Return the covariance matrix corresponding to the input (potentially weighted) correlation matrix or return the biased sample covariance matrix of the input array of shape (ndim, nsam) or (nsam, ndim) or a pair of (potentially weighted) time series x(1:nsam) and y(1:nsam) where ndim is the number of data dimensions (the number of data attributes) and nsam is the number of data points.
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Detailed Description

Return the covariance matrix corresponding to the input (potentially weighted) correlation matrix or return the biased sample covariance matrix of the input array of shape (ndim, nsam) or (nsam, ndim) or a pair of (potentially weighted) time series x(1:nsam) and y(1:nsam) where ndim is the number of data dimensions (the number of data attributes) and nsam is the number of data points.

This generic interface performs either of the following two computational tasks:

compute the covariance matrix corresponding to an input correlation matrix and vector of standard deviations in arbitrary ndim dimensions.
compute the sample covariance matrix of a random sample of nsam observations in arbitrary ndim dimensional space.

See the documentation of the parent module pm_sampleCov for algorithmic details and sample covariance matrix definition.

Parameters

[in,out]	cov	: The output positive semi-definite square matrix of shape `(1:ndim, 1:ndim)` of, type `complex` of kind any supported by the processor (e.g., CK, CK32, CK64, or CK128), type `real` of kind any supported by the processor (e.g., RK, RK32, RK64, or RK128), representing the covariance matrix corresponding to the input `sample` or `cor`, whichever is present. On output, If the input arguments `x` and `y` are present, then `ndim == 2` holds by definition and the output covariance is of the form \begin{equation} \ms{cov} = \begin{bmatrix} \sigma_{xx} && \sigma_{yx} \\ \sigma_{xy} && \sigma_{yy} \end{bmatrix} \end{equation} Otherwise, is `subset` is present, then only the specified input `subset` will be overwritten with the covariance matrix. Any elements not in the specified input `subset` remain intact.
[in]	subset	: The input scalar constant argument that can be any of the following: The constant lowDia, implying that only the lower-diagonal subset of the output covariance matrix must be computed. The constant uppDia, implying that only the upper-diagonal subset of the output covariance matrix must be computed. This input argument is merely serves to resolve the different procedures of this generic interface from each other at compile-time. (optional. It must be present if and only if the input arguments `x` and `y` are both missing.)
[in]	cor	: The input positive semi-definite square matrix of shape `(1:ndim, 1:ndim)` of the same type and kind as the input `cov`, representing the correlation matrix based upon which the output covariance matrix `cov` is to be computed. (optional. It must be present if and only if the input arguments `std` and `subsetr` are present and the rest are missing.)
[in]	subsetr	: The input scalar constant argument that can be any of the following: The constant lowDia, implying that only the lower-diagonal subset of the input correlation matrix must be used. The constant uppDia, implying that only the upper-diagonal subset of the input correlation matrix must be used. This input argument is merely serves to resolve the different procedures of this generic interface from each other at compile-time. Although the allowed subset constants imply the use of the diagonal elements, they are by definition assumed to be `1` and therefore are not referenced in the algorithm. (optional. It must be present if and only if the input arguments `subset`, `cor`, and `std` are present and the rest are missing.)
[in]	std	: The input positive vector of shape `(1:ndim)` of type `real` of the same kind as the input `cov`, containing the standard deviations of the `ndim` data attributes based upon which the output covariance matrix `cov` is to be computed. (optional. It must be present if and only if the input arguments `cor` and `subsetr` are present and the rest are missing.)
[in]	mean	: The input `contiguous` vector of shape `(ndim)` of the same type and kind as the output `cov` containing the `sample` mean. If the input `sample` is missing and `x` and `y` are present, then `mean` must be a vector of size `ndim = 2`. If the input `sample` is present and is a 2D array, then `mean` must be a vector of size `ndim = size(sample, 3 - dim)` (i.e., computed along the specified input dimension `dim`). The mean vector can be readily computed via getMean or setMean. (optional. default = getFilled(0., ndim) or `[0., 0.]` depending on the rank of the input `sample` and `dim` or the presence of `x` and `y`.)
[in]	x	: The input `contiguous` vector of shape `(nsam)` of the same type and kind as the output `cov`, containing the first attribute `x` of the observational sample, where `nsam` is the number of observations in the sample. (optional. It must be present if and only if the input argument `y` is present and `cor` and `sample` are missing.)
[in]	y	: The input `contiguous` vector of shape `(nsam)` of the same type and kind as the output `cov`, containing the second attribute `x` of the observational sample, where `nsam` is the number of observations in the sample. (optional. It must be present if and only if the input argument `x` is present and `cor` and `sample` are missing.)
[in]	sample	: The input `contiguous` array of shape `(ndim, nsam)` or `(nsam, ndim)` of the same type and kind as the output `cov`, containing the sample comprised of `nsam` observations each with `ndim` attributes. If `sample` is a matrix, then the direction along which the covariance matrix is computed is dictated by the input argument `dim`. (optional. It must be present if and only if the input argument `dim` is present and `x` and `y` are missing.)
[in]	dim	: The input scalar `integer` of default kind IK indicating the dimension of `sample` along which the covariance matrix must be computed. If `dim = 1`, the input `sample` is assumed to have the shape `(nsam, ndim)`. If `dim = 2`, the input `sample` is assumed to have the shape `(ndim, nsam)`. (optional. It must be present if and only if the input argument `sample` is present.)
[in]	weight	: The `contiguous` vector of length `nsam` of, type `integer` of default kind IK, or type `real` of the same kind as the kind of the output `cov`, containing the corresponding weights of individual `nsam` observations in `sample`. (optional. default = getFilled(1, nsam). It can be present if and only if the input arguments `sample` or `x` and `y` are present.)
[in]	weisum	: The input scalar of the same type and kind as the input `weight` containing `sum(weight)`. This quantity is a byproduct of computing the mean of a sample and is automatically returned by setMean. (optional. It must be present if and only if the `weight` argument is present and the input arguments `cor`, `subsetr`, and `std` are missing.)

Possible calling interfaces ⛓

: use pm_sampleCov, only: setCov, lowDia, uppDia

! correlation matrix to covariance matrix.

call setCov(cov(1:ndim, 1:ndim), subset, cor(1:ndim, 1:ndim), subsetr, std(1:ndim))

! XY time series covariance matrix.

call setCov(cov(1:2, 1:2) , x(1:nsam), y(1:nsam))

call setCov(cov(1:2, 1:2), mean(1:2), x(1:nsam), y(1:nsam))

call setCov(cov(1:2, 1:2) , x(1:nsam), y(1:nsam), weight(1:nsam))

call setCov(cov(1:2, 1:2), mean(1:2), x(1:nsam), y(1:nsam), weight(1:nsam), weisum)

! sample covariance matrix.

call setCov(cov(1:ndim, 1:ndim), subset , sample(:,:), dim)

call setCov(cov(1:ndim, 1:ndim), subset, mean(1:ndim), sample(:,:), dim)

call setCov(cov(1:ndim, 1:ndim), subset, , sample(:,:), dim, weight(1:nsam))

call setCov(cov(1:ndim, 1:ndim), subset, mean(1:ndim), sample(:,:), dim, weight(1:nsam), weisum)

pm_sampleCov::setCov
Return the covariance matrix corresponding to the input (potentially weighted) correlation matrix or ...
Definition: pm_sampleCov.F90:1694

pm_sampleCov
This module contains classes and procedures for computing the properties related to the covariance ma...
Definition: pm_sampleCov.F90:170

Warning: The condition any(x(1) /= x) must hold for the corresponding input arguments.
The condition any(y(1) /= x) must hold for the corresponding input arguments.
The input sample must contain at least two unique values per sample attribute.
The condition 0 < sum(weight) must hold for the corresponding input arguments.
The condition all(0. <= weight) must hold for the corresponding input arguments.
The condition 1 < size(sample, dim) must hold for the corresponding input arguments.
The condition 0 < size(sample, 3 - dim) must hold for the corresponding input arguments.
The condition all(shape(cov) == shape(cor)) must hold for the corresponding input arguments.
The condition 1 <= dim .and. dim <= rank(sample) must hold for the corresponding input arguments.
The condition size(sample, 3 - dim) == size(variance) must hold for the corresponding input arguments.
The condition size(sample, 3 - dim) == size(mean) must hold for the corresponding input arguments.
The condition size(sample, dim) == size(weight) must hold for the corresponding input arguments.
The condition size(x) == size(weight) must hold for the corresponding input arguments.
The condition size(x) == size(y) 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.

Note: Note the effects of bias-correction in computing the covariance matrix become noticeable only for very sample sample sizes (i.e., when nsam is small).

See also: getCor
setCor
getCov
setCov
getVar
setVar
getMean
setMean
getCovMerged
setCovMerged
setCovMeanMerged
getVarCorrection
getMeanMerged
setMeanMerged
getVarMerged
setVarMerged

Example usage ⛓

: 1program example

2

3 use pm_kind, only: SK, IK, LK

4 use pm_sampleMean, only: getMean

5 use pm_sampleMean, only: setMean

6 use pm_sampleShift, only: getShifted

7 use pm_arraySpace, only: getLinSpace

8 use pm_arrayResize, only: setResized

9 use pm_distUnif, only: getUnifRand

10 use pm_arrayFill, only: getFilled

11 use pm_sampleCov, only: uppDia

12 use pm_sampleCov, only: lowDia

13 use pm_sampleCov, only: setCov

14 use pm_io, only: display_type

15 use pm_io, only: getFormat

16

17 implicit none

18

19 integer(IK) :: itry, ntry = 10

20 type(display_type) :: disp

21 character(:), allocatable :: format

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

23

24 call disp%skip()

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

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

27 call disp%show("!Convert correlation matrix and standard deviation to covariance matrix.")

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

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

30 call disp%skip()

31

32 block

33 use pm_kind, only: TKG => RKS ! All other real types are also supported.

