Return a (collection) of random vector(s) of size ndim
from the ndim
dimensional MultiVariate Normal (MVN) distribution, optionally with the specified input mean(1:ndim)
and the specified subset
of the Cholesky Factorization of the Covariance matrix of the MVN distribution.
 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.
(optional, default = rngf_type.) 
[out]  rand  : The output contiguous array of rank 1 of length ndim of

type
real of kind any supported by the processor (e.g., RK, RK32, RK64, or RK128),
containing the Multivariate Normal distributed random output vector.

[in]  mean  : The input contiguous vector of the same type, kind, rank, and size as rand , representing the mean of the Multivariate Normal distribution.
(optional, default = [(0., i = 1, size(rand))] . It must be present if the input argument chol is missing.) 
[in]  chol  : The input contiguous matrix of shape (ndim, ndim) whose specified triangular subset contains the Cholesky Factorization of the covariance matrix of the MVN distribution.
(optional, the default is the Identity matrix of rank ndim . It must be present if and only if the input argument subset is also present.) 
[in]  subset  : The input scalar constant that can be any of the following:

the constant uppDia or an object of type uppDia_type implying that the upperdiagonal triangular block of the input
chol must be used while the lower subset is not referenced.

the constant lowDia or an object of type lowDia_type implying that the lowerdiagonal triangular block of the input
chol must be used while the upper subset is not referenced.
This argument is merely a convenience to differentiate the different procedure functionalities within this generic interface.
(optional. It must be present if and only if the input argument chol is present.) 
Possible calling interfaces ⛓
call setMultiNormRand(rng, rand(
1:ndim), mean(
1:ndim), chol(
1:ndim,
1:ndim), subset)
call setMultiNormRand(rand(
1:ndim,
1:nsam), mean(
1:ndim), chol(
1:ndim,
1:ndim), subset)
call setMultiNormRand(rng, rand(
1:ndim,
1:nsam), mean(
1:ndim), chol(
1:ndim,
1:ndim), subset)
Return a (collection) of random vector(s) of size ndim from the ndimdimensional MultiVariate Normal ...
This module contains classes and procedures for computing various statistical quantities related to t...
 Warning
 The condition
all(shape(chol) == size(rand, 1))
must hold for the corresponding input arguments.
The condition size(mean, 1, IK) == size(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
.
 See also
 getNormRand
setNormRand
getNormLogPDF
Example usage ⛓
12 real(RKG),
allocatable :: mean(:), cov(:,:), chol(:,:), rand(:)
15 type(display_type) :: disp
19 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
20 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
21 call disp%show(
"! Generate random numbers from the (Standard) Multivariate Normal distribution.")
22 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
23 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
27 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
28 call disp%show(
"! Multivariate Normal random vector from a Standard distribution.")
29 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
33 call disp%show(
"call setResized(rand, 3_IK)")
35 call disp%show(
"call setMultiNormRand(rand)")
42 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
43 call disp%show(
"! Multivariate Normal random vector with a particular mean and Identity covariance matrix.")
44 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
52 call disp%show(
"call setResized(rand, size(mean, 1, IK))")
54 call disp%show(
"call setMultiNormRand(rand, mean)")
61 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
62 call disp%show(
"! Multivariate Normal random vector with zero mean and Covariance matrix specified via the Cholesky Lower Triangle.")
63 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
67 call disp%show(
"cov = reshape([1., 1., 1., 4.], shape = [2, 2])")
68 cov
= reshape([
1.,
1.,
1.,
4.], shape
= [
2,
2])
71 call disp%show(
"chol = getMatChol(cov, uppDia)")
73 call disp%show(
"call setResized(rand, size(chol, 1, IK))")
75 call disp%show(
"call setMultiNormRand(rand, chol, uppDia)")
82 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
83 call disp%show(
"! Multivariate Normal random vector with given mean and Covariance matrix specified via the Cholesky Lower Triangle.")
84 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
92 call disp%show(
"cov = reshape([1., 1., 1., 4.], shape = [2, 2])")
93 cov
= reshape([
1.,
1.,
1.,
4.], shape
= [
2,
2])
96 call disp%show(
"chol = getMatChol(cov, uppDia)")
98 call disp%show(
"call setResized(rand, size(chol, 1, IK))")
