ParaMonte Fortran 2.0.0
Parallel Monte Carlo and Machine Learning Library
See the latest version documentation.
pm_sampleCov::setCovMean Interface Reference

Return the covariance matrix and mean vector corresponding to the input (potentially weighted) input sample 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.
More...

Detailed Description

Return the covariance matrix and mean vector corresponding to the input (potentially weighted) input sample 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.

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,
  1. type complex of kind any supported by the processor (e.g., CK, CK32, CK64, or CK128),
  2. 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 the pair x and y, whichever is present.
On output,
  1. 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}

  2. 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:
  1. The constant lowDia, implying that only the lower-diagonal subset of the output covariance matrix must be computed.
  2. 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.)
[out]mean: The output contiguous vector of shape (ndim) of the same type and kind as the output cov containing the sample mean.
  1. If the input sample is missing and x and y are present, then mean must be a vector of size ndim = 2.
  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).
[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 sample is 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 sample is 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.
  1. If dim = 1, the input sample is assumed to have the shape (nsam, ndim).
  2. 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,
  1. type integer of default kind IK, or
  2. 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.)
[out]weisum: The output scalar of the same type and kind as the input weight, containing sum(weight).
(optional. It must be present if and only if the weight argument is present.)
[in]meang: The input vector of the same type, kind, and rank as the output cov of shape (1:ndim) containing the best guess for the sample mean.
If no good guess is known a priori, meang can be set to any observation in sample.
See the example usage below.


