Return a (collection) of random vector(s) of size ndim from the ndim-dimensional MultiVariate Uniform Ellipsoidal (MVUE) distribution, optionally with the specified input mean(1:ndim) and the specified subset of the Cholesky Factorization of the Gramian matrix of the MVUE distribution. More...

Detailed Description

Return a (collection) of random vector(s) of size ndim from the ndim-dimensional MultiVariate Uniform Ellipsoidal (MVUE) distribution, optionally with the specified input mean(1:ndim) and the specified subset of the Cholesky Factorization of the Gramian matrix of the MVUE 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` vector of shape `(1:ndim)` or matrix of shape `(1:ndim, 1:nsam)` of type `real` of kind any supported by the processor (e.g., RK, RK32, RK64, or RK128), containing the random output vector(s).
[in]	mean	: The input `contiguous` vector of shape `(1:ndim)`, of the same type and kind as the output `rand`, representing the center of the distribution. (optional, default = `[(0., i = 1, size(rand))]`.)
[in]	chol	: The input `contiguous` matrix of shape `(ndim, ndim)` whose specified triangular `subset` contains the Cholesky Factorization of the Gramian matrix of the MVUE 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 upper-diagonal 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 lower-diagonal 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 ⛓

: use pm_distUnifEll, only: setUnifEllRand

! single vector, using default rng

call setUnifEllRand(rand(1:ndim), mean(1:ndim))

call setUnifEllRand(rand(1:ndim), chol(1:ndim, 1:ndim), subset)

call setUnifEllRand(rand(1:ndim), mean(1:ndim), chol(1:ndim, 1:ndim), subset)

! single vector, using custom rng

call setUnifEllRand(rng, rand(1:ndim), mean(1:ndim))

call setUnifEllRand(rng, rand(1:ndim), chol(1:ndim, 1:ndim), subset)

call setUnifEllRand(rng, rand(1:ndim), mean(1:ndim), chol(1:ndim, 1:ndim), subset)

! collection of `nsam` vectors, using default rng

call setUnifEllRand(rand(1:ndim, 1:nsam), mean(1:ndim))

call setUnifEllRand(rand(1:ndim, 1:nsam), chol(1:ndim, 1:ndim), subset)

call setUnifEllRand(rand(1:ndim, 1:nsam), mean(1:ndim), chol(1:ndim, 1:ndim), subset)

! collection of `nsam` vectors, using custom rng

call setUnifEllRand(rng, rand(1:ndim, 1:nsam), mean(1:ndim))

call setUnifEllRand(rng, rand(1:ndim, 1:nsam), chol(1:ndim, 1:ndim), subset)

call setUnifEllRand(rng, rand(1:ndim, 1:nsam), mean(1:ndim), chol(1:ndim, 1:ndim), subset)

pm_distUnifEll::setUnifEllRand
Return a (collection) of random vector(s) of size ndim from the ndim-dimensional MultiVariate Uniform...
Definition: pm_distUnifEll.F90:2714

pm_distUnifEll
This module contains classes and procedures for computing various statistical quantities related to t...
Definition: pm_distUnifEll.F90:144

Warning: The condition size(mean, 1) == size(rand, 1) must hold for the corresponding input arguments.
The condition all(shape(chol) == size(rand, 1)) must hold for the corresponding input arguments.
These conditions are verified only if the library is built with the preprocessor macro CHECK_ENABLED=1.

Remarks: The procedures under discussion are impure. The procedures of this generic interface are pure when the input argument rng is set to xoshiro256ssw_type and the compile-time macro CHECK_ENABLED is set to 0 or is undefined.; The procedures under discussion are impure.

See also: getNormRand
setNormRand
getMultiNormLogPDF

Example usage ⛓

: 1program example

2

3 use pm_kind, only: SK

4 use pm_kind, only: IK, LK, RK ! all real kinds are supported.

5 use pm_io, only: display_type

6 use pm_matrixChol, only: getMatChol, uppDia, lowDia

7 use pm_distUnifEll, only: setUnifEllRand

8

9 implicit none

10

11 integer(IK), parameter :: NDIM = 2_IK ! The number of dimensions of the MVN distribution.