34 use pm_matrixCopy, only: setMatCopy, rdpack

35 use pm_distCov, only: getCovRand

36 integer(IK) :: ndim

37 real(TKG), allocatable :: cov(:,:), cor(:,:), std(:)

38 format = getFormat(mold = [0._TKG], ed = SK_"es", signed = .true._LK)

39 do itry = 1, ntry

40 call disp%skip()

41 call disp%show("ndim = getUnifRand(1, 7)")

42 ndim = getUnifRand(1, 7)

43 call disp%show("ndim")

44 call disp%show( ndim )

45 call disp%show("std = getUnifRand(1, 10, ndim)")

46 std = getUnifRand(1, 10, ndim)

47 call disp%show("std")

48 call disp%show( std , format = format )

49 call disp%show("cor = getCovRand(1., ndim)")

50 cor = getCovRand(1., ndim)

51 call disp%show("cor")

52 call disp%show( cor , format = format )

53 call disp%show("cov = getFilled(0._TKG, ndim, ndim + 1)")

54 cov = getFilled(0._TKG, ndim, ndim + 1)

55 call disp%show("call setMatCopy(cov(1:ndim, 1:ndim), rdpack, cor, rdpack, lowDia)")

56 call setMatCopy(cov(1:ndim, 1:ndim), rdpack, cor, rdpack, lowDia)

57 call disp%show("cov")

58 call disp%show( cov , format = format )

59 call disp%show("call setCov(cov(1:ndim, 2:ndim+1), uppDia, cov(1:ndim, 1:ndim), lowDia, std) ! convert lower-triangle correlation matrix to upper-triangle covariance matrix.")

60 call setCov(cov(1:ndim, 2:ndim+1), uppDia, cov(1:ndim, 1:ndim), lowDia, std)

61 call disp%show("cov")

62 call disp%show( cov , format = format )

63 call disp%skip()

64 call disp%show("cov = getFilled(0._TKG, ndim, ndim + 1)")

65 cov = getFilled(0._TKG, ndim, ndim + 1)

66 call disp%show("call setMatCopy(cov(1:ndim, 2:ndim+1), rdpack, cor, rdpack, uppDia)")

67 call setMatCopy(cov(1:ndim, 2:ndim+1), rdpack, cor, rdpack, uppDia)

68 call disp%show("cov")

69 call disp%show( cov , format = format )

70 call disp%show("call setCov(cov(1:ndim, 1:ndim), lowDia, cov(1:ndim, 2:ndim+1), uppDia, std) ! convert upper-triangle correlation matrix to lower-triangle covariance matrix.")

71 call setCov(cov(1:ndim, 1:ndim), lowDia, cov(1:ndim, 2:ndim+1), uppDia, std)

72 call disp%show("cov")

73 call disp%show( cov , format = format )

74 call disp%skip()

75 end do

76 end block

77

78 call disp%skip()

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

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

81 call disp%show("!Compute the covariance matrix of a 2-D sample.")

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

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

84 call disp%skip()

85

86 block

87 use pm_kind, only: TKG => RKS ! All other real types are also supported.

88 real(TKG), allocatable :: sample(:,:), samShifted(:,:), cov(:,:), mean(:)

89 integer(IK) :: ndim, nsam, dim

90 call disp%show("ndim = 2; nsam = 10; dim = 2")

91 ndim = 2; nsam = 10; dim = 2

92 call disp%show("sample = reshape(getUnifRand(1, 20, ndim * nsam), shape = [ndim, nsam], order = [2, 1])")

93 sample = reshape(getUnifRand(1, 20, ndim * nsam), shape = [ndim, nsam], order = [2, 1])

94 call disp%show("sample")

95 call disp%show( sample )

96 call disp%show("mean = getMean(sample, dim)")

97 mean = getMean(sample, dim)

98 call disp%show("mean")

99 call disp%show( mean )

100 call disp%show("samShifted = getShifted(sample, dim, -mean)")

101 samShifted = getShifted(sample, dim, -mean)

102 call disp%show("samShifted")

103 call disp%show( samShifted )

104 call disp%show("cov = getFilled(0., ndim, ndim)")

105 cov = getFilled(0., ndim, ndim)

106 call disp%show("call setCov(cov, uppDia, mean, sample, dim)")

107 call setCov(cov, uppDia, mean, sample, dim)

108 call disp%show("cov")

109 call disp%show( cov )

110 call disp%show("cov = getFilled(0., ndim, ndim)")

111 cov = getFilled(0., ndim, ndim)

112 call disp%show("call setCov(cov, uppDia, samShifted, dim) ! same result as above.")

113 call setCov(cov, uppDia, samShifted, dim)

114 call disp%show("cov")

115 call disp%show( cov )

116 call disp%skip()

117 call disp%show("cov = getFilled(0., ndim, ndim)")

118 cov = getFilled(0., ndim, ndim)

119 call disp%show("call setCov(cov, lowDia, mean, sample, dim)")

120 call setCov(cov, lowDia, mean, sample, dim)

121 call disp%show("cov")

122 call disp%show( cov )

123 call disp%show("cov = getFilled(0., ndim, ndim)")

124 cov = getFilled(0., ndim, ndim)

125 call disp%show("call setCov(cov, lowDia, samShifted, dim) ! same result as above.")

126 call setCov(cov, lowDia, samShifted, dim)

127 call disp%show("cov")

128 call disp%show( cov )

129 call disp%skip()

130 call disp%show("Compute the sample covariance along the first dimension.", deliml = SK_'''')

131 call disp%skip()

132 call disp%show("dim = 1")

133 dim = 1

134 call disp%show("cov = getFilled(0., ndim, ndim)")

135 cov = getFilled(0., ndim, ndim)

136 call disp%show("call setCov(cov, uppDia, mean, transpose(sample), dim)")

137 call setCov(cov, uppDia, mean, transpose(sample), dim)

138 call disp%show("cov")

139 call disp%show( cov )

140 call disp%show("cov = getFilled(0., ndim, ndim)")

141 cov = getFilled(0., ndim, ndim)

142 call disp%show("call setCov(cov, uppDia, transpose(samShifted), dim) ! same result as above.")

143 call setCov(cov, uppDia, transpose(samShifted), dim)

144 call disp%show("cov")

145 call disp%show( cov )

146 call disp%skip()

147 call disp%show("cov = getFilled(0., ndim, ndim)")

148 cov = getFilled(0., ndim, ndim)

149 call disp%show("call setCov(cov, lowDia, mean, transpose(sample), dim)")

150 call setCov(cov, lowDia, mean, transpose(sample), dim)

151 call disp%show("cov")

152 call disp%show( cov )

153 call disp%show("cov = getFilled(0., ndim, ndim)")

154 cov = getFilled(0., ndim, ndim)

155 call disp%show("call setCov(cov, lowDia, transpose(samShifted), dim) ! same result as above.")

156 call setCov(cov, lowDia, transpose(samShifted), dim)

157 call disp%show("cov")

158 call disp%show( cov )

159 call disp%skip()

160 call disp%show("Compute the full sample covariance for a pair of time series.", deliml = SK_'''')

161 call disp%skip()

162 call disp%show("cov = getFilled(0., ndim, ndim)")

163 cov = getFilled(0., ndim, ndim)

164 call disp%show("call setCov(cov, mean, sample(1,:), sample(2,:))")

165 call setCov(cov, mean, sample(1,:), sample(2,:))

166 call disp%show("cov")

167 call disp%show( cov )

168 call disp%show("cov = getFilled(0., ndim, ndim)")

169 cov = getFilled(0., ndim, ndim)

170 call disp%show("call setCov(cov, samShifted(1,:), samShifted(2,:)) ! same result as above.")

171 call setCov(cov, samShifted(1,:), samShifted(2,:))

172 call disp%show("cov")

173 call disp%show( cov )

174 call disp%skip()

175 end block

176

177 call disp%skip()

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

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

180 call disp%show("!Compute the biased covariance matrix of a weighted 2-D sample.")

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

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

183 call disp%skip()

184

185 block

186 use pm_kind, only: TKG => RKS ! All other real types are also supported.

187 real(TKG) :: rweisum

188 integer(IK) :: iweisum

189 real(TKG), allocatable :: rweight(:)

190 integer(IK), allocatable :: iweight(:)

191 real(TKG), allocatable :: sample(:,:), samShifted(:,:), cov(:,:), mean(:)

192 integer(IK) :: ndim, nsam, dim

193 call disp%show("ndim = 2; nsam = 10; dim = 2")

194 ndim = 2; nsam = 10; dim = 2

195 call disp%show("sample = reshape(getUnifRand(1, 20, ndim * nsam), shape = [ndim, nsam], order = [2, 1])")

196 sample = reshape(getUnifRand(1, 20, ndim * nsam), shape = [ndim, nsam], order = [2, 1])

197 call disp%show("sample")

198 call disp%show( sample )

199 call disp%show("call setResized(mean, ndim)")

200 call setResized(mean, ndim)

201 call disp%show("iweight = getUnifRand(1, 10, nsam) ! integer-valued weights.")

202 iweight = getUnifRand(1, 10, nsam) ! integer-valued weights.

203 call disp%show("iweight")

204 call disp%show( iweight )

205 call disp%show("call setMean(mean, sample, dim, iweight, iweisum)")

206 call setMean(mean, sample, dim, iweight, iweisum)

207 call disp%show("mean")

208 call disp%show( mean )

209 call disp%show("iweisum")

210 call disp%show( iweisum )

211 call disp%show("rweight = iweight ! or real-valued weights.")

212 rweight = iweight ! or real-valued weights.

213 call disp%show("iweight")

214 call disp%show( iweight )

215 call disp%show("call setMean(mean, sample, dim, rweight, rweisum)")

216 call setMean(mean, sample, dim, rweight, rweisum)

217 call disp%show("mean")

218 call disp%show( mean )

219 call disp%show("rweisum")

220 call disp%show( rweisum )

221

222 call disp%skip()

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

224 call disp%show("!Compute the covariance matrix integer weights.")

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

226 call disp%skip()

227

228 call disp%show("samShifted = getShifted(sample, dim, -mean)")

229 samShifted = getShifted(sample, dim, -mean)

230 call disp%show("samShifted")

231 call disp%show( samShifted )

232 call disp%show("cov = getFilled(0., ndim, ndim)")

233 cov = getFilled(0., ndim, ndim)

234 call disp%show("call setCov(cov, uppDia, mean, sample, dim, iweight, iweisum)")

235 call setCov(cov, uppDia, mean, sample, dim, iweight, iweisum)

236 call disp%show("cov")

237 call disp%show( cov )

238 call disp%show("cov = getFilled(0., ndim, ndim)")

239 cov = getFilled(0., ndim, ndim)

240 call disp%show("call setCov(cov, uppDia, samShifted, dim, iweight, sum(iweight)) ! same result as above.")