Possible calling interfaces

! XY time series covariance matrix.
call setCovMean(cov(1:2, 1:2), mean(1:2), x(1:nsam), y(1:nsam) , meang(1:ndim))
call setCovMean(cov(1:2, 1:2), mean(1:2), x(1:nsam), y(1:nsam), weight(1:nsam), weisum, meang(1:ndim))
! sample covariance matrix.
call setCovMean(cov(1:ndim, 1:ndim), subset, mean(1:ndim) , sample(:,:), dim , meang(1:ndim))
call setCovMean(cov(1:ndim, 1:ndim), subset, mean(1:ndim) , sample(:,:), dim, weight(1:nsam), weisum , meang(1:ndim))
Return the covariance matrix and mean vector corresponding to the input (potentially weighted) input ...
This module contains classes and procedures for computing the properties related to the covariance ma...
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.
The condition 1 < size(x) must hold for the corresponding input arguments.
The condition 1 < 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
7 use pm_distUnif, only: getUnifRand
8 use pm_arrayFill, only: getFilled
9 use pm_sampleCov, only: uppDia
10 use pm_sampleCov, only: lowDia
11 use pm_sampleCov, only: setCovMean
12 use pm_io, only: display_type
13 use pm_io, only: getFormat
14
15 implicit none
16
17 integer(IK) :: itry, ntry = 10
18 type(display_type) :: disp
19 character(:), allocatable :: format
20 disp = display_type(file = "main.out.F90")
21
22 call disp%skip()
23 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
24 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
25 call disp%show("!Compute the covariance matrix of a 2-D sample.")
26 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
27 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
28 call disp%skip()
29
30 block
31 use pm_kind, only: TKG => RKS ! All other real types are also supported.
32 real(TKG), allocatable :: sample(:,:), cov(:,:), mean(:), meang(:)
33 integer(IK) :: ndim, nsam, dim
34 call disp%show("ndim = 2; nsam = 10; dim = 2")
35 ndim = 2; nsam = 10; dim = 2
36 call disp%show("sample = reshape(getUnifRand(1, 20, ndim * nsam), shape = [ndim, nsam], order = [2, 1])")
37 sample = reshape(getUnifRand(1, 20, ndim * nsam), shape = [ndim, nsam], order = [2, 1])
38 call disp%show("sample")
39 call disp%show( sample )
40 call disp%show("meang = sample(:,1)")
41 meang = sample(:,1)
42 call disp%show("mean = getFilled(0., ndim)")
43 mean = getFilled(0., ndim)
44 call disp%show("cov = getFilled(0., ndim, ndim)")
45 cov = getFilled(0., ndim, ndim)
46 call disp%show("call setCovMean(cov, uppDia, mean, sample, dim, meang)")
47 call setCovMean(cov, uppDia, mean, sample, dim, meang)
48 call disp%show("mean")
49 call disp%show( mean )
50 call disp%show("cov")
51 call disp%show( cov )
52 call disp%skip()
53 call disp%show("mean = getFilled(0., ndim)")
54 mean = getFilled(0., ndim)
55 call disp%show("cov = getFilled(0., ndim, ndim)")
56 cov = getFilled(0., ndim, ndim)
57 call disp%show("call setCovMean(cov, lowDia, mean, sample, dim, meang)")
58 call setCovMean(cov, lowDia, mean, sample, dim, meang)
59 call disp%show("mean")
60 call disp%show( mean )
61 call disp%show("cov")
62 call disp%show( cov )
63 call disp%skip()
64 call disp%show("Compute the sample covariance along the first dimension.", deliml = SK_'''')
65 call disp%skip()
66 call disp%show("dim = 1")
67 dim = 1
68 call disp%show("mean = getFilled(0., ndim)")
69 mean = getFilled(0., ndim)
70 call disp%show("cov = getFilled(0., ndim, ndim)")
71 cov = getFilled(0., ndim, ndim)
72 call disp%show("call setCovMean(cov, uppDia, mean, transpose(sample), dim, meang)")
73 call setCovMean(cov, uppDia, mean, transpose(sample), dim, meang)
74 call disp%show("mean")
75 call disp%show( mean )
76 call disp%show("cov")
77 call disp%show( cov )
78 call disp%skip()
79 call disp%show("mean = getFilled(0., ndim)")
80 mean = getFilled(0., ndim)
81 call disp%show("cov = getFilled(0., ndim, ndim)")
82 cov = getFilled(0., ndim, ndim)
83 call disp%show("call setCovMean(cov, lowDia, mean, transpose(sample), dim, meang)")
84 call setCovMean(cov, lowDia, mean, transpose(sample), dim, meang)
85 call disp%show("mean")
86 call disp%show( mean )
87 call disp%show("cov")
88 call disp%show( cov )
89 call disp%skip()
90 call disp%show("Compute the full sample covariance for a pair of time series.", deliml = SK_'''')
91 call disp%skip()
