Return a random positive-definite power-law-distributed (correlation) matrix.
See the documentation of pm_distCov for details.
- Parameters
-
| [in,out] | rng | : The input/output scalar that can be an object of,
-
type rngf_type, implying the use of intrinsic Fortran uniform RNG.
-
type xoshiro256ssw_type, implying the use of xoshiro256** uniform RNG.
|
| [out] | rand | : The output 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),
containing a random (optionally power-law-distributed determinant) positive-definite matrix.
The output rand can of complex type if and only if the optional input argument method is missing.
|
| [in,out] | method | : The input/output scalar constant that can be one of the following:
-
The scalar input constant dvine implying the use of the Dvine algorithm for generating random covariance matrices whose determinants are power-law distributed with exponent
eta.
In this case, the argument method has intent(in).
-
A scalar output variable of type onion_type such as onion implying the use of the Onion algorithm for generating random covariance matrices whose determinants are power-law distributed with exponent
eta.
In this case, the argument method has intent(out).
If the Cholesky factorization within the Onion algorithm fails, methodinfo will be set to the order of the leading minor of the specified input subset of mat that is not positive definite, indicating the occurrence of an error and that the factorization could not be completed.
Otherwise, the info component of the onion method is set to 0.
The resulting matrix distribution from dvine and onion are identically distributed but onion method tends to have slightly faster runtime.
The larger eta is, the more the output random matrix looks like the Identity matrix.
Setting eta = 0. corresponds to a uniform distribution of the output matrix over the space of positive-definite correlation matrices.
See the description of the output argument rand for more information on the effects of eta on the off-diagonal elements of the output positive-definite matrix.
(optional. If missing the Gram method is used for random matrix generation. It must be missing for output rand of type complex.) |
| [in] | eta | : The input non-negative scalar of type real of the same kind as the output argument rand.
The larger eta is, the more the output random matrix looks like the Identity matrix.
Setting eta = 0. corresponds to a uniform distribution of the output matrix over the space of positive-definite correlation matrices.
See the description of the output argument rand for more information on the effects of eta on the off-diagonal elements of the output positive-definite matrix.
(optional. It must be present if and only if the input argument method is also present.) |
| [in] | scale | : The input scalar or contiguous vector of size ndim of type real of the same kind as the output argument rand, representing the scale of the matrix (e.g., the standard deviation of a covariance matrix) along each dimension.
(optional. default = 1.) |
Possible calling interfaces ⛓
call setCovRand(rng, rand(
1:ndim,
1:ndim),
scale(
1:ndim))
call setCovRand(rng, rand(
1:ndim,
1:ndim), method, eta)
call setCovRand(rng, rand(
1:ndim,
1:ndim), method, eta, scale)
call setCovRand(rng, rand(
1:ndim,
1:ndim), method, eta,
scale(
1:ndim))
Return a random positive-definite power-law-distributed (correlation) matrix.
This module contains classes and procedures for generating random matrices distributed on the space o...
- Warning
- The condition
0 <= eta must hold for the corresponding input arguments.
The condition all([0 < scale]) must hold for the corresponding input arguments.
The condition size(rand, 1) == size(rand, 2) must hold for the corresponding input arguments.
The condition rank(scale) == 0 .or. all(size(scale) == shape(rand)) must hold for the corresponding input arguments.
These conditions are verified only if the library is built with the preprocessor macro CHECK_ENABLED=1.
-
Beware that when the input argument
scale is missing, the diagonal elements of the output correlation matrix are not enforced to match 1.
As such, numerical matrix multiplication errors may lead to diagonal matrix values slightly deviating from 1. If you need such a guarantee on the diagonal elements of the output random correlation matrix, use getCovRand.
Example usage ⛓
17 integer(IK) :: itry, ndim
19 type(display_type) :: disp
23 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
24 call disp%show(
"!Gram method for real covariance.")