241 call setCov(cov, uppDia, samShifted, dim, iweight, sum(iweight))

242 call disp%show("cov")

243 call disp%show( cov )

244 call disp%skip()

245 call disp%show("cov = getFilled(0., ndim, ndim)")

246 cov = getFilled(0., ndim, ndim)

247 call disp%show("call setCov(cov, lowDia, mean, sample, dim, iweight, iweisum)")

248 call setCov(cov, lowDia, mean, sample, dim, iweight, iweisum)

249 call disp%show("cov")

250 call disp%show( cov )

251 call disp%show("cov = getFilled(0., ndim, ndim)")

252 cov = getFilled(0., ndim, ndim)

253 call disp%show("call setCov(cov, lowDia, samShifted, dim, iweight, sum(iweight)) ! same result as above.")

254 call setCov(cov, lowDia, samShifted, dim, iweight, sum(iweight))

255 call disp%show("cov")

256 call disp%show( cov )

257 call disp%skip()

258 call disp%show("Compute the sample covariance along the first dimension.", deliml = SK_'''')

259 call disp%skip()

260 call disp%show("dim = 1")

261 dim = 1

262 call disp%show("cov = getFilled(0., ndim, ndim)")

263 cov = getFilled(0., ndim, ndim)

264 call disp%show("call setCov(cov, uppDia, mean, transpose(sample), dim, iweight, iweisum)")

265 call setCov(cov, uppDia, mean, transpose(sample), dim, iweight, iweisum)

266 call disp%show("cov")

267 call disp%show( cov )

268 call disp%show("cov = getFilled(0., ndim, ndim)")

269 cov = getFilled(0., ndim, ndim)

270 call disp%show("call setCov(cov, uppDia, transpose(samShifted), dim, iweight, sum(iweight)) ! same result as above.")

271 call setCov(cov, uppDia, transpose(samShifted), dim, iweight, sum(iweight))

272 call disp%show("cov")

273 call disp%show( cov )

274 call disp%skip()

275 call disp%show("cov = getFilled(0., ndim, ndim)")

276 cov = getFilled(0., ndim, ndim)

277 call disp%show("call setCov(cov, lowDia, mean, transpose(sample), dim, iweight, iweisum)")

278 call setCov(cov, lowDia, mean, transpose(sample), dim, iweight, iweisum)

279 call disp%show("cov")

280 call disp%show( cov )

281 call disp%show("cov = getFilled(0., ndim, ndim)")

282 cov = getFilled(0., ndim, ndim)

283 call disp%show("call setCov(cov, lowDia, transpose(samShifted), dim, iweight, sum(iweight)) ! same result as above.")

284 call setCov(cov, lowDia, transpose(samShifted), dim, iweight, sum(iweight))

285 call disp%show("cov")

286 call disp%show( cov )

287 call disp%skip()

288 call disp%show("Compute the full sample covariance for a pair of time series.", deliml = SK_'''')

289 call disp%skip()

290 call disp%show("cov = getFilled(0., ndim, ndim)")

291 cov = getFilled(0., ndim, ndim)

292 call disp%show("call setCov(cov, mean, sample(1,:), sample(2,:), iweight, iweisum)")

293 call setCov(cov, mean, sample(1,:), sample(2,:), iweight, iweisum)

294 call disp%show("cov")

295 call disp%show( cov )

296 call disp%show("cov = getFilled(0., ndim, ndim)")

297 cov = getFilled(0., ndim, ndim)

298 call disp%show("call setCov(cov, samShifted(1,:), samShifted(2,:), iweight, sum(iweight)) ! same result as above.")

299 call setCov(cov, samShifted(1,:), samShifted(2,:), iweight, sum(iweight))

300 call disp%show("cov")

301 call disp%show( cov )

302 call disp%skip()

303

304 call disp%skip()

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

306 call disp%show("!Compute the covariance matrix real weights.")

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

308 call disp%skip()

309

310 call disp%show("dim = 2_IK")

311 dim = 2_IK

312 call disp%show("samShifted = getShifted(sample, dim, -mean)")

313 samShifted = getShifted(sample, dim, -mean)

314 call disp%show("samShifted")

315 call disp%show( samShifted )

316 call disp%show("cov = getFilled(0., ndim, ndim)")

317 cov = getFilled(0., ndim, ndim)

318 call disp%show("call setCov(cov, uppDia, mean, sample, dim, rweight, rweisum)")

319 call setCov(cov, uppDia, mean, sample, dim, rweight, rweisum)

320 call disp%show("cov")

321 call disp%show( cov )

322 call disp%show("cov = getFilled(0., ndim, ndim)")

323 cov = getFilled(0., ndim, ndim)

324 call disp%show("call setCov(cov, uppDia, samShifted, dim, rweight, sum(rweight)) ! same result as above.")

325 call setCov(cov, uppDia, samShifted, dim, rweight, sum(rweight))

326 call disp%show("cov")

327 call disp%show( cov )

328 call disp%skip()

329 call disp%show("cov = getFilled(0., ndim, ndim)")

330 cov = getFilled(0., ndim, ndim)

331 call disp%show("call setCov(cov, lowDia, mean, sample, dim, rweight, rweisum)")

332 call setCov(cov, lowDia, mean, sample, dim, rweight, rweisum)

333 call disp%show("cov")

334 call disp%show( cov )

335 call disp%show("cov = getFilled(0., ndim, ndim)")

336 cov = getFilled(0., ndim, ndim)

337 call disp%show("call setCov(cov, lowDia, samShifted, dim, rweight, sum(rweight)) ! same result as above.")

338 call setCov(cov, lowDia, samShifted, dim, rweight, sum(rweight))

339 call disp%show("cov")

340 call disp%show( cov )

341 call disp%skip()

342 call disp%show("Compute the sample covariance along the first dimension.", deliml = SK_'''')

343 call disp%skip()

344 call disp%show("dim = 1")

345 dim = 1

346 call disp%show("cov = getFilled(0., ndim, ndim)")

347 cov = getFilled(0., ndim, ndim)

348 call disp%show("call setCov(cov, uppDia, mean, transpose(sample), dim, rweight, rweisum)")

349 call setCov(cov, uppDia, mean, transpose(sample), dim, rweight, rweisum)

350 call disp%show("cov")

351 call disp%show( cov )

352 call disp%show("cov = getFilled(0., ndim, ndim)")

353 cov = getFilled(0., ndim, ndim)

354 call disp%show("call setCov(cov, uppDia, transpose(samShifted), dim, rweight, sum(rweight)) ! same result as above.")

355 call setCov(cov, uppDia, transpose(samShifted), dim, rweight, sum(rweight))

356 call disp%show("cov")

357 call disp%show( cov )

358 call disp%skip()

359 call disp%show("cov = getFilled(0., ndim, ndim)")

360 cov = getFilled(0., ndim, ndim)

361 call disp%show("call setCov(cov, lowDia, mean, transpose(sample), dim, rweight, rweisum)")

362 call setCov(cov, lowDia, mean, transpose(sample), dim, rweight, rweisum)

363 call disp%show("cov")

364 call disp%show( cov )

365 call disp%show("cov = getFilled(0., ndim, ndim)")

366 cov = getFilled(0., ndim, ndim)

367 call disp%show("call setCov(cov, lowDia, transpose(samShifted), dim, rweight, sum(rweight)) ! same result as above.")

368 call setCov(cov, lowDia, transpose(samShifted), dim, rweight, sum(rweight))

369 call disp%show("cov")

370 call disp%show( cov )

371 call disp%skip()

372 call disp%show("Compute the full sample covariance for a pair of time series.", deliml = SK_'''')

373 call disp%skip()

374 call disp%show("cov = getFilled(0., ndim, ndim)")

375 cov = getFilled(0., ndim, ndim)

376 call disp%show("call setCov(cov, mean, sample(1,:), sample(2,:), rweight, rweisum)")

377 call setCov(cov, mean, sample(1,:), sample(2,:), rweight, rweisum)

378 call disp%show("cov")

379 call disp%show( cov )

380 call disp%show("cov = getFilled(0., ndim, ndim)")

381 cov = getFilled(0., ndim, ndim)

382 call disp%show("call setCov(cov, samShifted(1,:), samShifted(2,:), rweight, sum(rweight)) ! same result as above.")

383 call setCov(cov, samShifted(1,:), samShifted(2,:), rweight, sum(rweight))

384 call disp%show("cov")

385 call disp%show( cov )

386 call disp%skip()

387 end block

388

389end 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_arraySpace::getLinSpace
Generate count evenly spaced points over the interval [x1, x2] if x1 < x2, or [x2,...
Definition: pm_arraySpace.F90:108

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::getFormat
Generate and return a generic or type/kind-specific IO format with the requested specifications that ...
Definition: pm_io.F90:18485

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_matrixCopy::setMatCopy
Copy a desired subset of the input source matrix of arbitrary shape (:) or (:,:) to the target subset...
Definition: pm_matrixCopy.F90:17748

pm_sampleMean::getMean
Generate and return the (weighted) mean of an input sample of nsam observations with ndim = 1 or 2 at...
Definition: pm_sampleMean.F90:229

pm_sampleMean::setMean
Return the (weighted) mean of a pair of time series or of an input sample of nsam observations with n...
Definition: pm_sampleMean.F90:2008

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

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::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_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_matrixCopy
This module contains procedures and generic interfaces relevant to copying (diagonal or upper/lower t...
Definition: pm_matrixCopy.F90:113

pm_sampleMean
This module contains classes and procedures for computing the first moment (i.e., the statistical mea...
Definition: pm_sampleMean.F90:104

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!Convert correlation matrix and standard deviation to covariance matrix.

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

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

7

8

9ndim = getUnifRand(1, 7)

10ndim

11+6

12std = getUnifRand(1, 10, ndim)

13std

14+8.000000E+00, +1.000000E+01, +6.000000E+00, +7.000000E+00, +7.000000E+00, +2.000000E+00

15cor = getCovRand(1., ndim)

16cor

17+1.000000E+00, -6.146030E-01, +6.040904E-01, +8.290617E-03, -3.783300E-01, +1.096828E-01

18-6.146030E-01, +1.000000E+00, -9.506434E-01, -4.666466E-01, -1.907399E-01, -5.286536E-01

19+6.040904E-01, -9.506434E-01, +1.000000E+00, +6.326929E-01, +1.303558E-01, +4.052165E-01

20+8.290617E-03, -4.666466E-01, +6.326929E-01, +1.000000E+00, +4.495044E-02, -3.255494E-02

21-3.783300E-01, -1.907399E-01, +1.303558E-01, +4.495044E-02, +1.000000E+00, +2.005435E-01

22+1.096828E-01, -5.286536E-01, +4.052165E-01, -3.255494E-02, +2.005435E-01, +1.000000E+00

23cov = getFilled(0._TKG, ndim, ndim + 1)

24call setMatCopy(cov(1:ndim, 1:ndim), rdpack, cor, rdpack, lowDia)

25cov

26+1.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00

27-6.146030E-01, +1.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00

28+6.040904E-01, -9.506434E-01, +1.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00

29+8.290617E-03, -4.666466E-01, +6.326929E-01, +1.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00

30-3.783300E-01, -1.907399E-01, +1.303558E-01, +4.495044E-02, +1.000000E+00, +0.000000E+00, +0.000000E+00

31+1.096828E-01, -5.286536E-01, +4.052165E-01, -3.255494E-02, +2.005435E-01, +1.000000E+00, +0.000000E+00

32call setCov(cov(1:ndim, 2:ndim+1), uppDia, cov(1:ndim, 1:ndim), lowDia, std) ! convert lower-triangle correlation matrix to upper-triangle covariance matrix.