92 call disp%show("mean = getFilled(0., ndim)")
93 mean = getFilled(0., ndim)
94 call disp%show("cov = getFilled(0., ndim, ndim)")
95 cov = getFilled(0., ndim, ndim)
96 call disp%show("call setCovMean(cov, mean, sample(1,:), sample(2,:), meang = sample(1:2,1))")
97 call setCovMean(cov, mean, sample(1,:), sample(2,:), meang = sample(1:2,1))
98 call disp%show("mean")
99 call disp%show( mean )
100 call disp%show("cov")
101 call disp%show( cov )
102 call disp%skip()
103 end block
104
105 call disp%skip()
106 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
107 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
108 call disp%show("!Compute the biased covariance matrix of a weighted 2-D sample.")
109 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
110 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
111 call disp%skip()
112
113 block
114 use pm_kind, only: TKG => RKS ! All other real types are also supported.
115 real(TKG) :: rweisum
116 integer(IK) :: iweisum
117 real(TKG), allocatable :: rweight(:)
118 integer(IK), allocatable :: iweight(:)
119 real(TKG), allocatable :: sample(:,:), cov(:,:), mean(:), meang(:)
120 integer(IK) :: ndim, nsam, dim
121 call disp%show("ndim = 2; nsam = 10; dim = 2")
122 ndim = 2; nsam = 10; dim = 2
123 call disp%show("sample = reshape(getUnifRand(1, 20, ndim * nsam), shape = [ndim, nsam], order = [2, 1])")
124 sample = reshape(getUnifRand(1, 20, ndim * nsam), shape = [ndim, nsam], order = [2, 1])
125 call disp%show("sample")
126 call disp%show( sample )
127 call disp%show("meang = sample(:,1)")
128 meang = sample(:,1)
129 call disp%show("call setResized(mean, ndim)")
130 call setResized(mean, ndim)
131 call disp%show("iweight = getUnifRand(1, 10, nsam) ! integer-valued weights.")
132 iweight = getUnifRand(1, 10, nsam) ! integer-valued weights.
133 call disp%show("iweight")
134 call disp%show( iweight )
135 call disp%show("rweight = iweight ! or real-valued weights.")
136 rweight = iweight ! or real-valued weights.
137 call disp%show("iweight")
138 call disp%show( iweight )
139
140 call disp%skip()
141 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
142 call disp%show("!Compute the covariance matrix integer weights.")
143 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
144 call disp%skip()
145
146 call disp%show("mean = getFilled(0., ndim)")
147 mean = getFilled(0., ndim)
148 call disp%show("cov = getFilled(0., ndim, ndim)")
149 cov = getFilled(0., ndim, ndim)
150 call disp%show("call setCovMean(cov, uppDia, mean, sample, dim, iweight, iweisum, meang)")
151 call setCovMean(cov, uppDia, mean, sample, dim, iweight, iweisum, meang)
152 call disp%show("iweisum")
153 call disp%show( iweisum )
154 call disp%show("mean")
155 call disp%show( mean )
156 call disp%show("cov")
157 call disp%show( cov )
158 call disp%skip()
159 call disp%show("mean = getFilled(0., ndim)")
160 mean = getFilled(0., ndim)
161 call disp%show("cov = getFilled(0., ndim, ndim)")
162 cov = getFilled(0., ndim, ndim)
163 call disp%show("call setCovMean(cov, lowDia, mean, sample, dim, iweight, iweisum, meang)")
164 call setCovMean(cov, lowDia, mean, sample, dim, iweight, iweisum, meang)
165 call disp%show("iweisum")
166 call disp%show( iweisum )
167 call disp%show("mean")
168 call disp%show( mean )
169 call disp%show("cov")
170 call disp%show( cov )
171 call disp%skip()
172 call disp%show("Compute the sample covariance along the first dimension.", deliml = SK_'''')
173 call disp%skip()
174 call disp%show("dim = 1")
175 dim = 1
176 call disp%show("mean = getFilled(0., ndim)")
177 mean = getFilled(0., ndim)
178 call disp%show("cov = getFilled(0., ndim, ndim)")
179 cov = getFilled(0., ndim, ndim)
180 call disp%show("call setCovMean(cov, uppDia, mean, transpose(sample), dim, iweight, iweisum, meang)")
181 call setCovMean(cov, uppDia, mean, transpose(sample), dim, iweight, iweisum, meang)
182 call disp%show("iweisum")
183 call disp%show( iweisum )
184 call disp%show("mean")
185 call disp%show( mean )
186 call disp%show("cov")
187 call disp%show( cov )
188 call disp%skip()
189 call disp%show("mean = getFilled(0., ndim)")
190 mean = getFilled(0., ndim)
191 call disp%show("cov = getFilled(0., ndim, ndim)")
192 cov = getFilled(0., ndim, ndim)