100 call disp%show(
"call setMultiNormRand(rand, mean, chol, uppDia)")
113 real(RKG) :: rand(
2,
5000)
116 if (
0 /= getErrTableWrite(
"setMultiNormRand.RK.txt", rand, trans))
error stop 'table write failed.'
119 if (
0 /= getErrTableWrite(
"setMultiNormRandMean.RK.txt", rand, trans))
error stop 'table write failed.'
122 if (
0 /= getErrTableWrite(
"setMultiNormRandChol.RK.txt", rand, trans))
error stop 'table write failed.'
125 if (
0 /= getErrTableWrite(
"setMultiNormRandMeanChol.RK.txt", rand, trans))
error stop 'table write failed.'
Allocate or resize (shrink or expand) an input allocatable scalar string or array of rank 1....
Generate and return the iostat code resulting from writing the input table of rank 1 or 2 to the spec...
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.
Generate and return the upper or the lower Cholesky factorization of the input symmetric positivedef...
This module contains procedures and generic interfaces for resizing allocatable arrays of various typ...
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 typeparameters 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 CFortran Interoper...
integer, parameter SK
The default character kind in the ParaMonte library: kind("a") in Fortran, c_char in CFortran Intero...
integer, parameter RKS
The singleprecision real kind in Fortran mode. On most platforms, this is an 32bit real kind.
This module contains procedures and generic interfaces for computing the Cholesky factorization of po...
Generate and return an object of type display_type.
Example Unix compile command via Intel ifort
compiler ⛓
3ifort fpp standardsemantics 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 /standardsemantics /O3 /I:..\..\..\include main.F90 ..\..\..\lib\libparamonte*.lib /exe:main.exe
Example Unix / MinGW compile command via GNU gfortran
compiler ⛓
3gfortran cpp ffreelinelengthnone O3 Wl,rpath,../../../lib I../../../inc main.F90 ../../../lib/libparamonte* o main.exe
Example output ⛓
170.251981348,
0.294494063,
+0.404360265
275.00000000,
5.00000000
315.56487083,
5.42903852
39cov
= reshape([
1.,
1.,
1.,
4.], shape
= [
2,
2])
41+1.00000000,
+1.00000000
42+1.00000000,
+4.00000000
471.49003351,
1.15621459
575.00000000,
5.00000000
58cov
= reshape([
1.,
1.,
1.,
4.], shape
= [
2,
2])
60+1.00000000,
+1.00000000
61+1.00000000,
+4.00000000
665.60389519,
5.55669832
Postprocessing of the example output ⛓
3import matplotlib.pyplot
as plt
14 pattern =
"*." + kind +
".txt"
15 fileList = glob.glob(pattern)
19 df = pd.read_csv(file, delimiter =
",", header =
None)
22 left, width = 0.1, 0.65
23 bottom, height = 0.1, 0.65
27 fig = plt.figure(figsize=(8, 8))
29 plt.rcParams.update({
'font.size': fontsize  2})
30 ax = fig.add_axes([left, bottom, width, height])
31 ax_histx = fig.add_axes([left, bottom + height + spacing, width, 0.2], sharex = ax)
32 ax_histy = fig.add_axes([left + width + spacing, bottom, 0.2, height], sharey = ax)
34 for axes
in [ax, ax_histx, ax_histy]:
35 axes.grid(visible =
True, which =
"both", axis =
"both", color =
"0.85", linestyle =
"")
36 axes.tick_params(axis =
"y", which =
"minor")
37 axes.tick_params(axis =
"x", which =
"minor")
40 ax_histy.tick_params(axis =
"y", labelleft =
False)
41 ax_histx.tick_params(axis =
"x", labelbottom =
False)
44 ax.scatter ( df.values[:, 0]
50 ax_histx.hist(df.values[:, 0], bins = 50, zorder = 1000)
51 ax_histy.hist(df.values[:,1], bins = 50, orientation =
"horizontal", zorder= 1000)
53 ax.set_xlabel(
"X", fontsize = 17)
54 ax.set_ylabel(
"Y", fontsize = 17)
56 plt.savefig(file.replace(
".txt",
".png"))
Visualization of the example output ⛓
 Test:
 test_pm_distMultiNorm
 Internal naming convention:
 The following illustrates the internal naming convention used for the procedures within this generic interface.
setMNR_RNGD_DM_DC_XXX_D1_RK5()
     
     
     
     The Kind of the output array.
     The type of the output array: R => Real.
    The Dimension of the output array.
   The subset of the Cholesky factor: D/U/L => Default/Upper/Lower.
  The Cholesky factor of the covariance matrix of the distribution: DC/AC => Default/Arbitrary Cholesky
 The mean of the distribution: DM/AM => Default/Arbitrary Mean
The random number generator: RNG D/F/X => Default/Fortran/Xoroshiro256++
 Todo:
 Normal Priority: The access pattern for the upperdiagonal subset of
chol
is non contiguous
in the current implementation.
The access pattern can be likely made contiguous
by an appropriate implementation.
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, April 23, 2017, 12:36 AM, Institute for Computational Engineering and Sciences (ICES), University of Texas at Austin
Definition at line 3264 of file pm_distMultiNorm.F90.