33cov

34+1.000000E+00, +6.400000E+01, -4.916824E+01, +2.899634E+01, +4.642745E-01, -2.118648E+01, +1.754925E+00

35-6.146030E-01, +1.000000E+00, +1.000000E+02, -5.703860E+01, -3.266526E+01, -1.335180E+01, -1.057307E+01

36+6.040904E-01, -9.506434E-01, +1.000000E+00, +3.600000E+01, +2.657310E+01, +5.474944E+00, +4.862597E+00

37+8.290617E-03, -4.666466E-01, +6.326929E-01, +1.000000E+00, +4.900000E+01, +2.202572E+00, -4.557692E-01

38-3.783300E-01, -1.907399E-01, +1.303558E-01, +4.495044E-02, +1.000000E+00, +4.900000E+01, +2.807609E+00

39+1.096828E-01, -5.286536E-01, +4.052165E-01, -3.255494E-02, +2.005435E-01, +1.000000E+00, +4.000000E+00

40

41cov = getFilled(0._TKG, ndim, ndim + 1)

42call setMatCopy(cov(1:ndim, 2:ndim+1), rdpack, cor, rdpack, uppDia)

43cov

44+0.000000E+00, +1.000000E+00, -6.146030E-01, +6.040904E-01, +8.290617E-03, -3.783300E-01, +1.096828E-01

45+0.000000E+00, +0.000000E+00, +1.000000E+00, -9.506434E-01, -4.666466E-01, -1.907399E-01, -5.286536E-01

46+0.000000E+00, +0.000000E+00, +0.000000E+00, +1.000000E+00, +6.326929E-01, +1.303558E-01, +4.052165E-01

47+0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +1.000000E+00, +4.495044E-02, -3.255494E-02

48+0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +1.000000E+00, +2.005435E-01

49+0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +1.000000E+00

50call setCov(cov(1:ndim, 1:ndim), lowDia, cov(1:ndim, 2:ndim+1), uppDia, std) ! convert upper-triangle correlation matrix to lower-triangle covariance matrix.

51cov

52+6.400000E+01, +1.000000E+00, -6.146030E-01, +6.040904E-01, +8.290617E-03, -3.783300E-01, +1.096828E-01

53-4.916824E+01, +1.000000E+02, +1.000000E+00, -9.506434E-01, -4.666466E-01, -1.907399E-01, -5.286536E-01

54+2.899634E+01, -5.703860E+01, +3.600000E+01, +1.000000E+00, +6.326929E-01, +1.303558E-01, +4.052165E-01

55+4.642745E-01, -3.266526E+01, +2.657310E+01, +4.900000E+01, +1.000000E+00, +4.495044E-02, -3.255494E-02

56-2.118648E+01, -1.335180E+01, +5.474944E+00, +2.202572E+00, +4.900000E+01, +1.000000E+00, +2.005435E-01

57+1.754925E+00, -1.057307E+01, +4.862597E+00, -4.557692E-01, +2.807609E+00, +4.000000E+00, +1.000000E+00

58

59

60ndim = getUnifRand(1, 7)

61ndim

62+5

63std = getUnifRand(1, 10, ndim)

64std

65+2.000000E+00, +1.000000E+00, +1.000000E+01, +5.000000E+00, +8.000000E+00

66cor = getCovRand(1., ndim)

67cor

68+1.000000E+00, +8.009845E-01, -2.793278E-01, +3.857585E-01, -5.876297E-01

69+8.009845E-01, +1.000000E+00, +2.067818E-02, +4.578368E-01, -6.384791E-01

70-2.793278E-01, +2.067818E-02, +1.000000E+00, +4.020308E-01, +4.063008E-01

71+3.857585E-01, +4.578368E-01, +4.020308E-01, +1.000000E+00, -4.943134E-01

72-5.876297E-01, -6.384791E-01, +4.063008E-01, -4.943134E-01, +1.000000E+00

73cov = getFilled(0._TKG, ndim, ndim + 1)

74call setMatCopy(cov(1:ndim, 1:ndim), rdpack, cor, rdpack, lowDia)

75cov

76+1.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00

77+8.009845E-01, +1.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00

78-2.793278E-01, +2.067818E-02, +1.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00

79+3.857585E-01, +4.578368E-01, +4.020308E-01, +1.000000E+00, +0.000000E+00, +0.000000E+00

80-5.876297E-01, -6.384791E-01, +4.063008E-01, -4.943134E-01, +1.000000E+00, +0.000000E+00

81call setCov(cov(1:ndim, 2:ndim+1), uppDia, cov(1:ndim, 1:ndim), lowDia, std) ! convert lower-triangle correlation matrix to upper-triangle covariance matrix.

82cov

83+1.000000E+00, +4.000000E+00, +1.601969E+00, -5.586555E+00, +3.857584E+00, -9.402076E+00

84+8.009845E-01, +1.000000E+00, +1.000000E+00, +2.067818E-01, +2.289184E+00, -5.107833E+00

85-2.793278E-01, +2.067818E-02, +1.000000E+00, +1.000000E+02, +2.010154E+01, +3.250406E+01

86+3.857585E-01, +4.578368E-01, +4.020308E-01, +1.000000E+00, +2.500000E+01, -1.977254E+01

87-5.876297E-01, -6.384791E-01, +4.063008E-01, -4.943134E-01, +1.000000E+00, +6.400000E+01

88

89cov = getFilled(0._TKG, ndim, ndim + 1)

90call setMatCopy(cov(1:ndim, 2:ndim+1), rdpack, cor, rdpack, uppDia)

91cov

92+0.000000E+00, +1.000000E+00, +8.009845E-01, -2.793278E-01, +3.857585E-01, -5.876297E-01

93+0.000000E+00, +0.000000E+00, +1.000000E+00, +2.067818E-02, +4.578368E-01, -6.384791E-01

94+0.000000E+00, +0.000000E+00, +0.000000E+00, +1.000000E+00, +4.020308E-01, +4.063008E-01

95+0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +1.000000E+00, -4.943134E-01

96+0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +1.000000E+00

97call setCov(cov(1:ndim, 1:ndim), lowDia, cov(1:ndim, 2:ndim+1), uppDia, std) ! convert upper-triangle correlation matrix to lower-triangle covariance matrix.

98cov

99+4.000000E+00, +1.000000E+00, +8.009845E-01, -2.793278E-01, +3.857585E-01, -5.876297E-01

100+1.601969E+00, +1.000000E+00, +1.000000E+00, +2.067818E-02, +4.578368E-01, -6.384791E-01

101-5.586555E+00, +2.067818E-01, +1.000000E+02, +1.000000E+00, +4.020308E-01, +4.063008E-01

102+3.857584E+00, +2.289184E+00, +2.010154E+01, +2.500000E+01, +1.000000E+00, -4.943134E-01

103-9.402076E+00, -5.107833E+00, +3.250406E+01, -1.977254E+01, +6.400000E+01, +1.000000E+00

104

105

106ndim = getUnifRand(1, 7)

107ndim

108+1

109std = getUnifRand(1, 10, ndim)

110std

111+7.000000E+00

112cor = getCovRand(1., ndim)

113cor

114+1.000000E+00

115cov = getFilled(0._TKG, ndim, ndim + 1)

116call setMatCopy(cov(1:ndim, 1:ndim), rdpack, cor, rdpack, lowDia)

117cov

118+1.000000E+00, +0.000000E+00

119call setCov(cov(1:ndim, 2:ndim+1), uppDia, cov(1:ndim, 1:ndim), lowDia, std) ! convert lower-triangle correlation matrix to upper-triangle covariance matrix.

120cov

121+1.000000E+00, +4.900000E+01

122

123cov = getFilled(0._TKG, ndim, ndim + 1)

124call setMatCopy(cov(1:ndim, 2:ndim+1), rdpack, cor, rdpack, uppDia)

125cov

126+0.000000E+00, +1.000000E+00

127call setCov(cov(1:ndim, 1:ndim), lowDia, cov(1:ndim, 2:ndim+1), uppDia, std) ! convert upper-triangle correlation matrix to lower-triangle covariance matrix.

128cov

129+4.900000E+01, +1.000000E+00

130

131

132ndim = getUnifRand(1, 7)

133ndim

134+6

135std = getUnifRand(1, 10, ndim)

136std

137+3.000000E+00, +9.000000E+00, +2.000000E+00, +9.000000E+00, +5.000000E+00, +8.000000E+00

138cor = getCovRand(1., ndim)

139cor

140+1.000000E+00, +2.053825E-01, +3.161243E-01, -1.263403E-03, +2.475277E-01, -3.331965E-01

141+2.053825E-01, +1.000000E+00, +7.257905E-01, -1.875798E-02, +4.492319E-01, +4.462033E-01

142+3.161243E-01, +7.257905E-01, +1.000000E+00, -4.131553E-01, -3.410673E-02, +5.687057E-01

143-1.263403E-03, -1.875798E-02, -4.131553E-01, +1.000000E+00, -1.294024E-01, -1.136455E-01

144+2.475277E-01, +4.492319E-01, -3.410673E-02, -1.294024E-01, +1.000000E+00, -1.098759E-01

145-3.331965E-01, +4.462033E-01, +5.687057E-01, -1.136455E-01, -1.098759E-01, +1.000000E+00

146cov = getFilled(0._TKG, ndim, ndim + 1)

147call setMatCopy(cov(1:ndim, 1:ndim), rdpack, cor, rdpack, lowDia)

148cov

149+1.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00

150+2.053825E-01, +1.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00

151+3.161243E-01, +7.257905E-01, +1.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00

152-1.263403E-03, -1.875798E-02, -4.131553E-01, +1.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00

153+2.475277E-01, +4.492319E-01, -3.410673E-02, -1.294024E-01, +1.000000E+00, +0.000000E+00, +0.000000E+00

154-3.331965E-01, +4.462033E-01, +5.687057E-01, -1.136455E-01, -1.098759E-01, +1.000000E+00, +0.000000E+00

155call setCov(cov(1:ndim, 2:ndim+1), uppDia, cov(1:ndim, 1:ndim), lowDia, std) ! convert lower-triangle correlation matrix to upper-triangle covariance matrix.

