single-dimensional point one sample centered at mean.")

112 call setDisMahalSq(mahalSq(1, 1), point(1, 1), invCov(1,1,1), mean(1,1))

113 call disp%show("mahalSq(1, 1)")

114 call disp%show( mahalSq(1, 1) )

115

116 end block

117

118 call disp%skip()

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

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

121 call disp%show("! Compute the Mahalanobis distance squared for complex-valued arguments.")

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

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

124 call disp%skip()

125

126 block

127

128 use pm_kind, only: CKG => CKS ! all other complex kinds are also supported.

129 complex(CKG), allocatable :: point(:,:), mahalSq(:,:), mean(:,:), invCov(:,:,:)

130

131 call disp%show("ndim = 3; npnt = 5; nsam = 2")

132 ndim = 3; npnt = 5; nsam = 2

133 call disp%show("point = getUnifRand(0., 1., ndim, npnt)")

134 point = getUnifRand(0., 1., ndim, npnt)

135 call disp%show("point")

136 call disp%show( point )

137 call disp%show("call setResized(mean, [ndim, nsam])")

138 call setResized(mean, [ndim, nsam])

139 call disp%show("call setResized(invCov, [ndim, ndim, nsam])")

140 call setResized(invCov, [ndim, ndim, nsam])

141 call disp%show("do isam = 1, nsam")

142 do isam = 1, nsam

143 call disp%show("invCov(:,:,isam) = getCovRand(mold = (1., 1.), ndim = ndim)")

144 invCov(:,:,isam) = getCovRand(mold = (1., 1.), ndim = ndim)

145 call disp%show("mean(:,isam) = getFilled(isam - 1, ndim)")

146 mean(:,isam) = getFilled(isam - 1, ndim)

147 call disp%show("end do")

148 end do

149 call disp%show("invCov")

150 call disp%show( invCov )

151 call disp%show("mean")

152 call disp%show( mean )

153 call disp%show("call setResized(mahalSq, [nsam, npnt])")

154 call setResized(mahalSq, [nsam, npnt])

155

156

157 call disp%show("call setDisMahalSq(mahalSq, point, invCov) ! for a set of ndim-dimensional points against nsam independent samples centered at origin.")

158 call setDisMahalSq(mahalSq, point, invCov)

159 call disp%show("mahalSq")

160 call disp%show( mahalSq )

161 call disp%show("call setDisMahalSq(mahalSq, point, invCov, mean) ! for a set of ndim-dimensional points against nsam independent samples centered at mean.")

162 call setDisMahalSq(mahalSq, point, invCov, mean)

163 call disp%show("mahalSq")

164 call disp%show( mahalSq )

165

166 call disp%show("call setDisMahalSq(mahalSq(1:nsam, 1), point(1:ndim, 1), invCov) ! for a single ndim-dimensional point against nsam independent samples centered at origin.")

167 call setDisMahalSq(mahalSq(1:nsam, 1), point(1:ndim, 1), invCov)

168 call disp%show("mahalSq(1:nsam, 1)")

169 call disp%show( mahalSq(1:nsam, 1) )

170 call disp%show("call setDisMahalSq(mahalSq(1:nsam, 1), point(1:ndim, 1), invCov, mean) ! for a single ndim-dimensional point against nsam independent samples centered at mean.")

171 call setDisMahalSq(mahalSq(1:nsam, 1), point(1:ndim, 1), invCov, mean)

172 call disp%show("mahalSq(1:nsam, 1)")

173 call disp%show( mahalSq(1:nsam, 1) )

174

175

176 call disp%show("call setDisMahalSq(mahalSq(1, 1:npnt), point, invCov(:,:,1)) ! for a set of ndim-dimensional points against one sample centered at origin.")

177 call setDisMahalSq(mahalSq(1, 1:npnt), point, invCov(:,:,1))

178 call disp%show("mahalSq(1, 1:npnt)")

179 call disp%show( mahalSq(1, 1:npnt) )

180 call disp%show("call setDisMahalSq(mahalSq(1, 1:npnt), point, invCov(:,:,1), mean(:,1)) ! for a set of ndim-dimensional points against one sample centered at mean.")