193 call disp%show("call setCovMean(cov, lowDia, mean, transpose(sample), dim, iweight, iweisum, meang)")
194 call setCovMean(cov, lowDia, mean, transpose(sample), dim, iweight, iweisum, meang)
195 call disp%show("iweisum")
196 call disp%show( iweisum )
197 call disp%show("mean")
198 call disp%show( mean )
199 call disp%show("cov")
200 call disp%show( cov )
201 call disp%skip()
202 call disp%show("Compute the full sample covariance for a pair of time series.", deliml = SK_'''')
203 call disp%skip()
204 call disp%show("mean = getFilled(0., ndim)")
205 mean = getFilled(0., ndim)
206 call disp%show("cov = getFilled(0., ndim, ndim)")
207 cov = getFilled(0., ndim, ndim)
208 call disp%show("call setCovMean(cov, mean, sample(1,:), sample(2,:), iweight, iweisum, meang = sample(1:2,1))")
209 call setCovMean(cov, mean, sample(1,:), sample(2,:), iweight, iweisum, meang = sample(1:2,1))
210 call disp%show("iweisum")
211 call disp%show( iweisum )
212 call disp%show("mean")
213 call disp%show( mean )
214 call disp%show("cov")
215 call disp%show( cov )
216 call disp%skip()
217
218 call disp%skip()
219 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
220 call disp%show("!Compute the covariance matrix real weights.")
221 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
222 call disp%skip()
223
224 call disp%show("dim = 2_IK")
225 dim = 2_IK
226 call disp%show("meang = sample(:,1)")
227 meang = sample(:,1)
228 call disp%show("mean = getFilled(0., ndim)")
229 mean = getFilled(0., ndim)
230 call disp%show("cov = getFilled(0., ndim, ndim)")
231 cov = getFilled(0., ndim, ndim)
232 call disp%show("call setCovMean(cov, uppDia, mean, sample, dim, rweight, rweisum, meang)")
233 call setCovMean(cov, uppDia, mean, sample, dim, rweight, rweisum, meang)
234 call disp%show("rweisum")
235 call disp%show( rweisum )
236 call disp%show("mean")
237 call disp%show( mean )
238 call disp%show("cov")
239 call disp%show( cov )
240 call disp%skip()
241 call disp%show("mean = getFilled(0., ndim)")
242 mean = getFilled(0., ndim)
243 call disp%show("cov = getFilled(0., ndim, ndim)")
244 cov = getFilled(0., ndim, ndim)
245 call disp%show("call setCovMean(cov, lowDia, mean, sample, dim, rweight, rweisum, meang)")
246 call setCovMean(cov, lowDia, mean, sample, dim, rweight, rweisum, meang)
247 call disp%show("rweisum")
248 call disp%show( rweisum )
249 call disp%show("mean")
250 call disp%show( mean )
251 call disp%show("cov")
252 call disp%show( cov )
253 call disp%skip()
254 call disp%show("Compute the sample covariance along the first dimension.", deliml = SK_'''')
255 call disp%skip()
256 call disp%show("dim = 1")
257 dim = 1
258 call disp%show("mean = getFilled(0., ndim)")
259 mean = getFilled(0., ndim)
260 call disp%show("cov = getFilled(0., ndim, ndim)")
261 cov = getFilled(0., ndim, ndim)
262 call disp%show("call setCovMean(cov, uppDia, mean, transpose(sample), dim, rweight, rweisum, meang)")
263 call setCovMean(cov, uppDia, mean, transpose(sample), dim, rweight, rweisum, meang)
264 call disp%show("rweisum")
265 call disp%show( rweisum )
266 call disp%show("mean")
267 call disp%show( mean )
268 call disp%show("cov")
269 call disp%show( cov )
270 call disp%skip()
271 call disp%show("mean = getFilled(0., ndim)")
272 mean = getFilled(0., ndim)
273 call disp%show("cov = getFilled(0., ndim, ndim)")
274 cov = getFilled(0., ndim, ndim)
275 call disp%show("call setCovMean(cov, lowDia, mean, transpose(sample), dim, rweight, rweisum, meang)")
276 call setCovMean(cov, lowDia, mean, transpose(sample), dim, rweight, rweisum, meang)
277 call disp%show("rweisum")
278 call disp%show( rweisum )
279 call disp%show("mean")
280 call disp%show( mean )
281 call disp%show("cov")
282 call disp%show( cov )
283 call disp%skip()
284 call disp%show("Compute the full sample covariance for a pair of time series.", deliml = SK_'''')
285 call disp%skip()
286 call disp%show("mean = getFilled(0., ndim)")
287 mean = getFilled(0., ndim)
288 call disp%show("cov = getFilled(0., ndim, ndim)")
289 cov = getFilled(0., ndim, ndim)
290 call disp%show("call setCovMean(cov, mean, sample(1,:), sample(2,:), rweight, rweisum, meang = sample(1:2,1))")
291 call setCovMean(cov, mean, sample(1,:), sample(2,:), rweight, rweisum, meang = sample(1:2,1))
292 call disp%show("rweisum")
293 call disp%show( rweisum )