156cov

157+1.000000E+00, +9.000000E+00, +5.545328E+00, +1.896746E+00, -3.411189E-02, +3.712915E+00, -7.996715E+00

158+2.053825E-01, +1.000000E+00, +8.100000E+01, +1.306423E+01, -1.519396E+00, +2.021544E+01, +3.212664E+01

159+3.161243E-01, +7.257905E-01, +1.000000E+00, +4.000000E+00, -7.436795E+00, -3.410673E-01, +9.099292E+00

160-1.263403E-03, -1.875798E-02, -4.131553E-01, +1.000000E+00, +8.100000E+01, -5.823108E+00, -8.182474E+00

161+2.475277E-01, +4.492319E-01, -3.410673E-02, -1.294024E-01, +1.000000E+00, +2.500000E+01, -4.395036E+00

162-3.331965E-01, +4.462033E-01, +5.687057E-01, -1.136455E-01, -1.098759E-01, +1.000000E+00, +6.400000E+01

163

164cov = getFilled(0._TKG, ndim, ndim + 1)

165call setMatCopy(cov(1:ndim, 2:ndim+1), rdpack, cor, rdpack, uppDia)

166cov

167+0.000000E+00, +1.000000E+00, +2.053825E-01, +3.161243E-01, -1.263403E-03, +2.475277E-01, -3.331965E-01

168+0.000000E+00, +0.000000E+00, +1.000000E+00, +7.257905E-01, -1.875798E-02, +4.492319E-01, +4.462033E-01

169+0.000000E+00, +0.000000E+00, +0.000000E+00, +1.000000E+00, -4.131553E-01, -3.410673E-02, +5.687057E-01

170+0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +1.000000E+00, -1.294024E-01, -1.136455E-01

171+0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +1.000000E+00, -1.098759E-01

172+0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +1.000000E+00

173call setCov(cov(1:ndim, 1:ndim), lowDia, cov(1:ndim, 2:ndim+1), uppDia, std) ! convert upper-triangle correlation matrix to lower-triangle covariance matrix.

174cov

175+9.000000E+00, +1.000000E+00, +2.053825E-01, +3.161243E-01, -1.263403E-03, +2.475277E-01, -3.331965E-01

176+5.545328E+00, +8.100000E+01, +1.000000E+00, +7.257905E-01, -1.875798E-02, +4.492319E-01, +4.462033E-01

177+1.896746E+00, +1.306423E+01, +4.000000E+00, +1.000000E+00, -4.131553E-01, -3.410673E-02, +5.687057E-01

178-3.411189E-02, -1.519396E+00, -7.436795E+00, +8.100000E+01, +1.000000E+00, -1.294024E-01, -1.136455E-01

179+3.712915E+00, +2.021544E+01, -3.410673E-01, -5.823108E+00, +2.500000E+01, +1.000000E+00, -1.098759E-01

180-7.996715E+00, +3.212664E+01, +9.099292E+00, -8.182474E+00, -4.395036E+00, +6.400000E+01, +1.000000E+00

181

182

183ndim = getUnifRand(1, 7)

184ndim

185+6

186std = getUnifRand(1, 10, ndim)

187std

188+9.000000E+00, +4.000000E+00, +9.000000E+00, +6.000000E+00, +1.000000E+01, +7.000000E+00

189cor = getCovRand(1., ndim)

190cor

191+1.000000E+00, +8.902969E-01, +7.304340E-01, +5.993081E-01, -7.672507E-01, -6.349715E-01

192+8.902969E-01, +1.000000E+00, +3.396693E-01, +3.598257E-01, -6.172136E-01, -5.159491E-01

193+7.304340E-01, +3.396693E-01, +1.000000E+00, +6.754510E-01, -6.397194E-01, -5.528387E-01

194+5.993081E-01, +3.598257E-01, +6.754510E-01, +1.000000E+00, -9.078873E-01, -1.553995E-01

195-7.672507E-01, -6.172136E-01, -6.397194E-01, -9.078873E-01, +1.000000E+00, +1.096865E-01

196-6.349715E-01, -5.159491E-01, -5.528387E-01, -1.553995E-01, +1.096865E-01, +1.000000E+00

197cov = getFilled(0._TKG, ndim, ndim + 1)

198call setMatCopy(cov(1:ndim, 1:ndim), rdpack, cor, rdpack, lowDia)

199cov

200+1.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00

201+8.902969E-01, +1.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00

202+7.304340E-01, +3.396693E-01, +1.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00

203+5.993081E-01, +3.598257E-01, +6.754510E-01, +1.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00

204-7.672507E-01, -6.172136E-01, -6.397194E-01, -9.078873E-01, +1.000000E+00, +0.000000E+00, +0.000000E+00

205-6.349715E-01, -5.159491E-01, -5.528387E-01, -1.553995E-01, +1.096865E-01, +1.000000E+00, +0.000000E+00

206call setCov(cov(1:ndim, 2:ndim+1), uppDia, cov(1:ndim, 1:ndim), lowDia, std) ! convert lower-triangle correlation matrix to upper-triangle covariance matrix.

207cov

208+1.000000E+00, +8.100000E+01, +3.205069E+01, +5.916515E+01, +3.236264E+01, -6.905257E+01, -4.000320E+01

209+8.902969E-01, +1.000000E+00, +1.600000E+01, +1.222809E+01, +8.635817E+00, -2.468855E+01, -1.444658E+01

210+7.304340E-01, +3.396693E-01, +1.000000E+00, +8.100000E+01, +3.647436E+01, -5.757475E+01, -3.482884E+01

211+5.993081E-01, +3.598257E-01, +6.754510E-01, +1.000000E+00, +3.600000E+01, -5.447324E+01, -6.526778E+00

212-7.672507E-01, -6.172136E-01, -6.397194E-01, -9.078873E-01, +1.000000E+00, +1.000000E+02, +7.678058E+00

213-6.349715E-01, -5.159491E-01, -5.528387E-01, -1.553995E-01, +1.096865E-01, +1.000000E+00, +4.900000E+01

214

215cov = getFilled(0._TKG, ndim, ndim + 1)

216call setMatCopy(cov(1:ndim, 2:ndim+1), rdpack, cor, rdpack, uppDia)

217cov

218+0.000000E+00, +1.000000E+00, +8.902969E-01, +7.304340E-01, +5.993081E-01, -7.672507E-01, -6.349715E-01

219+0.000000E+00, +0.000000E+00, +1.000000E+00, +3.396693E-01, +3.598257E-01, -6.172136E-01, -5.159491E-01

220+0.000000E+00, +0.000000E+00, +0.000000E+00, +1.000000E+00, +6.754510E-01, -6.397194E-01, -5.528387E-01

221+0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +1.000000E+00, -9.078873E-01, -1.553995E-01

222+0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +1.000000E+00, +1.096865E-01

223+0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +1.000000E+00

224call setCov(cov(1:ndim, 1:ndim), lowDia, cov(1:ndim, 2:ndim+1), uppDia, std) ! convert upper-triangle correlation matrix to lower-triangle covariance matrix.

225cov

226+8.100000E+01, +1.000000E+00, +8.902969E-01, +7.304340E-01, +5.993081E-01, -7.672507E-01, -6.349715E-01

227+3.205069E+01, +1.600000E+01, +1.000000E+00, +3.396693E-01, +3.598257E-01, -6.172136E-01, -5.159491E-01

228+5.916515E+01, +1.222809E+01, +8.100000E+01, +1.000000E+00, +6.754510E-01, -6.397194E-01, -5.528387E-01

229+3.236264E+01, +8.635817E+00, +3.647436E+01, +3.600000E+01, +1.000000E+00, -9.078873E-01, -1.553995E-01

230-6.905257E+01, -2.468855E+01, -5.757475E+01, -5.447324E+01, +1.000000E+02, +1.000000E+00, +1.096865E-01

231-4.000320E+01, -1.444658E+01, -3.482884E+01, -6.526778E+00, +7.678058E+00, +4.900000E+01, +1.000000E+00

232

233

234ndim = getUnifRand(1, 7)

235ndim

236+2

237std = getUnifRand(1, 10, ndim)

238std

239+2.000000E+00, +1.000000E+01

240cor = getCovRand(1., ndim)

241cor

242+1.000000E+00, +6.533102E-01

243+6.533102E-01, +1.000000E+00

244cov = getFilled(0._TKG, ndim, ndim + 1)

245call setMatCopy(cov(1:ndim, 1:ndim), rdpack, cor, rdpack, lowDia)

246cov

247+1.000000E+00, +0.000000E+00, +0.000000E+00

248+6.533102E-01, +1.000000E+00, +0.000000E+00

249call setCov(cov(1:ndim, 2:ndim+1), uppDia, cov(1:ndim, 1:ndim), lowDia, std) ! convert lower-triangle correlation matrix to upper-triangle covariance matrix.

250cov

251+1.000000E+00, +4.000000E+00, +1.306620E+01

252+6.533102E-01, +1.000000E+00, +1.000000E+02

253

254cov = getFilled(0._TKG, ndim, ndim + 1)

255call setMatCopy(cov(1:ndim, 2:ndim+1), rdpack, cor, rdpack, uppDia)

256cov

257+0.000000E+00, +1.000000E+00, +6.533102E-01

258+0.000000E+00, +0.000000E+00, +1.000000E+00

259call setCov(cov(1:ndim, 1:ndim), lowDia, cov(1:ndim, 2:ndim+1), uppDia, std) ! convert upper-triangle correlation matrix to lower-triangle covariance matrix.