181 call setDisMahalSq(mahalSq(1, 1:npnt), point, invCov(:,:,1), mean(:,1))

182 call disp%show("mahalSq(1, 1:npnt)")

183 call disp%show( mahalSq(1, 1:npnt) )

184

185 call disp%show("call setDisMahalSq(mahalSq(1, 1), point(1:ndim, 1), invCov(:,:,1)) ! for a single ndim-dimensional point one sample centered at origin.")

186 call setDisMahalSq(mahalSq(1, 1), point(1:ndim, 1), invCov(:,:,1))

187 call disp%show("mahalSq(1, 1)")

188 call disp%show( mahalSq(1, 1) )

189 call disp%show("call setDisMahalSq(mahalSq(1, 1), point(1:ndim, 1), invCov(:,:,1), mean(:,1)) ! for a single ndim-dimensional point one sample centered at mean.")

190 call setDisMahalSq(mahalSq(1, 1), point(1:ndim, 1), invCov(:,:,1), mean(:,1))

191 call disp%show("mahalSq(1, 1)")

192 call disp%show( mahalSq(1, 1) )

193

194

195 call disp%show("call setDisMahalSq(mahalSq(1, 1:npnt), point(1, 1:npnt), invCov(1,1,1)) ! for a set of single-dimensional points against one sample centered at origin.")

196 call setDisMahalSq(mahalSq(1, 1:npnt), point(1, 1:npnt), invCov(1,1,1))

197 call disp%show("mahalSq(1, 1:npnt)")

198 call disp%show( mahalSq(1, 1:npnt) )

199 call disp%show("call setDisMahalSq(mahalSq(1, 1:npnt), point(1, 1:npnt), invCov(1,1,1), mean(1,1)) ! for a set of single-dimensional points against one sample centered at mean.")

200 call setDisMahalSq(mahalSq(1, 1:npnt), point(1, 1:npnt), invCov(1,1,1), mean(1,1))

201 call disp%show("mahalSq(1, 1:npnt)")

202 call disp%show( mahalSq(1, 1:npnt) )

203

204 call disp%show("call setDisMahalSq(mahalSq(1, 1), point(1, 1), invCov(1,1,1)) ! for a single single-dimensional point one sample centered at origin.")

205 call setDisMahalSq(mahalSq(1, 1), point(1, 1), invCov(1,1,1))

206 call disp%show("mahalSq(1, 1)")

207 call disp%show( mahalSq(1, 1) )

208 call disp%show("call setDisMahalSq(mahalSq(1, 1), point(1, 1), invCov(1,1,1), mean(1,1)) ! for a single single-dimensional point one sample centered at mean.")

209 call setDisMahalSq(mahalSq(1, 1), point(1, 1), invCov(1,1,1), mean(1,1))

210 call disp%show("mahalSq(1, 1)")

211 call disp%show( mahalSq(1, 1) )

212

213 end block

214

215end program example

pm_arrayFill::getFilled
Generate and return an array of the specified rank and shape of arbitrary intrinsic type and kind wit...
Definition: pm_arrayFill.F90:132

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_distUnif::getUnifRand
Generate and return a scalar or a contiguous array of rank 1 of length s1 of randomly uniformly distr...
Definition: pm_distUnif.F90:4150

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_matrixInit::getMatInit
Generate and return a matrix of shape (shape(1), shape(2)) with the upper/lower triangle and the diag...
Definition: pm_matrixInit.F90:289

pm_sampleShift::getShifted
Generate a sample of shape (nsam), or (ndim, nsam) or (nsam, ndim) that is shifted by the specified i...
Definition: pm_sampleShift.F90:197

pm_arrayFill
This module contains procedures and generic interfaces for convenient allocation and filling of array...
Definition: pm_arrayFill.F90:42

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_distUnif
This module contains classes and procedures for computing various statistical quantities related to t...
Definition: pm_distUnif.F90:274

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::CKS
integer, parameter CKS
The single-precision complex kind in Fortran mode. On most platforms, this is a 32-bit real kind.
Definition: pm_kind.F90:570

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_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_matrixInit
This module contains procedures and generic interfaces relevant to generating and initializing matric...
Definition: pm_matrixInit.F90:116

pm_sampleShift
This module contains classes and procedures for shifting univariate or multivariate samples by arbitr...
Definition: pm_sampleShift.F90:59

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 Mahalanobis distance squared for real-valued arguments.