12 real(RK) :: mean(NDIM), gramian(NDIM, NDIM), choLow(NDIM, NDIM), rand(NDIM)

13 integer(IK) :: info

14

15 type(display_type) :: disp

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

17

18 call disp%skip()

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

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

21 call disp%show("! Generate random numbers from the (Standard) Multivariate Uniform Ellipsoidal distribution.")

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

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

24 call disp%skip()

25

26 call disp%skip()

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

28 call disp%show("! Multivariate Uniform Ellipsoidal random vector from a Standard distribution.")

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

30 call disp%skip()

31

32 call disp%skip()

33 call disp%show("call setUnifEllRand(rand)")

34 call setUnifEllRand(rand)

35 call disp%show("rand")

36 call disp%show( rand )

37 call disp%skip()

38

39 call disp%skip()

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

41 call disp%show("! Multivariate Uniform Ellipsoidal random vector with a particular mean and Identity Gramian matrix.")

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

43 call disp%skip()

44

45 call disp%skip()

46 call disp%show("mean = [-5., -5.]")

47 mean = [-5., -5.]

48 call disp%show("call setUnifEllRand(rand, mean)")

49 call setUnifEllRand(rand, mean)

50 call disp%show("rand")

51 call disp%show( rand )

52 call disp%skip()

53

54 call disp%skip()

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

56 call disp%show("! Multivariate Uniform Ellipsoidal random vector with zero mean and Gramian matrix specified via the Cholesky Lower Triangle.")

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

58 call disp%skip()

59

60 call disp%skip()

61 call disp%show("gramian = reshape([1., 1., 1., 4.], shape = [NDIM, NDIM])")

62 gramian = reshape([1., 1., 1., 4.], shape = [NDIM, NDIM])

63 call disp%show("choLow = getMatChol(gramian, lowDia)")

64 choLow = getMatChol(gramian, lowDia)

65 call disp%show("choLow")

66 call disp%show( choLow )

67 call disp%show("call setUnifEllRand(rand, choLow, lowDia)")

68 call setUnifEllRand(rand, choLow, lowDia)

69 call disp%show("rand")

70 call disp%show( rand )

71 call disp%skip()

72

73 call disp%skip()

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

75 call disp%show("! Multivariate Uniform Ellipsoidal random vector with given mean and Gramian matrix specified via the Cholesky Lower Triangle.")

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

77 call disp%skip()

78

79 call disp%skip()

80 call disp%show("mean")

81 call disp%show( mean )

82 call disp%show("choLow")

83 call disp%show( choLow )

84 call disp%show("call setUnifEllRand(rand, mean, choLow, lowDia)")

85 call setUnifEllRand(rand, mean, choLow, lowDia)

86 call disp%show("rand")

87 call disp%show( rand )

88 call disp%skip()

89

90 !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

91 ! Output an example rand array for visualization.

92 !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

93

94 block

95 integer(IK) :: fileUnit, i

96 integer(IK), parameter :: NVEC = 5000_IK

97 open(newunit = fileUnit, file = "setUnifEllRand.RK.txt")

98 do i = 1, NVEC

99 call setUnifEllRand(rand)

100 write(fileUnit,"(*(g0,:,','))") rand

101 end do

102 close(fileUnit)

103 open(newunit = fileUnit, file = "setUnifEllRandMean.RK.txt")

104 do i = 1, NVEC

105 call setUnifEllRand(rand, mean)

106 write(fileUnit,"(*(g0,:,','))") rand

107 end do

108 close(fileUnit)

109 open(newunit = fileUnit, file = "setUnifEllRandChol.RK.txt")

110 do i = 1, NVEC

111 call setUnifEllRand(rand, choLow, lowDia)

112 write(fileUnit,"(*(g0,:,','))") rand

113 end do

114 close(fileUnit)

115 open(newunit = fileUnit, file = "setUnifEllRandMeanChol.RK.txt")

116 do i = 1, NVEC

117 call setUnifEllRand(rand, mean, choLow, lowDia)

118 write(fileUnit,"(*(g0,:,','))") rand

119 end do

120 close(fileUnit)

121 end block

122

123end program example

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_matrixChol::getMatChol
Generate and return the upper or the lower Cholesky factorization of the input symmetric positive-def...
Definition: pm_matrixChol.F90:368

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::RK
integer, parameter RK
The default real kind in the ParaMonte library: real64 in Fortran, c_double in C-Fortran Interoperati...
Definition: pm_kind.F90:543

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_matrixChol
This module contains procedures and generic interfaces for computing the Cholesky factorization of po...
Definition: pm_matrixChol.F90:133

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! Generate random numbers from the (Standard) Multivariate Uniform Ellipsoidal distribution.