25 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
31 real(TKG),
allocatable ::
scale(:)
32 real(TKG),
allocatable :: rand(:,:)
36 call disp%show(
"ndim = getUnifRand(3, 9)")
40 call disp%show(
"call setResized(rand, [ndim, ndim])")
42 call disp%show(
"scale = getUnifRand(1, 10, ndim)")
48 call disp%show(
"call setCovRand(rngf, rand)")
52 call disp%show(
"isMatClass(rand, posdefmat)")
54 call disp%show(
"isMatClass(rand, hermitian)")
59 call disp%show(
"call setCovRand(rngf, rand, scale(1))")
63 call disp%show(
"isMatClass(rand, posdefmat)")
65 call disp%show(
"isMatClass(rand, hermitian)")
71 call disp%show(
"call setCovRand(rngf, rand, scale)")
75 call disp%show(
"isMatClass(rand, posdefmat)")
77 call disp%show(
"isMatClass(rand, hermitian)")
88 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
89 call disp%show(
"!Gram method for complex covariance.")
90 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
96 real(TKG),
allocatable ::
scale(:)
97 complex(TKG),
allocatable :: rand(:,:)
101 call disp%show(
"ndim = getUnifRand(3, 9)")
105 call disp%show(
"call setResized(rand, [ndim, ndim])")
107 call disp%show(
"scale = getUnifRand(1, 10, ndim)")
113 call disp%show(
"call setCovRand(rngf, rand)")
117 call disp%show(
"isMatClass(rand, posdefmat)")
119 call disp%show(
"isMatClass(rand, hermitian)")
124 call disp%show(
"call setCovRand(rngf, rand, scale(1))")
128 call disp%show(
"isMatClass(rand, posdefmat)")
130 call disp%show(
"isMatClass(rand, hermitian)")
136 call disp%show(
"call setCovRand(rngf, rand, scale)")
140 call disp%show(
"isMatClass(rand, posdefmat)")
142 call disp%show(
"isMatClass(rand, hermitian)")
153 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%")
154 call disp%show(
"!Dvine and Onion methods.")
155 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%")
161 real(TKG),
allocatable :: rand(:,:)
162 real(TKG) :: eta, scale
166 call disp%show(
"eta = getUnifRand(1, 10)")
168 call disp%show(
"ndim = getUnifRand(2, 5)")
170 call disp%show(
"call setResized(rand, [ndim, ndim])")
172 call disp%show(
"scale = getUnifRand(1, 10)")
176 call disp%show(
"call setCovRand(rngf, rand, dvine, eta)")
182 call disp%show(
"isMatClass(rand, posdefmat)")
184 call disp%show(
"isMatClass(rand, hermitian)")
189 call disp%show(
"call setCovRand(rngf, rand, onion, eta)")
195 call disp%show(
"isMatClass(rand, posdefmat)")
197 call disp%show(
"isMatClass(rand, hermitian)")
202 call disp%show(
"call setCovRand(rngf, rand, dvine, eta, scale)")
206 call disp%show(
"isMatClass(rand, posdefmat)")
208 call disp%show(
"isMatClass(rand, hermitian)")
214 call disp%show(
"call setCovRand(rngf, rand, onion, eta, scale)")
220 call disp%show(
"isMatClass(rand, posdefmat)")
222 call disp%show(
"isMatClass(rand, hermitian)")
231 call disp%show(
"ndim = getUnifRand(2, 10)")
233 call disp%show(
"call setResized(rand, [ndim, ndim])")
237 call disp%show(
"call setCovRand(rngf, rand, dvine, eta = 0._TKG, scale = [(real(itry, TKG), itry = 1, ndim)])")
238 call setCovRand(
rngf, rand,
dvine, eta
= 0._TKG, scale
= [(
real(itry, TKG), itry
= 1, ndim)])
241 call disp%show(
"isMatClass(rand, posdefmat)")
245 call disp%show(
"call setCovRand(rngf, rand, onion, eta = 0._TKG, scale = [(real(itry, TKG), itry = 1, ndim)])")
246 call setCovRand(
rngf, rand,
onion, eta
= 0._TKG, scale
= [(
real(itry, TKG), itry
= 1, ndim)])
251 call disp%show(
"isMatClass(rand, posdefmat)")
Allocate or resize (shrink or expand) an input allocatable scalar string or array of rank 1....
Generate and return a (collection) of random vector(s) of size ndim from the ndim-dimensional MultiVa...
Generate and return a scalar or a contiguous array of rank 1 of length s1 of randomly uniformly distr...
This is a generic method of the derived type display_type with pass attribute.
This is a generic method of the derived type display_type with pass attribute.
[LEGACY code] Return the lower-triangle of the Cholesky factorization of the symmetric positive-def...