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

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

7

8ndim = 3; npnt = 5; nsam = 2

9point = getUnifRand(0., 1., ndim, npnt)

10point

11+0.108310223, +0.163552523, +0.154336512, +0.882765174, +0.135043442

12+0.837170243, +0.339359999, +0.212025642E-2, +0.179435432, +0.957951069

13+0.714514554, +0.269619286, +0.664380789E-1, +0.684940279, +0.811459124

14call setResized(mean, [ndim, nsam])

15call setResized(invCov, [ndim, ndim, nsam])

16do isam = 1, nsam

17invCov(:,:,isam) = getCovRand(mold = 1., ndim = ndim)

18mean(:,isam) = getFilled(isam - 1, ndim)

19end do

20invCov(:,:,isam) = getCovRand(mold = 1., ndim = ndim)

21mean(:,isam) = getFilled(isam - 1, ndim)

22end do

23invCov

24slice(:,:,1) =

25+1.00000000, +0.873463392, -0.295291334

26+0.873463392, +1.00000000, -0.700885773

27-0.295291334, -0.700885773, +1.00000000

28slice(:,:,2) =

29+1.00000000, +0.988503277, +0.758755982

30+0.988503277, +1.00000000, +0.776153624

31+0.758755982, +0.776153624, +1.00000000

32mean

33+0.00000000, +1.00000000

34+0.00000000, +1.00000000

35+0.00000000, +1.00000000

36call setResized(mahalSq, [nsam, npnt])

37call setDisMahalSq(mahalSq, point, invCov) ! for a set of ndim-dimensional points against nsam independent samples centered at origin.

38mahalSq

39+0.497314304, +0.157267034, +0.225567333E-1, +1.02795458, +0.665996194

40+2.44836283, +0.533289969, +0.446641780E-1, +2.70210433, +3.22308826

41call setDisMahalSq(mahalSq, point, invCov, mean) ! for a set of ndim-dimensional points against nsam independent samples centered at mean.

42mahalSq

43+0.497314304, +0.157267034, +0.225567333E-1, +1.02795458, +0.665996194

44+1.64863944, +4.43812466, +6.89493227, +1.43388224, +1.11715221

45call setDisMahalSq(mahalSq(1:nsam, 1), point(1:ndim, 1), invCov) ! for a single ndim-dimensional point against nsam independent samples centered at origin.

46mahalSq(1:nsam, 1)

47+0.497314304, +2.44836283

48call setDisMahalSq(mahalSq(1:nsam, 1), point(1:ndim, 1), invCov, mean) ! for a single ndim-dimensional point against nsam independent samples centered at mean.

49mahalSq(1:nsam, 1)

50+0.497314304, +1.64863944

51call setDisMahalSq(mahalSq(1, 1:npnt), point, invCov(:,:,1)) ! for a set of ndim-dimensional points against one sample centered at origin.

52mahalSq(1, 1:npnt)

53+0.497314304, +0.157267034, +0.225567333E-1, +1.02795458, +0.665996194

54call setDisMahalSq(mahalSq(1, 1:npnt), point, invCov(:,:,1), mean(:,1)) ! for a set of ndim-dimensional points against one sample centered at mean.

55mahalSq(1, 1:npnt)

56+0.497314304, +0.157267034, +0.225567333E-1, +1.02795458, +0.665996194

57call setDisMahalSq(mahalSq(1, 1), point(1:ndim, 1), invCov(:,:,1)) ! for a single ndim-dimensional point one sample centered at origin.

58mahalSq(1, 1)

59+0.497314304

60call setDisMahalSq(mahalSq(1, 1), point(1:ndim, 1), invCov(:,:,1), mean(:,1)) ! for a single ndim-dimensional point one sample centered at mean.

61mahalSq(1, 1)

62+0.497314304

63call setDisMahalSq(mahalSq(1, 1:npnt), point(1, 1:npnt), invCov(1,1,1)) ! for a set of single-dimensional points against one sample centered at origin.