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

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

7

8

9!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

10! Multivariate Uniform Ellipsoidal random vector from a Standard distribution.

11!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

12

13

14call setUnifEllRand(rand)

15rand

16-0.66924052351860452, -0.39463295180884450

17

18

19!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

20! Multivariate Uniform Ellipsoidal random vector with a particular mean and Identity Gramian matrix.

21!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

22

23

24mean = [-5., -5.]

25call setUnifEllRand(rand, mean)

26rand

27-5.0544090255341958, -4.0071909973918673

28

29

30!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

31! Multivariate Uniform Ellipsoidal random vector with zero mean and Gramian matrix specified via the Cholesky Lower Triangle.

32!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

33

34

35gramian = reshape([1., 1., 1., 4.], shape = [NDIM, NDIM])

36choLow = getMatChol(gramian, lowDia)

37choLow

38+1.0000000000000000, +0.0000000000000000

39+1.0000000000000000, +1.7320508075688772

40call setUnifEllRand(rand, choLow, lowDia)

41rand

42-0.28254481136513249, -1.6790197729545753

43

44

45!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

46! Multivariate Uniform Ellipsoidal random vector with given mean and Gramian matrix specified via the Cholesky Lower Triangle.

47!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

48

49

50mean

51-5.0000000000000000, -5.0000000000000000

52choLow

53+1.0000000000000000, +0.0000000000000000

54+1.0000000000000000, +1.7320508075688772

55call setUnifEllRand(rand, mean, choLow, lowDia)

56rand

57-5.8651233976746084, -6.7217395072616135

58

59

Postprocessing of the example output ⛓
1#!/usr/bin/env python

2

3import matplotlib.pyplot as plt

4import pandas as pd

5import numpy as np

6import glob

7import sys

8

9linewidth = 2

10fontsize = 17

11

12for kind in ["RK"]:

13

14 pattern = "*." + kind + ".txt"

15 fileList = glob.glob(pattern)

16

17 for file in fileList:

18

19 df = pd.read_csv(file, delimiter = ",", header = None)

20

21 # definitions for the axes

22 left, width = 0.1, 0.65

23 bottom, height = 0.1, 0.65

24 spacing = 0.015

25

26 # start with a square Figure

27 fig = plt.figure(figsize = (8, 8))

28

29 plt.rcParams.update({'font.size': fontsize - 2})

30 ax = fig.add_axes([left, bottom, width, height]) # scatter plot

31 ax_histx = fig.add_axes([left, bottom + height + spacing, width, 0.2], sharex = ax) # histx

32 ax_histy = fig.add_axes([left + width + spacing, bottom, 0.2, height], sharey = ax) # histy

33

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")

38

39 # no labels

40 ax_histy.tick_params(axis = "y", labelleft = False)

41 ax_histx.tick_params(axis = "x", labelbottom = False)

42

43 # the scatter plot:

44 ax.scatter ( df.values[:, 0]

45 , df.values[:, 1]

46 , s = 8

47 , zorder = 1000

48 )

49

50 ax_histx.hist(df.values[:, 0], bins = 50, zorder = 1000)

51 ax_histy.hist(df.values[:, 1], bins = 50, orientation = "horizontal", zorder = 1000)

52

53 ax.set_xlabel("X", fontsize = 17)

54 ax.set_ylabel("Y", fontsize = 17)

55

56 plt.savefig(file.replace(".txt",".png"))

Visualization of the example output ⛓

Test:: test_pm_distUnifEll

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 2714 of file pm_distUnifEll.F90.

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

src/fortran/main/pm_distUnifEll.F90