Generate and return .true. if and only if the input matrix is of the specified input class.
Generate and return the determinant of the input general square matrix.
This module contains procedures and generic interfaces for resizing allocatable arrays of various typ...
type(onion_type) onion
The scalar module variable object of type onion_type implying the use of the Onion algorithm for gene...
type(dvine_type), parameter dvine
The scalar constant of type dvine_type implying the use of the Dvine algorithm for generating random ...
This module contains classes and procedures for computing various statistical quantities related to t...
This module contains classes and procedures for computing various statistical quantities related to t...
type(rngf_type) rngf
The scalar constant object of type rngf_type whose presence signified the use of the Fortran intrinsi...
This module contains classes and procedures for input/output (IO) or generic display operations on st...
type(display_type) disp
This is a scalar module variable an object of type display_type for general display.
This module defines the relevant Fortran kind type-parameters frequently used in the ParaMonte librar...
integer, parameter LK
The default logical kind in the ParaMonte library: kind(.true.) in Fortran, kind(....
integer, parameter IK
The default integer kind in the ParaMonte library: int32 in Fortran, c_int32_t in C-Fortran Interoper...
integer, parameter RKD
The double precision real kind in Fortran mode. On most platforms, this is an 64-bit real kind.
integer, parameter SK
The default character kind in the ParaMonte library: kind("a") in Fortran, c_char in C-Fortran Intero...
integer, parameter RKS
The single-precision real kind in Fortran mode. On most platforms, this is an 32-bit real kind.
This module contains procedures and generic interfaces for computing the Cholesky factorization of po...
This module contains abstract and concrete derived types that are required for compile-time resolutio...
type(posdefmat_type), parameter posdefmat
This is a scalar parameter object of type hermitian_type that is exclusively used to signify the Herm...
type(hermitian_type), parameter hermitian
This is a scalar parameter object of type hermitian_type that is exclusively used to signify the Herm...
This module contains procedures and generic interfaces relevant to the computation of the determinant...
Generate and return an object of type display_type.
The derived type that can be used for constructing containers of format or left and right delimiters ...
Example Unix compile command via Intel ifort compiler ⛓
3ifort -fpp -standard-semantics -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
Example Windows Batch compile command via Intel ifort compiler ⛓
2set PATH=..\..\..\lib;%PATH%
3ifort /fpp /standard-semantics /O3 /I:..\..\..\include main.F90 ..\..\..\lib\libparamonte*.lib /exe:main.exe
Example Unix / MinGW compile command via GNU gfortran compiler ⛓
3gfortran -cpp -ffree-line-length-none -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
Example output ⛓
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+64.0000000,
-85.0748978
745+9.11215687,
-85.0748978,
+64.0000000
763+1.00000000,
-0.432123780
764-0.432123780,
+1.00000000
775+1.00000000,
+0.155968428
776+0.155968428,
+1.00000000
785+49.0000000,
-3.63959265
786-3.63959265,
+49.0000000
798+49.0000000,
-4.23254156
799-4.23254156,
+49.0000000
817+1.00000000,
-0.174169600,
+0.262220860
818-0.174169600,
+1.00000000,
-0.186970353
819+0.262220860,
-0.186970353,
+1.00000000
830+1.00000000,