64mahalSq(1, 1:npnt)

65+0.117311040E-1, +0.267494284E-1, +0.238197595E-1, +0.779274344, +0.182367321E-1

66call setDisMahalSq(mahalSq(1, 1:npnt), point(1, 1:npnt), invCov(1,1,1), mean(1,1)) ! for a set of single-dimensional points against one sample centered at mean.

67mahalSq(1, 1:npnt)

68+0.117311040E-1, +0.267494284E-1, +0.238197595E-1, +0.779274344, +0.182367321E-1

69call setDisMahalSq(mahalSq(1, 1), point(1, 1), invCov(1,1,1)) ! for a single single-dimensional point one sample centered at origin.

70mahalSq(1, 1)

71+0.117311040E-1

72call setDisMahalSq(mahalSq(1, 1), point(1, 1), invCov(1,1,1), mean(1,1)) ! for a single single-dimensional point one sample centered at mean.

73mahalSq(1, 1)

74+0.117311040E-1

75

76!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

77!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

78! Compute the Mahalanobis distance squared for complex-valued arguments.

79!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

80!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

81

82ndim = 3; npnt = 5; nsam = 2

83point = getUnifRand(0., 1., ndim, npnt)

84point

85(+0.375551045, +0.00000000), (+0.126413524, +0.00000000), (+0.876723468, +0.00000000), (+0.668731630, +0.00000000), (+0.109771073, +0.00000000)

86(+0.368765593E-1, +0.00000000), (+0.132633448E-1, +0.00000000), (+0.931067467, +0.00000000), (+0.403245270, +0.00000000), (+0.686878026, +0.00000000)

87(+0.635960162, +0.00000000), (+0.819039464, +0.00000000), (+0.738346100, +0.00000000), (+0.984976768, +0.00000000), (+0.862937272, +0.00000000)

88call setResized(mean, [ndim, nsam])

89call setResized(invCov, [ndim, ndim, nsam])

90do isam = 1, nsam

91invCov(:,:,isam) = getCovRand(mold = (1., 1.), ndim = ndim)

92mean(:,isam) = getFilled(isam - 1, ndim)

93end do

94invCov(:,:,isam) = getCovRand(mold = (1., 1.), ndim = ndim)

95mean(:,isam) = getFilled(isam - 1, ndim)

96end do

97invCov

98slice(:,:,1) =

99(+1.00000000, +0.00000000), (-0.561239839, -0.408925712), (-0.210951618E-2, -0.613281988E-1)

100(-0.561239839, +0.408925712), (+1.00000000, +0.00000000), (+0.477281570, +0.271481603)

101(-0.210951618E-2, +0.613281988E-1), (+0.477281570, -0.271481603), (+1.00000000, +0.00000000)

102slice(:,:,2) =

103(+1.00000000, +0.00000000), (+0.502396882, -0.224990189), (-0.244474053, +0.524210095)

104(+0.502396882, +0.224990189), (+1.00000000, +0.00000000), (+0.224503934, +0.455577910)

105(-0.244474053, -0.524210095), (+0.224503934, -0.455577910), (+1.00000000, +0.00000000)

106mean

107(+0.00000000, +0.00000000), (+1.00000000, +0.00000000)

108(+0.00000000, +0.00000000), (+1.00000000, +0.00000000)

109(+0.00000000, +0.00000000), (+1.00000000, +0.00000000)

110call setResized(mahalSq, [nsam, npnt])

111call setDisMahalSq(mahalSq, point, invCov) ! for a set of ndim-dimensional points against nsam independent samples centered at origin.

112mahalSq

113(+0.552677333, -0.931322575E-9), (+0.695032716, +0.00000000), (+1.91790152, +0.00000000), (+1.65365815, -0.745058060E-8), (+1.70927870, +0.00000000)

114(+0.454511195, -0.149011612E-7), (+0.642919779, +0.00000000), (+2.99304914, +0.00000000), (+1.70722079, +0.596046448E-7), (+1.52409840, +0.298023224E-7)

115call setDisMahalSq(mahalSq, point, invCov, mean) ! for a set of ndim-dimensional points against nsam independent samples centered at mean.