260cov

261+4.000000E+00, +1.000000E+00, +6.533102E-01

262+1.306620E+01, +1.000000E+02, +1.000000E+00

263

264

265ndim = getUnifRand(1, 7)

266ndim

267+3

268std = getUnifRand(1, 10, ndim)

269std

270+4.000000E+00, +2.000000E+00, +4.000000E+00

271cor = getCovRand(1., ndim)

272cor

273+1.000000E+00, +2.043380E-01, +4.009339E-01

274+2.043380E-01, +1.000000E+00, +9.075730E-01

275+4.009339E-01, +9.075730E-01, +1.000000E+00

276cov = getFilled(0._TKG, ndim, ndim + 1)

277call setMatCopy(cov(1:ndim, 1:ndim), rdpack, cor, rdpack, lowDia)

278cov

279+1.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00

280+2.043380E-01, +1.000000E+00, +0.000000E+00, +0.000000E+00

281+4.009339E-01, +9.075730E-01, +1.000000E+00, +0.000000E+00

282call setCov(cov(1:ndim, 2:ndim+1), uppDia, cov(1:ndim, 1:ndim), lowDia, std) ! convert lower-triangle correlation matrix to upper-triangle covariance matrix.

283cov

284+1.000000E+00, +1.600000E+01, +1.634704E+00, +6.414942E+00

285+2.043380E-01, +1.000000E+00, +4.000000E+00, +7.260584E+00

286+4.009339E-01, +9.075730E-01, +1.000000E+00, +1.600000E+01

287

288cov = getFilled(0._TKG, ndim, ndim + 1)

289call setMatCopy(cov(1:ndim, 2:ndim+1), rdpack, cor, rdpack, uppDia)

290cov

291+0.000000E+00, +1.000000E+00, +2.043380E-01, +4.009339E-01

292+0.000000E+00, +0.000000E+00, +1.000000E+00, +9.075730E-01

293+0.000000E+00, +0.000000E+00, +0.000000E+00, +1.000000E+00

294call setCov(cov(1:ndim, 1:ndim), lowDia, cov(1:ndim, 2:ndim+1), uppDia, std) ! convert upper-triangle correlation matrix to lower-triangle covariance matrix.

295cov

296+1.600000E+01, +1.000000E+00, +2.043380E-01, +4.009339E-01

297+1.634704E+00, +4.000000E+00, +1.000000E+00, +9.075730E-01

298+6.414942E+00, +7.260584E+00, +1.600000E+01, +1.000000E+00

299

300

301ndim = getUnifRand(1, 7)

302ndim

303+3

304std = getUnifRand(1, 10, ndim)

305std

306+1.000000E+00, +1.000000E+01, +1.000000E+00

307cor = getCovRand(1., ndim)

308cor

309+1.000000E+00, +6.648872E-01, -3.382753E-01

310+6.648872E-01, +1.000000E+00, -1.525386E-01

311-3.382753E-01, -1.525386E-01, +1.000000E+00

312cov = getFilled(0._TKG, ndim, ndim + 1)

313call setMatCopy(cov(1:ndim, 1:ndim), rdpack, cor, rdpack, lowDia)

314cov

315+1.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00

316+6.648872E-01, +1.000000E+00, +0.000000E+00, +0.000000E+00

317-3.382753E-01, -1.525386E-01, +1.000000E+00, +0.000000E+00

318call setCov(cov(1:ndim, 2:ndim+1), uppDia, cov(1:ndim, 1:ndim), lowDia, std) ! convert lower-triangle correlation matrix to upper-triangle covariance matrix.

319cov

320+1.000000E+00, +1.000000E+00, +6.648872E+00, -3.382753E-01

321+6.648872E-01, +1.000000E+00, +1.000000E+02, -1.525386E+00

322-3.382753E-01, -1.525386E-01, +1.000000E+00, +1.000000E+00

323

324cov = getFilled(0._TKG, ndim, ndim + 1)

325call setMatCopy(cov(1:ndim, 2:ndim+1), rdpack, cor, rdpack, uppDia)

326cov

327+0.000000E+00, +1.000000E+00, +6.648872E-01, -3.382753E-01

328+0.000000E+00, +0.000000E+00, +1.000000E+00, -1.525386E-01

329+0.000000E+00, +0.000000E+00, +0.000000E+00, +1.000000E+00

330call setCov(cov(1:ndim, 1:ndim), lowDia, cov(1:ndim, 2:ndim+1), uppDia, std) ! convert upper-triangle correlation matrix to lower-triangle covariance matrix.

331cov

332+1.000000E+00, +1.000000E+00, +6.648872E-01, -3.382753E-01

333+6.648872E+00, +1.000000E+02, +1.000000E+00, -1.525386E-01

334-3.382753E-01, -1.525386E+00, +1.000000E+00, +1.000000E+00

335

336

337ndim = getUnifRand(1, 7)

338ndim

339+7

340std = getUnifRand(1, 10, ndim)

341std

342+8.000000E+00, +4.000000E+00, +1.000000E+01, +8.000000E+00, +5.000000E+00, +4.000000E+00, +9.000000E+00

343cor = getCovRand(1., ndim)

344cor

345+1.000000E+00, +9.621283E-01, -6.292422E-01, -7.141774E-01, +3.926840E-01, +2.214031E-01, -4.519387E-01

346+9.621283E-01, +1.000000E+00, -7.547417E-01, -8.737954E-01, +4.880659E-01, +1.197928E-01, -5.189089E-01

347-6.292422E-01, -7.547417E-01, +1.000000E+00, +8.021672E-01, -7.937289E-01, -2.737021E-01, +4.582914E-01

348-7.141774E-01, -8.737954E-01, +8.021672E-01, +1.000000E+00, -5.158389E-01, +1.213096E-01, +5.098246E-01

349+3.926840E-01, +4.880659E-01, -7.937289E-01, -5.158389E-01, +1.000000E+00, +9.855510E-02, -5.040165E-01

350+2.214031E-01, +1.197928E-01, -2.737021E-01, +1.213096E-01, +9.855510E-02, +1.000000E+00, -1.514607E-01

351-4.519387E-01, -5.189089E-01, +4.582914E-01, +5.098246E-01, -5.040165E-01, -1.514607E-01, +1.000000E+00

352cov = getFilled(0._TKG, ndim, ndim + 1)

353call setMatCopy(cov(1:ndim, 1:ndim), rdpack, cor, rdpack, lowDia)

354cov

355+1.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00

356+9.621283E-01, +1.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00

357-6.292422E-01, -7.547417E-01, +1.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00

358-7.141774E-01, -8.737954E-01, +8.021672E-01, +1.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00

359+3.926840E-01, +4.880659E-01, -7.937289E-01, -5.158389E-01, +1.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00

360+2.214031E-01, +1.197928E-01, -2.737021E-01, +1.213096E-01, +9.855510E-02, +1.000000E+00, +0.000000E+00, +0.000000E+00

361-4.519387E-01, -5.189089E-01, +4.582914E-01, +5.098246E-01, -5.040165E-01, -1.514607E-01, +1.000000E+00, +0.000000E+00

362call setCov(cov(1:ndim, 2:ndim+1), uppDia, cov(1:ndim, 1:ndim), lowDia, std) ! convert lower-triangle correlation matrix to upper-triangle covariance matrix.

363cov

364+1.000000E+00, +6.400000E+01, +3.078811E+01, -5.033938E+01, -4.570735E+01, +1.570736E+01, +7.084900E+00, -3.253959E+01

365+9.621283E-01, +1.000000E+00, +1.600000E+01, -3.018967E+01, -2.796145E+01, +9.761318E+00, +1.916685E+00, -1.868072E+01

366-6.292422E-01, -7.547417E-01, +1.000000E+00, +1.000000E+02, +6.417337E+01, -3.968645E+01, -1.094808E+01, +4.124623E+01

367-7.141774E-01, -8.737954E-01, +8.021672E-01, +1.000000E+00, +6.400000E+01, -2.063355E+01, +3.881907E+00, +3.670737E+01

368+3.926840E-01, +4.880659E-01, -7.937289E-01, -5.158389E-01, +1.000000E+00, +2.500000E+01, +1.971102E+00, -2.268074E+01

369+2.214031E-01, +1.197928E-01, -2.737021E-01, +1.213096E-01, +9.855510E-02, +1.000000E+00, +1.600000E+01, -5.452584E+00

370-4.519387E-01, -5.189089E-01, +4.582914E-01, +5.098246E-01, -5.040165E-01, -1.514607E-01, +1.000000E+00, +8.100000E+01

371

372cov = getFilled(0._TKG, ndim, ndim + 1)

373call setMatCopy(cov(1:ndim, 2:ndim+1), rdpack, cor, rdpack, uppDia)

374cov

375+0.000000E+00, +1.000000E+00, +9.621283E-01, -6.292422E-01, -7.141774E-01, +3.926840E-01, +2.214031E-01, -4.519387E-01

376+0.000000E+00, +0.000000E+00, +1.000000E+00, -7.547417E-01, -8.737954E-01, +4.880659E-01, +1.197928E-01, -5.189089E-01

377+0.000000E+00, +0.000000E+00, +0.000000E+00, +1.000000E+00, +8.021672E-01, -7.937289E-01, -2.737021E-01, +4.582914E-01

378+0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +1.000000E+00, -5.158389E-01, +1.213096E-01, +5.098246E-01

379+0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +1.000000E+00, +9.855510E-02, -5.040165E-01

380+0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +1.000000E+00, -1.514607E-01

381+0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +0.000000E+00, +1.000000E+00

382call setCov(cov(1:ndim, 1:ndim), lowDia, cov(1:ndim, 2:ndim+1), uppDia, std) ! convert upper-triangle correlation matrix to lower-triangle covariance matrix.

383cov

384+6.400000E+01, +1.000000E+00, +9.621283E-01, -6.292422E-01, -7.141774E-01, +3.926840E-01, +2.214031E-01, -4.519387E-01

385+3.078811E+01, +1.600000E+01, +1.000000E+00, -7.547417E-01, -8.737954E-01, +4.880659E-01, +1.197928E-01, -5.189089E-01

386-5.033938E+01, -3.018967E+01, +1.000000E+02, +1.000000E+00, +8.021672E-01, -7.937289E-01, -2.737021E-01, +4.582914E-01

387-4.570735E+01, -2.796145E+01, +6.417337E+01, +6.400000E+01, +1.000000E+00, -5.158389E-01, +1.213096E-01, +5.098246E-01

388+1.570736E+01, +9.761318E+00, -3.968645E+01, -2.063355E+01, +2.500000E+01, +1.000000E+00, +9.855510E-02, -5.040165E-01

389+7.084900E+00, +1.916685E+00, -1.094808E+01, +3.881907E+00, +1.971102E+00, +1.600000E+01, +1.000000E+00, -1.514607E-01

390-3.253959E+01, -1.868072E+01, +4.124623E+01, +3.670737E+01, -2.268074E+01, -5.452584E+00, +8.100000E+01, +1.000000E+00

391

392

393ndim = getUnifRand(1, 7)

394ndim

395+1

396std = getUnifRand(1, 10, ndim)

397std

398+1.000000E+01

399cor = getCovRand(1., ndim)

400cor

401+1.000000E+00

402cov = getFilled(0._TKG, ndim, ndim + 1)

403call setMatCopy(cov(1:ndim, 1:ndim), rdpack, cor, rdpack, lowDia)

404cov

405+1.000000E+00, +0.000000E+00

406call setCov(cov(1:ndim, 2:ndim+1), uppDia, cov(1:ndim, 1:ndim), lowDia, std) ! convert lower-triangle correlation matrix to upper-triangle covariance matrix.