+0.522613525,
+0.670238212E-1
831+0.522613525,
+1.00000000,
-0.122490957
832+0.670238212E-1,
-0.122490957,
+1.00000000
841+25.0000000,
-5.14816952,
+1.68904364
842-5.14816952,
+25.0000000,
-0.894779325
843+1.68904364,
-0.894779325,
+25.0000000
855+25.0000000,
+6.03189182,
+4.14547825
856+6.03189182,
+25.0000000,
-17.3841953
857+4.14547825,
-17.3841953,
+25.0000000
875+1.00000000,
-0.327304602E-1
876-0.327304602E-1,
+1.00000000
887+1.00000000,
+0.193414688E-1
888+0.193414688E-1,
+1.00000000
897+36.0000000,
-2.42128372
898-2.42128372,
+36.0000000
910+36.0000000,
+16.2747421
911+16.2747421,
+36.0000000
929+1.00000000,
-0.201402664
930-0.201402664,
+1.00000000
941+1.00000000,
-0.232133389
942-0.232133389,
+1.00000000
951+36.0000000,
+0.454919815
952+0.454919815,
+36.0000000
964+36.0000000,
+20.6934013
965+20.6934013,
+36.0000000
983+1.00000000,
-0.372969985,
-0.374505222,
+0.109540582
984-0.372969985,
+1.00000000,
+0.165067613E-1,
+0.218678236
985-0.374505222,
+0.165067613E-1,
+1.00000000,
-0.583541036
986+0.109540582,
+0.218678236,
-0.583541036,
+1.00000000
997+1.00000000,
-0.114963531,
-0.298758388,
-0.104634173
998-0.114963531,
+1.00000000,
-0.330263108,
-0.511745036
999-0.298758388,
-0.330263108,
+1.00000000,
-0.540939808
1000-0.104634173,
-0.511745036,
-0.540939808,
+1.00000000
1009+81.0000000,
+26.6339645,
+10.7979622,
+31.2080002
1010+26.6339645,
+81.0000000,
+2.52577424,
+32.4428215
1011+10.7979622,
+2.52577424,
+81.0000000,
+14.3352013
1012+31.2080002,
+32.4428215,
+14.3352013,
+81.0000000
1024+1.00000000,
-0.240146160,
-0.407497138E-1,
+31.2080002
1025+26.6339645,
+1.00000000,
-1.17803288,
+32.4428215
1026+10.7979622,
+2.52577424,
+1.00000000,
+14.3352013
1027+31.2080002,
+32.4428215,
+14.3352013,
+81.0000000
1045+1.00000000,
+0.127711296,
+0.515108705,
-0.743637085E-1
1046+0.127711296,
+1.00000000,
+0.865820795E-2,
-0.664231703E-1
1047+0.515108705,
+0.865820795E-2,
+1.00000000,
+0.640287697E-1
1048-0.743637085E-1,
-0.664231703E-1,
+0.640287697E-1,
+1.00000000
1059+1.00000000,
-0.393271923,
-0.177916437,
+0.103122868
1060-0.393271923,
+1.00000000,
-0.244356006,
-0.391173720
1061-0.177916437,
-0.244356006,
+1.00000000,
+0.610120595E-3
1062+0.103122868,
-0.391173720,
+0.610120595E-3,
+1.00000000
1071+81.0000000,
+31.7667694,
-19.5949402,
+9.43964005
1072+31.7667694,
+81.0000000,
-48.8774033,
+18.2937279
1073-19.5949402,
-48.8774033,
+81.0000000,
+20.7111053
1074+9.43964005,
+18.2937279,
+20.7111053,
+81.0000000
1086+1.00000000,
+0.143210888E-1,
+0.346610427,
+9.43964005
1087+31.7667694,
+1.00000000,
+11.1167889,
+18.2937279
1088-19.5949402,
-48.8774033,
+1.00000000,
+20.7111053
1089+9.43964005,
+18.2937279,
+20.7111053,
+81.0000000
1107+1.00000000,
-0.801032782E-1,
+0.163490295
1108-0.801032782E-1,
+1.00000000,
+0.158034563
1109+0.163490295,
+0.158034563,
+1.00000000
1120+1.00000000,
-0.379363418,
+0.492337257
1121-0.379363418,
+1.00000000,
-0.303344548
1122+0.492337257,
-0.303344548,
+1.00000000
1131+36.0000000,
-13.0127354,
-2.69560933
1132-13.0127354,
+36.0000000,
+2.05713034
1133-2.69560933,
+2.05713034,
+36.0000000
1145+36.0000000,
+1.91980505,
+3.56779456
1146+1.91980505,
+36.0000000,
-34.6922646
1147+3.56779456,
-34.6922646,
+36.0000000
1159call setCovRand(
rngf, rand,
dvine, eta
= 0._TKG, scale
= [(
real(itry, TKG), itry
= 1, ndim)])
1161+1.00000000,
+0.197524786,
-1.08652520,
-0.899771214,
+1.60182416,
+0.350251436,
+1.98801780,
-4.89459229,
-5.81478357
1162+0.197524786,
+4.00000000,
-2.07108426,
+1.63782883,
+2.10013771,
-5.61528635,
-2.64949918,
-4.22243834,
-6.41157246
1163-1.08652520,
-2.07108426,