116mahalSq

117(+0.552677333, -0.931322575E-9), (+0.695032716, +0.00000000), (+1.91790152, +0.00000000), (+1.65365815, -0.745058060E-8), (+1.70927870, +0.00000000)

118(+2.10065198, -0.298023224E-7), (+2.63856125, -0.149011612E-7), (+0.892771631E-1, +0.00000000), (+0.666306376, -0.325962901E-8), (+1.14903581, +0.745058060E-8)

119call setDisMahalSq(mahalSq(1:nsam, 1), point(1:ndim, 1), invCov) ! for a single ndim-dimensional point against nsam independent samples centered at origin.

120mahalSq(1:nsam, 1)

121(+0.552677333, -0.931322575E-9), (+0.454511195, -0.149011612E-7)

122call setDisMahalSq(mahalSq(1:nsam, 1), point(1:ndim, 1), invCov, mean) ! for a single ndim-dimensional point against nsam independent samples centered at mean.

123mahalSq(1:nsam, 1)

124(+0.552677333, -0.931322575E-9), (+2.10065198, -0.298023224E-7)

125call setDisMahalSq(mahalSq(1, 1:npnt), point, invCov(:,:,1)) ! for a set of ndim-dimensional points against one sample centered at origin.

126mahalSq(1, 1:npnt)

127(+0.552677333, -0.931322575E-9), (+0.695032716, +0.00000000), (+1.91790152, +0.00000000), (+1.65365815, -0.745058060E-8), (+1.70927870, +0.00000000)

128call setDisMahalSq(mahalSq(1, 1:npnt), point, invCov(:,:,1), mean(:,1)) ! for a set of ndim-dimensional points against one sample centered at mean.

129mahalSq(1, 1:npnt)

130(+0.552677333, -0.931322575E-9), (+0.695032716, +0.00000000), (+1.91790152, +0.00000000), (+1.65365815, -0.745058060E-8), (+1.70927870, +0.00000000)

131call setDisMahalSq(mahalSq(1, 1), point(1:ndim, 1), invCov(:,:,1)) ! for a single ndim-dimensional point one sample centered at origin.

132mahalSq(1, 1)

133(+0.552677333, -0.931322575E-9)

134call setDisMahalSq(mahalSq(1, 1), point(1:ndim, 1), invCov(:,:,1), mean(:,1)) ! for a single ndim-dimensional point one sample centered at mean.

135mahalSq(1, 1)

136(+0.552677333, -0.931322575E-9)

137call setDisMahalSq(mahalSq(1, 1:npnt), point(1, 1:npnt), invCov(1,1,1)) ! for a set of single-dimensional points against one sample centered at origin.

138mahalSq(1, 1:npnt)

139(+0.141038582, +0.00000000), (+0.159803797E-1, +0.00000000), (+0.768644035, +0.00000000), (+0.447201997, +0.00000000), (+0.120496880E-1, +0.00000000)

140call setDisMahalSq(mahalSq(1, 1:npnt), point(1, 1:npnt), invCov(1,1,1), mean(1,1)) ! for a set of single-dimensional points against one sample centered at mean.

141mahalSq(1, 1:npnt)

142(+0.141038582, +0.00000000), (+0.159803797E-1, +0.00000000), (+0.768644035, +0.00000000), (+0.447201997, +0.00000000), (+0.120496880E-1, +0.00000000)

143call setDisMahalSq(mahalSq(1, 1), point(1, 1), invCov(1,1,1)) ! for a single single-dimensional point one sample centered at origin.

144mahalSq(1, 1)

145(+0.141038582, +0.00000000)

146call setDisMahalSq(mahalSq(1, 1), point(1, 1), invCov(1,1,1), mean(1,1)) ! for a single single-dimensional point one sample centered at mean.

147mahalSq(1, 1)

148(+0.141038582, +0.00000000)

149

Test:: test_pm_distanceMahal

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

Todo:: High Priority: The performance of the implementation for complex input can be improved by using do_product on columns of invCov instead of the current implementation working with rows of invCov in matmul.

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.
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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 1533 of file pm_distanceMahal.F90.

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

src/fortran/main/pm_distanceMahal.F90