407cov

408+1.000000E+00, +1.000000E+02

409

410cov = getFilled(0._TKG, ndim, ndim + 1)

411call setMatCopy(cov(1:ndim, 2:ndim+1), rdpack, cor, rdpack, uppDia)

412cov

413+0.000000E+00, +1.000000E+00

414call setCov(cov(1:ndim, 1:ndim), lowDia, cov(1:ndim, 2:ndim+1), uppDia, std) ! convert upper-triangle correlation matrix to lower-triangle covariance matrix.

415cov

416+1.000000E+02, +1.000000E+00

417

418

419!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

420!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

421!Compute the covariance matrix of a 2-D sample.

422!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

423!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

424

425ndim = 2; nsam = 10; dim = 2

426sample = reshape(getUnifRand(1, 20, ndim * nsam), shape = [ndim, nsam], order = [2, 1])

427sample

428+5.00000000, +8.00000000, +4.00000000, +2.00000000, +15.0000000, +20.0000000, +4.00000000, +18.0000000, +7.00000000, +1.00000000

429+12.0000000, +17.0000000, +8.00000000, +13.0000000, +3.00000000, +19.0000000, +10.0000000, +9.00000000, +3.00000000, +16.0000000

430mean = getMean(sample, dim)

431mean

432+8.40000057, +11.0000000

433samShifted = getShifted(sample, dim, -mean)

434samShifted

435-3.40000057, -0.400000572, -4.40000057, -6.40000057, +6.59999943, +11.5999994, -4.40000057, +9.59999943, -1.40000057, -7.40000057

436+1.00000000, +6.00000000, -3.00000000, +2.00000000, -8.00000000, +8.00000000, -1.00000000, -2.00000000, -8.00000000, +5.00000000

437cov = getFilled(0., ndim, ndim)

438call setCov(cov, uppDia, mean, sample, dim)

439cov

440+41.8400002, -0.599999845

441+0.00000000, +27.2000008

442cov = getFilled(0., ndim, ndim)

443call setCov(cov, uppDia, samShifted, dim) ! same result as above.

444cov

445+41.8400002, -0.599999845

446+0.00000000, +27.2000008

447

448cov = getFilled(0., ndim, ndim)

449call setCov(cov, lowDia, mean, sample, dim)

450cov

451+41.8400002, +0.00000000

452-0.599999845, +27.2000008

453cov = getFilled(0., ndim, ndim)

454call setCov(cov, lowDia, samShifted, dim) ! same result as above.

455cov

456+41.8400002, +0.00000000

457-0.599999845, +27.2000008

458

459'Compute the sample covariance along the first dimension.'

460

461dim = 1

462cov = getFilled(0., ndim, ndim)

463call setCov(cov, uppDia, mean, transpose(sample), dim)

464cov

465+41.8400002, -0.599999845

466+0.00000000, +27.2000008

467cov = getFilled(0., ndim, ndim)

468call setCov(cov, uppDia, transpose(samShifted), dim) ! same result as above.

469cov

470+41.8400002, -0.599999845

471+0.00000000, +27.2000008

472

473cov = getFilled(0., ndim, ndim)

474call setCov(cov, lowDia, mean, transpose(sample), dim)

475cov

476+41.8400002, +0.00000000

477-0.599999845, +27.2000008

478cov = getFilled(0., ndim, ndim)

479call setCov(cov, lowDia, transpose(samShifted), dim) ! same result as above.

480cov

481+41.8400002, +0.00000000

482-0.599999845, +27.2000008

483

484'Compute the full sample covariance for a pair of time series.'

485

486cov = getFilled(0., ndim, ndim)

487call setCov(cov, mean, sample(1,:), sample(2,:))

488cov

489+41.8400002, -0.599999845

490-0.599999845, +27.2000008

491cov = getFilled(0., ndim, ndim)

492call setCov(cov, samShifted(1,:), samShifted(2,:)) ! same result as above.

493cov

494+41.8400002, -0.599999845

495-0.599999845, +27.2000008

496

497

498!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

499!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

500!Compute the biased covariance matrix of a weighted 2-D sample.

501!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

502!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

503

504ndim = 2; nsam = 10; dim = 2

505sample = reshape(getUnifRand(1, 20, ndim * nsam), shape = [ndim, nsam], order = [2, 1])

506sample

507+18.0000000, +11.0000000, +5.00000000, +18.0000000, +17.0000000, +13.0000000, +13.0000000, +9.00000000, +9.00000000, +17.0000000

508+16.0000000, +11.0000000, +12.0000000, +19.0000000, +5.00000000, +17.0000000, +15.0000000, +20.0000000, +11.0000000, +16.0000000

509call setResized(mean, ndim)

510iweight = getUnifRand(1, 10, nsam) ! integer-valued weights.

511iweight

512+6, +3, +7, +1, +7, +7, +1, +1, +3, +2

513call setMean(mean, sample, dim, iweight, iweisum)

514mean

515+12.8157892, +12.7894735

516iweisum

517+38

518rweight = iweight ! or real-valued weights.

519iweight

520+6, +3, +7, +1, +7, +7, +1, +1, +3, +2

521call setMean(mean, sample, dim, rweight, rweisum)

522mean

523+12.8157892, +12.7894735

524rweisum

525+38.0000000

526

527!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

528!Compute the covariance matrix integer weights.

529!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

530

531samShifted = getShifted(sample, dim, -mean)

532samShifted

533+5.18421078, -1.81578922, -7.81578922, +5.18421078, +4.18421078, +0.184210777, +0.184210777, -3.81578922, -3.81578922, +4.18421078

534+3.21052647, -1.78947353, -0.789473534, +6.21052647, -7.78947353, +4.21052647, +2.21052647, +7.21052647, -1.78947353, +3.21052647

535cov = getFilled(0., ndim, ndim)

536call setCov(cov, uppDia, mean, sample, dim, iweight, iweisum)

537cov

538+22.1502762, -0.459834158

539+0.00000000, +19.7451534

540cov = getFilled(0., ndim, ndim)

541call setCov(cov, uppDia, samShifted, dim, iweight, sum(iweight)) ! same result as above.

542cov

543+22.1502762, -0.459834158

544+0.00000000, +19.7451534

545

546cov = getFilled(0., ndim, ndim)

547call setCov(cov, lowDia, mean, sample, dim, iweight, iweisum)

548cov

549+22.1502762, +0.00000000

550-0.459833741, +19.7451534

551cov = getFilled(0., ndim, ndim)

552call setCov(cov, lowDia, samShifted, dim, iweight, sum(iweight)) ! same result as above.

553cov

554+22.1502762, +0.00000000

555-0.459833741, +19.7451534

556

557'Compute the sample covariance along the first dimension.'

558

559dim = 1

560cov = getFilled(0., ndim, ndim)

561call setCov(cov, uppDia, mean, transpose(sample), dim, iweight, iweisum)

562cov

563+22.1502762, -0.459833443

564+0.00000000, +19.7451534

565cov = getFilled(0., ndim, ndim)

566call setCov(cov, uppDia, transpose(samShifted), dim, iweight, sum(iweight)) ! same result as above.

567cov

568+22.1502762, -0.459833443

569+0.00000000, +19.7451534

570

571cov = getFilled(0., ndim, ndim)

572call setCov(cov, lowDia, mean, transpose(sample), dim, iweight, iweisum)

573cov

574+22.1502762, +0.00000000

575-0.459833443, +19.7451534

576cov = getFilled(0., ndim, ndim)

577call setCov(cov, lowDia, transpose(samShifted), dim, iweight, sum(iweight)) ! same result as above.

578cov

579+22.1502762, +0.00000000

580-0.459833443, +19.7451534

581

582'Compute the full sample covariance for a pair of time series.'

583

584cov = getFilled(0., ndim, ndim)

585call setCov(cov, mean, sample(1,:), sample(2,:), iweight, iweisum)

586cov

587+22.1502762, -0.459834158

588-0.459834158, +19.7451534

589cov = getFilled(0., ndim, ndim)

590call setCov(cov, samShifted(1,:), samShifted(2,:), iweight, sum(iweight)) ! same result as above.

591cov

592+22.1502762, -0.459834158

593-0.459834158, +19.7451534

594

595

596!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

597!Compute the covariance matrix real weights.

598!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

599

600dim = 2_IK

601samShifted = getShifted(sample, dim, -mean)

602samShifted

603+5.18421078, -1.81578922, -7.81578922, +5.18421078, +4.18421078, +0.184210777, +0.184210777, -3.81578922, -3.81578922, +4.18421078

604+3.21052647, -1.78947353, -0.789473534, +6.21052647, -7.78947353, +4.21052647, +2.21052647, +7.21052647, -1.78947353, +3.21052647

605cov = getFilled(0., ndim, ndim)

606call setCov(cov, uppDia, mean, sample, dim, rweight, rweisum)

607cov

608+22.1502762, -0.459834158

609+0.00000000, +19.7451534

610cov = getFilled(0., ndim, ndim)

611call setCov(cov, uppDia, samShifted, dim, rweight, sum(rweight)) ! same result as above.

612cov

613+22.1502762, -0.459834158

614+0.00000000, +19.7451534

615

616cov = getFilled(0., ndim, ndim)

617call setCov(cov, lowDia, mean, sample, dim, rweight, rweisum)

618cov

619+22.1502762, +0.00000000

620-0.459833741, +19.7451534

621cov = getFilled(0., ndim, ndim)

622call setCov(cov, lowDia, samShifted, dim, rweight, sum(rweight)) ! same result as above.

623cov

624+22.1502762, +0.00000000

625-0.459833741, +19.7451534

626

627'Compute the sample covariance along the first dimension.'

628

629dim = 1

630cov = getFilled(0., ndim, ndim)

631call setCov(cov, uppDia, mean, transpose(sample), dim, rweight, rweisum)

632cov

633+22.1502762, -0.459833443

634+0.00000000, +19.7451534

635cov = getFilled(0., ndim, ndim)

636call setCov(cov, uppDia, transpose(samShifted), dim, rweight, sum(rweight)) ! same result as above.