294 call disp%show("mean")
295 call disp%show( mean )
296 call disp%show("cov")
297 call disp%show( cov )
298 call disp%skip()
299 end block
300
301end program example
Generate and return an array of the specified rank and shape of arbitrary intrinsic type and kind wit...
Allocate or resize (shrink or expand) an input allocatable scalar string or array of rank 1....
Generate count evenly spaced points over the interval [x1, x2] if x1 < x2, or [x2,...
Generate and return a scalar or a contiguous array of rank 1 of length s1 of randomly uniformly distr...
Generate and return a generic or type/kind-specific IO format with the requested specifications that ...
Definition: pm_io.F90:18485
This is a generic method of the derived type display_type with pass attribute.
Definition: pm_io.F90:11726
This is a generic method of the derived type display_type with pass attribute.
Definition: pm_io.F90:11508
Generate a sample of shape (nsam), or (ndim, nsam) or (nsam, ndim) that is shifted by the specified i...
This module contains procedures and generic interfaces for convenient allocation and filling of array...
This module contains procedures and generic interfaces for resizing allocatable arrays of various typ...
This module contains procedures and generic interfaces for generating arrays with linear or logarithm...
This module contains classes and procedures for computing various statistical quantities related to t...
This module contains classes and procedures for input/output (IO) or generic display operations on st...
Definition: pm_io.F90:252
type(display_type) disp
This is a scalar module variable an object of type display_type for general display.
Definition: pm_io.F90:11393
This module defines the relevant Fortran kind type-parameters frequently used in the ParaMonte librar...
Definition: pm_kind.F90:268
integer, parameter LK
The default logical kind in the ParaMonte library: kind(.true.) in Fortran, kind(....
Definition: pm_kind.F90:541
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
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
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
This module contains classes and procedures for shifting univariate or multivariate samples by arbitr...
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 covariance matrix of a 2-D sample.
5!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
6!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
7
8ndim = 2; nsam = 10; dim = 2
9sample = reshape(getUnifRand(1, 20, ndim * nsam), shape = [ndim, nsam], order = [2, 1])
10sample
11+10.0000000, +5.00000000, +1.00000000, +20.0000000, +7.00000000, +8.00000000, +2.00000000, +5.00000000, +19.0000000, +19.0000000
12+3.00000000, +12.0000000, +8.00000000, +20.0000000, +1.00000000, +18.0000000, +14.0000000, +15.0000000, +4.00000000, +13.0000000
13meang = sample(:,1)
14mean = getFilled(0., ndim)
15cov = getFilled(0., ndim, ndim)
16call setCovMean(cov, uppDia, mean, sample, dim, meang)
17mean
18+9.60000038, +10.8000002
19cov
20+46.8400002, +3.82000017
21+0.00000000, +38.1599998
22
23mean = getFilled(0., ndim)
24cov = getFilled(0., ndim, ndim)
25call setCovMean(cov, lowDia, mean, sample, dim, meang)
26mean
27+9.60000038, +10.8000002
28cov
29+46.8400002, +0.00000000
30+3.82000017, +38.1599998
31
32'Compute the sample covariance along the first dimension.'
33
34dim = 1
35mean = getFilled(0., ndim)
36cov = getFilled(0., ndim, ndim)
37call setCovMean(cov, uppDia, mean, transpose(sample), dim, meang)
38mean
39+9.60000038, +10.8000002
40cov
41+46.8400002, +3.82000017
42+0.00000000, +38.1599998
43
44mean = getFilled(0., ndim)
45cov = getFilled(0., ndim, ndim)
46call setCovMean(cov, lowDia, mean, transpose(sample), dim, meang)
47mean
48+9.60000038, +10.8000002
49cov
50+46.8400002, +0.00000000
51+3.82000017, +38.1599998
52
53'Compute the full sample covariance for a pair of time series.'
54
55mean = getFilled(0., ndim)
56cov = getFilled(0., ndim, ndim)
57call setCovMean(cov, mean, sample(1,:), sample(2,:), meang = sample(1:2,1))
58mean
59+9.60000038, +10.8000002
60cov
61+46.8400002, +3.82000017
62+3.82000017, +38.1599998
63
64
65!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
66!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