+9.00000000,
-3.22147989,
-0.655206084,
+6.93056011,
-7.32249546,
+3.19860888,
+11.3008451
1164-0.899771214,
+1.63782883,
-3.22147989,
+16.0000000,
+2.31580687,
-9.37684059,
-7.61445665,
-1.33025646,
+2.76267791
1165+1.60182416,
+2.10013771,
-0.655206084,
+2.31580687,
+25.0000000,
-5.97565365,
+7.52172184,
-21.8173752,
+4.94239283
1166+0.350251436,
-5.61528635,
+6.93056011,
-9.37684059,
-5.97565365,
+36.0000000,
+15.2414961,
+13.5586033,
+20.8919735
1167+1.98801780,
-2.64949918,
-7.32249546,
-7.61445665,
+7.52172184,
+15.2414961,
+49.0000000,
-11.2109098,
+13.1339788
1168-4.89459229,
-4.22243834,
+3.19860888,
-1.33025646,
-21.8173752,
+13.5586033,
-11.2109098,
+64.0000000,
+16.8706017
1169-5.81478357,
-6.41157246,
+11.3008451,
+2.76267791,
+4.94239283,
+20.8919735,
+13.1339788,
+16.8706017,
+81.0000000
1173call setCovRand(
rngf, rand,
onion, eta
= 0._TKG, scale
= [(
real(itry, TKG), itry
= 1, ndim)])
1177+1.00000000,
+0.860533714E-1,
+0.167458639,
-0.874029770E-1,
-0.472449660,
+0.350251436,
+1.98801780,
-4.89459229,
-5.81478357
1178+0.197524786,
+1.00000000,
-0.565134287E-1,
+0.320119888,
-0.114935264,
-5.61528635,
-2.64949918,
-4.22243834,
-6.41157246
1179-1.08652520,
-2.07108426,
+1.00000000,
-0.381791294,
+0.627015650,
+6.93056011,
-7.32249546,
+3.19860888,
+11.3008451
1180-0.899771214,
+1.63782883,
-3.22147989,
+1.00000000,
-1.98168969,
-9.37684059,
-7.61445665,
-1.33025646,
+2.76267791
1181+1.60182416,
+2.10013771,
-0.655206084,
+2.31580687,
+1.00000000,
-5.97565365,
+7.52172184,
-21.8173752,
+4.94239283
1182+0.350251436,
-5.61528635,
+6.93056011,
-9.37684059,
-5.97565365,
+36.0000000,
+15.2414961,
+13.5586033,
+20.8919735
1183+1.98801780,
-2.64949918,
-7.32249546,
-7.61445665,
+7.52172184,
+15.2414961,
+49.0000000,
-11.2109098,
+13.1339788
1184-4.89459229,
-4.22243834,
+3.19860888,
-1.33025646,
-21.8173752,
+13.5586033,
-11.2109098,
+64.0000000,
+16.8706017
1185-5.81478357,
-6.41157246,
+11.3008451,
+2.76267791,
+4.94239283,
+20.8919735,
+13.1339788,
+16.8706017,
+81.0000000
- Test:
- test_pm_distCov
- Todo:
- High Priority: The current implementation of this generic interface uses a naive method of computing the Cholesky factorization with a default matrix packing for the Onion method.
The RFP packing format must be also implemented for this generic interface.
- Todo:
- High Priority: The current implementation of the Gram method can be significantly improved, both computationally and functionally.
Final Remarks ⛓
If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.
For details on the naming abbreviations, see this page.
For details on the naming conventions, see this page.
This software is distributed under the MIT license with additional terms outlined below.
-
If you use any parts or concepts from this library to any extent, please acknowledge the usage by citing the relevant publications of the ParaMonte library.
-
If you regenerate any parts/ideas from this library in a programming environment other than those currently supported by this ParaMonte library (i.e., other than C, C++, Fortran, MATLAB, Python, R), please also ask the end users to cite this original ParaMonte library.
This software is available to the public under a highly permissive license.
Help us justify its continued development and maintenance by acknowledging its benefit to society, distributing it, and contributing to it.
- Copyright
- Computational Data Science Lab
- Author:
- Amir Shahmoradi, Monday March 6, 2017, 3:22 pm, Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin.
Definition at line 787 of file pm_distCov.F90.
1.9.3