637cov

638+22.1502762, -0.459833443

639+0.00000000, +19.7451534

640

641cov = getFilled(0., ndim, ndim)

642call setCov(cov, lowDia, mean, transpose(sample), dim, rweight, rweisum)

643cov

644+22.1502762, +0.00000000

645-0.459833443, +19.7451534

646cov = getFilled(0., ndim, ndim)

647call setCov(cov, lowDia, transpose(samShifted), dim, rweight, sum(rweight)) ! same result as above.

648cov

649+22.1502762, +0.00000000

650-0.459833443, +19.7451534

651

652'Compute the full sample covariance for a pair of time series.'

653

654cov = getFilled(0., ndim, ndim)

655call setCov(cov, mean, sample(1,:), sample(2,:), rweight, rweisum)

656cov

657+22.1502762, -0.459834158

658-0.459834158, +19.7451534

659cov = getFilled(0., ndim, ndim)

660call setCov(cov, samShifted(1,:), samShifted(2,:), rweight, sum(rweight)) ! same result as above.

661cov

662+22.1502762, -0.459834158

663-0.459834158, +19.7451534

664

665

Benchmarks:

Benchmark :: The runtime performance of setCov along different sample dimensions. ⛓

! Test the performance of Cholesky factorization computation using an assumed-shape interface vs. explicit-shape interface.
program benchmark
 
    use pm_kind, only: IK, LK, RKG => RKD, SK
    use pm_sampleCov, only: uppDia
    use pm_bench, only: bench_type
 
    implicit none
 
    integer(IK)                         :: itry, ntry
    integer(IK)                         :: i
    integer(IK)                         :: iarr
    integer(IK)                         :: fileUnit
    integer(IK)     , parameter         :: NARR = 18_IK
    integer(IK)     , allocatable       :: rperm(:)
    real(RKG)       , allocatable       :: samdim1(:,:)
    real(RKG)       , allocatable       :: samdim2(:,:)
    type(bench_type), allocatable       :: bench(:)
    integer(IK)     , parameter         :: nsammax = 2**NARR
    integer(IK)     , parameter         :: ndim = 5_IK
    real(RKG)                           :: mean(ndim), cov(ndim, ndim)
    integer(IK)                         :: idim, jdim, isam, nsam
    real(RKG)                           :: dumm
 
    bench = [ bench_type(name = SK_"intrinsicDIM1", exec = intrinsicDIM1, overhead = setOverhead) &
            , bench_type(name = SK_"intrinsicDIM2", exec = intrinsicDIM2, overhead = setOverhead) &
            , bench_type(name = SK_"setCovDIM1", exec = setCovDIM1, overhead = setOverhead) &
            , bench_type(name = SK_"setCovDIM2", exec = setCovDIM2, overhead = setOverhead) &
            ]
 
    write(*,"(*(g0,:,' '))")
    write(*,"(*(g0,:,' '))") "sample covariance benchmarking..."
    write(*,"(*(g0,:,' '))")
 
    open(newunit = fileUnit, file = "main.out", status = "replace")
 
        write(fileUnit, "(*(g0,:,','))") "nsam", (bench(i)%name, i = 1, size(bench))
 
        dumm = 0._RKG
        loopOverMatrixSize: do iarr = 1, NARR - 1
 
            nsam = 2**iarr
            ntry = nsammax / nsam
            allocate(samdim1(nsam, ndim), samdim2(ndim, nsam))
            write(*,"(*(g0,:,' '))") "Benchmarking setCovDIM1() vs. setCovDIM2()", nsam, ntry
 
            do i = 1, size(bench)
                bench(i)%timing = bench(i)%getTiming()
            end do
 
            write(fileUnit,"(*(g0,:,','))") nsam, (bench(i)%timing%mean / ntry, i = 1, size(bench))
            deallocate(samdim1, samdim2)
 
        end do loopOverMatrixSize
        write(*,"(*(g0,:,' '))") dumm
 
    close(fileUnit)
 
contains
 
    !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    ! procedure wrappers.
    !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
    subroutine setOverhead()
        do itry = 1, ntry
            call random_number(mean)
            dumm = dumm - mean(1)
        end do
    end subroutine
 
    subroutine setSamDIM1()
        do isam = 1, nsam
            call random_number(samdim1(isam, 1 : ndim))
        end do
    end subroutine
 
    subroutine setSamDIM2()
        do isam = 1, nsam
            call random_number(samdim2(1 : ndim, isam))
        end do
    end subroutine
 
    subroutine setCovDIM1()
        block
            use pm_sampleCov, only: setCov
            do itry = 1, ntry
                call setSamDIM1()
                call setCov(cov, uppDia, samdim1, dim = 1_IK)
                dumm = dumm + cov(1,1)
            end do
        end block
    end subroutine
 
    subroutine setCovDIM2()
        block
            use pm_sampleCov, only: setCov
            do itry = 1, ntry
                call setSamDIM2()
                call setCov(cov, uppDia, samdim2, dim = 2_IK)
                dumm = dumm + cov(1,1)
            end do
        end block
    end subroutine
 
    subroutine intrinsicDIM1()
        do itry = 1, ntry
            call setSamDIM1()
            do jdim = 1, ndim
                do idim = 1, jdim
                    cov(idim, jdim) = dot_product(samdim1(1 : nsam, idim), samdim1(1 : nsam, jdim)) / size(samdim1, dim = 1)
                end do
            end do
            dumm = dumm + cov(1,1)
        end do
    end subroutine
 
    subroutine intrinsicDIM2()
        do itry = 1, ntry
            call setSamDIM2()
            do jdim = 1, ndim
                do idim = 1, jdim
                    cov(idim, jdim) = dot_product(samdim2(idim, 1 : nsam), samdim2(jdim, 1 : nsam)) / size(samdim2, dim = 2)
                end do
            end do
            dumm = dumm + cov(1,1)
        end do
    end subroutine
 
end program benchmark

Example Unix compile command via Intel ifort compiler ⛓

#!/usr/bin/env sh
rm main.exe
ifort -fpp -standard-semantics -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
./main.exe

Example Windows Batch compile command via Intel ifort compiler ⛓

del main.exe
set PATH=..\..\..\lib;%PATH%
ifort /fpp /standard-semantics /O3 /I:..\..\..\include main.F90 ..\..\..\lib\libparamonte*.lib /exe:main.exe
main.exe

Example Unix / MinGW compile command via GNU gfortran compiler ⛓

#!/usr/bin/env sh
rm main.exe
gfortran -cpp -ffree-line-length-none -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
./main.exe

Postprocessing of the benchmark output ⛓

#!/usr/bin/env python
 
import matplotlib.pyplot as plt
import pandas as pd
import numpy as np
 
import os
dirname = os.path.basename(os.getcwd()) 
 
fontsize = 14
 
df = pd.read_csv("main.out", delimiter = ",")
colnames = list(df.columns.values)
 
 
 
ax = plt.figure(figsize = 1.25 * np.array([6.4,4.6]), dpi = 200)
ax = plt.subplot()
 
for colname in colnames[1:]:
    plt.plot( df[colnames[0]].values
            , df[colname].values
            , linewidth = 2
            )
 
plt.xticks(fontsize = fontsize)
plt.yticks(fontsize = fontsize)
ax.set_xlabel(colnames[0], fontsize = fontsize)
ax.set_ylabel("Runtime [ seconds ]", fontsize = fontsize)
ax.set_title(" vs. ".join(colnames[1:])+"\nLower is better.", fontsize = fontsize)
ax.set_xscale("log")
ax.set_yscale("log")
plt.minorticks_on()
plt.grid(visible = True, which = "both", axis = "both", color = "0.85", linestyle = "-")
ax.tick_params(axis = "y", which = "minor")
ax.tick_params(axis = "x", which = "minor")
ax.legend   ( colnames[1:]
           #, loc='center left'
           #, bbox_to_anchor=(1, 0.5)
            , fontsize = fontsize
            )
 
plt.tight_layout()
plt.savefig("benchmark." + dirname + ".runtime.png")
 
 
 
ax = plt.figure(figsize = 1.25 * np.array([6.4,4.6]), dpi = 200)
ax = plt.subplot()
 
plt.plot( df[colnames[0]].values
        , np.ones(len(df[colnames[0]].values))
        , linestyle = "--"
       #, color = "black"
        , linewidth = 2
        )
for colname in colnames[2:]:
    plt.plot( df[colnames[0]].values
            , df[colname].values / df[colnames[1]].values
            , linewidth = 2
            )
 
plt.xticks(fontsize = fontsize)
plt.yticks(fontsize = fontsize)
ax.set_xlabel(colnames[0], fontsize = fontsize)
ax.set_ylabel("Runtime compared to {}".format(colnames[1]), fontsize = fontsize)
ax.set_title("Runtime Ratio Comparison. Lower means faster.\nLower than 1 means faster than {}().".format(colnames[1]), fontsize = fontsize)
ax.set_xscale("log")
ax.set_yscale("log")
plt.minorticks_on()
plt.grid(visible = True, which = "both", axis = "both", color = "0.85", linestyle = "-")
ax.tick_params(axis = "y", which = "minor")
ax.tick_params(axis = "x", which = "minor")
ax.legend   ( colnames[1:]
           #, bbox_to_anchor = (1, 0.5)
           #, loc = "center left"
            , fontsize = fontsize
            )
 
plt.tight_layout()
plt.savefig("benchmark." + dirname + ".runtime.ratio.png")

Visualization of the benchmark output ⛓

Benchmark moral ⛓

The procedures under the generic interface setCov can compute the covariance under two different sample axes.
Recall that C is a row-major language while Fortran is a column-major language.
As such, one would expect the computations for a sample whose observations are stored along the second axis would be faster in the Fortran programming language.
However, such an expectation does not appear to hold at all times and appears to depend significantly on the computing architecture and the number of data attributes involved.
The higher the number of data attributes, the more likely the computations along the second axis of sample will be faster.
Note that for small number of data attributes, the computations along the second data axis involve a small loop that has significant computational cost due to the implicit branching involved in the loop.

Test:: test_pm_sampleCov

Todo:: Normal Priority: The examples of this generic interface should be extended to corrected weighted covariance matrices.

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.
Fatemeh Bagheri, Monday 02:15 AM, September 27, 2021, Dallas, TX

Definition at line 1694 of file pm_sampleCov.F90.

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

src/fortran/main/pm_sampleCov.F90