67!Compute the biased covariance matrix of a weighted 2-D sample.
68!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
69!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
70
71ndim = 2; nsam = 10; dim = 2
72sample = reshape(getUnifRand(1, 20, ndim * nsam), shape = [ndim, nsam], order = [2, 1])
73sample
74+18.0000000, +2.00000000, +16.0000000, +7.00000000, +18.0000000, +16.0000000, +16.0000000, +13.0000000, +16.0000000, +11.0000000
75+20.0000000, +16.0000000, +16.0000000, +14.0000000, +20.0000000, +5.00000000, +2.00000000, +11.0000000, +19.0000000, +12.0000000
76meang = sample(:,1)
77call setResized(mean, ndim)
78iweight = getUnifRand(1, 10, nsam) ! integer-valued weights.
79iweight
80+6, +6, +7, +3, +2, +7, +2, +4, +6, +7
81rweight = iweight ! or real-valued weights.
82iweight
83+6, +6, +7, +3, +2, +7, +2, +4, +6, +7
84
85!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
86!Compute the covariance matrix integer weights.
87!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
88
89mean = getFilled(0., ndim)
90cov = getFilled(0., ndim, ndim)
91call setCovMean(cov, uppDia, mean, sample, dim, iweight, iweisum, meang)
92iweisum
93+50
94mean
95+13.1599998, +13.8199997
96cov
97+25.1744041, +0.168801263
98+0.00000000, +28.1476021
99
100mean = getFilled(0., ndim)
101cov = getFilled(0., ndim, ndim)
102call setCovMean(cov, lowDia, mean, sample, dim, iweight, iweisum, meang)
103iweisum
104+50
105mean
106+13.1599998, +13.8199997
107cov
108+25.1744041, +0.00000000
109+0.168801263, +28.1476021
110
111'Compute the sample covariance along the first dimension.'
112
113dim = 1
114mean = getFilled(0., ndim)
115cov = getFilled(0., ndim, ndim)
116call setCovMean(cov, uppDia, mean, transpose(sample), dim, iweight, iweisum, meang)
117iweisum
118+50
119mean
120+13.1599998, +13.8199997
121cov
122+25.1743984, +0.168801263
123+0.00000000, +28.1476021
124
125mean = getFilled(0., ndim)
126cov = getFilled(0., ndim, ndim)
127call setCovMean(cov, lowDia, mean, transpose(sample), dim, iweight, iweisum, meang)
128iweisum
129+50
130mean
131+13.1599998, +13.8199997
132cov
133+25.1744041, +0.00000000
134+0.168801263, +28.1475964
135
136'Compute the full sample covariance for a pair of time series.'
137
138mean = getFilled(0., ndim)
139cov = getFilled(0., ndim, ndim)
140call setCovMean(cov, mean, sample(1,:), sample(2,:), iweight, iweisum, meang = sample(1:2,1))
141iweisum
142+50
143mean
144+13.1599998, +13.8199997
145cov
146+25.1744041, +0.168801263
147+0.168801263, +28.1476021
148
149
150!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
151!Compute the covariance matrix real weights.
152!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
153
154dim = 2_IK
155meang = sample(:,1)
156mean = getFilled(0., ndim)
157cov = getFilled(0., ndim, ndim)
158call setCovMean(cov, uppDia, mean, sample, dim, rweight, rweisum, meang)
159rweisum
160+50.0000000
161mean
162+13.1599998, +13.8199997
163cov
164+25.1744041, +0.168801263
165+0.00000000, +28.1476021
166
167mean = getFilled(0., ndim)
168cov = getFilled(0., ndim, ndim)
169call setCovMean(cov, lowDia, mean, sample, dim, rweight, rweisum, meang)
170rweisum
171+50.0000000
172mean
173+13.1599998, +13.8199997
174cov
175+25.1744041, +0.00000000
176+0.168801263, +28.1476021
177
178'Compute the sample covariance along the first dimension.'
179
180dim = 1
181mean = getFilled(0., ndim)
182cov = getFilled(0., ndim, ndim)
183call setCovMean(cov, uppDia, mean, transpose(sample), dim, rweight, rweisum, meang)
184rweisum
185+50.0000000
186mean
187+13.1599998, +13.8199997
188cov
189+25.1743984, +0.168801263
190+0.00000000, +28.1476021
191
192mean = getFilled(0., ndim)
193cov = getFilled(0., ndim, ndim)
194call setCovMean(cov, lowDia, mean, transpose(sample), dim, rweight, rweisum, meang)
195rweisum
196+50.0000000
197mean
198+13.1599998, +13.8199997
199cov
200+25.1744041, +0.00000000
201+0.168801263, +28.1475964
202
203'Compute the full sample covariance for a pair of time series.'
204
205mean = getFilled(0., ndim)
206cov = getFilled(0., ndim, ndim)
207call setCovMean(cov, mean, sample(1,:), sample(2,:), rweight, rweisum, meang = sample(1:2,1))
208rweisum
209+50.0000000
210mean
211+13.1599998, +13.8199997
212cov
213+25.1744041, +0.168801263
214+0.168801263, +28.1476021
215
216
Benchmarks:


Benchmark :: The runtime performance of setCovMean along different sample dimensions.

1! Test the performance of Cholesky factorization computation using an assumed-shape interface vs. explicit-shape interface.
2program benchmark
3
4 use pm_kind, only: IK, LK, RKG => RKD, SK
5 use pm_sampleCov, only: uppDia
6 use pm_bench, only: bench_type
7
8 implicit none
9
10 integer(IK) :: itry, ntry
11 integer(IK) :: i
12 integer(IK) :: iarr
13 integer(IK) :: fileUnit
14 integer(IK) , parameter :: NARR = 18_IK
15 real(RKG) , allocatable :: samdim1(:,:)
16 real(RKG) , allocatable :: samdim2(:,:)
17 type(bench_type), allocatable :: bench(:)
18 integer(IK) , parameter :: nsammax = 2**NARR
19 integer(IK) , parameter :: ndim = 5_IK
20 real(RKG) :: cov(ndim, ndim), mean(ndim)
21 integer(IK) :: isam, nsam
22 real(RKG) :: dumm
23
24 bench = [ bench_type(name = SK_"setCovMeanDIM1", exec = setCovMeanDIM1, overhead = setOverhead) &
25 , bench_type(name = SK_"setCovMeanDIM2", exec = setCovMeanDIM2, overhead = setOverhead) &
26 ]
27
28 write(*,"(*(g0,:,' '))")
29 write(*,"(*(g0,:,' '))") "sample covariance benchmarking..."
30 write(*,"(*(g0,:,' '))")
31
32 open(newunit = fileUnit, file = "main.out", status = "replace")
33
34 write(fileUnit, "(*(g0,:,','))") "nsam", (bench(i)%name, i = 1, size(bench))
35
36 dumm = 0._RKG
37 loopOverMatrixSize: do iarr = 1, NARR - 1
38
39 nsam = 2**iarr
40 ntry = nsammax / nsam
41 allocate(samdim1(nsam, ndim), samdim2(ndim, nsam))
42 write(*,"(*(g0,:,' '))") "Benchmarking setCovMeanDIM1() vs. setCovMeanDIM2()", nsam, ntry
43
44 do i = 1, size(bench)
45 bench(i)%timing = bench(i)%getTiming()
46 end do
47
48 write(fileUnit,"(*(g0,:,','))") nsam, (bench(i)%timing%mean / ntry, i = 1, size(bench))
49 deallocate(samdim1, samdim2)
50
51 end do loopOverMatrixSize
52 write(*,"(*(g0,:,' '))") dumm
53
54 close(fileUnit)
55
56contains
57
58 !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
59 ! procedure wrappers.
60 !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
61
62 subroutine setOverhead()
63 do itry = 1, ntry
64 call random_number(mean)
65 dumm = dumm - mean(1)
66 end do
67 end subroutine
68
69 subroutine setSamDIM1()
70 do isam = 1, nsam
71 call random_number(samdim1(isam, 1 : ndim))
72 end do
73 end subroutine
74
75 subroutine setSamDIM2()
76 do isam = 1, nsam
77 call random_number(samdim2(1 : ndim, isam))
78 end do
79 end subroutine
80
81 subroutine setCovMeanDIM1()
82 block
83 use pm_sampleCov, only: setCovMean
84 do itry = 1, ntry
85 call setSamDIM1()
86 call setCovMean(cov, uppDia, mean, samdim1, dim = 1_IK, meang = samdim1(1,:))
87 dumm = dumm + cov(1,1) + mean(1)
88 end do
89 end block
90 end subroutine
91
92 subroutine setCovMeanDIM2()
93 block
94 use pm_sampleCov, only: setCovMean
95 do itry = 1, ntry
96 call setSamDIM2()
97 call setCovMean(cov, uppDia, mean, samdim2, dim = 2_IK, meang = samdim2(:,1))
98 dumm = dumm + cov(1,1) + mean(1)
99 end do
100 end block
101 end subroutine
102
103end program benchmark
Generate and return an object of type timing_type containing the benchmark timing information and sta...
Definition: pm_bench.F90:574
This module contains abstract interfaces and types that facilitate benchmarking of different procedur...
Definition: pm_bench.F90:41
integer, parameter RKD
The double precision real kind in Fortran mode. On most platforms, this is an 64-bit real kind.
Definition: pm_kind.F90:568
This is the class for creating benchmark and performance-profiling objects.
Definition: pm_bench.F90:386

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

Postprocessing of the benchmark output
1#!/usr/bin/env python
2
3import matplotlib.pyplot as plt
4import pandas as pd
5import numpy as np
6
7import os
8dirname = os.path.basename(os.getcwd())
9
10fontsize = 14
11
12df = pd.read_csv("main.out", delimiter = ",")
13colnames = list(df.columns.values)
14
15
18
19ax = plt.figure(figsize = 1.25 * np.array([6.4,4.6]), dpi = 200)
20ax = plt.subplot()
21
22for colname in colnames[1:]:
23 plt.plot( df[colnames[0]].values
24 , df[colname].values
25 , linewidth = 2
26 )
27
28plt.xticks(fontsize = fontsize)
29plt.yticks(fontsize = fontsize)
30ax.set_xlabel(colnames[0], fontsize = fontsize)
31ax.set_ylabel("Runtime [ seconds ]", fontsize = fontsize)
32ax.set_title(" vs. ".join(colnames[1:])+"\nLower is better.", fontsize = fontsize)
33ax.set_xscale("log")
34ax.set_yscale("log")
35plt.minorticks_on()
36plt.grid(visible = True, which = "both", axis = "both", color = "0.85", linestyle = "-")
37ax.tick_params(axis = "y", which = "minor")
38ax.tick_params(axis = "x", which = "minor")
39ax.legend ( colnames[1:]
40 #, loc='center left'
41 #, bbox_to_anchor=(1, 0.5)
42 , fontsize = fontsize
43 )
44
45plt.tight_layout()
46plt.savefig("benchmark." + dirname + ".runtime.png")
47
48
51
52ax = plt.figure(figsize = 1.25 * np.array([6.4,4.6]), dpi = 200)
53ax = plt.subplot()
54
55plt.plot( df[colnames[0]].values
56 , np.ones(len(df[colnames[0]].values))
57 , linestyle = "--"
58 #, color = "black"
59 , linewidth = 2
60 )
61for colname in colnames[2:]:
62 plt.plot( df[colnames[0]].values
63 , df[colname].values / df[colnames[1]].values
64 , linewidth = 2
65 )
66
67plt.xticks(fontsize = fontsize)
68plt.yticks(fontsize = fontsize)
69ax.set_xlabel(colnames[0], fontsize = fontsize)
70ax.set_ylabel("Runtime compared to {}".format(colnames[1]), fontsize = fontsize)
71ax.set_title("Runtime Ratio Comparison. Lower means faster.\nLower than 1 means faster than {}().".format(colnames[1]), fontsize = fontsize)
72ax.set_xscale("log")
73ax.set_yscale("log")
74plt.minorticks_on()
75plt.grid(visible = True, which = "both", axis = "both", color = "0.85", linestyle = "-")
76ax.tick_params(axis = "y", which = "minor")
77ax.tick_params(axis = "x", which = "minor")
78ax.legend ( colnames[1:]
79 #, bbox_to_anchor = (1, 0.5)
80 #, loc = "center left"
81 , fontsize = fontsize
82 )
83
84plt.tight_layout()
85plt.savefig("benchmark." + dirname + ".runtime.ratio.png")

Visualization of the benchmark output

Benchmark moral
  1. The procedures under the generic interface setCovMean can compute the covariance under two different sample axes.
  2. Recall that C is a row-major language while Fortran is a column-major language.
  3. 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.
  4. 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.
  5. The higher the number of data attributes, the more likely the computations along the second axis of sample will be faster.
  6. 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.

  1. 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.
  2. If you regenerate any parts/ideas from this library in a programming environment other than those currently supported by this ParaMonte library (i.e., other than C, C++, Fortran, MATLAB, Python, R), please also ask the end users to cite this original ParaMonte library.

This software is available to the public under a highly permissive license.
Help us justify its continued development and maintenance by acknowledging its benefit to society, distributing it, and contributing to it.

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 5298 of file pm_sampleCov.F90.


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