pm_distUnifElls::getUnifEllsLogPDF Interface Reference

Generate and return the natural logarithm of an approximation of the Probability Density Function (PDF) of the Multiple MultiVariate Uniformly Ellipsoidal (MMVUE) Distribution.
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Detailed Description

Generate and return the natural logarithm of an approximation of the Probability Density Function (PDF) of the Multiple MultiVariate Uniformly Ellipsoidal (MMVUE) Distribution.

See the documentation of pm_distUnifElls for details of the definition of the PDF.

Note

Note that the procedures of this generic interface do not require an input X value representing a location within the domain of the density function.
This is fine and intentional by design because the PDF is uniform across the entire support of the PDF.

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.
[in]	mean	: The input contiguous matrix of shape (1:ndim, 1:nell), of the same type and kind as the output logPDF, representing the centers of the ellipsoids in the distribution.
[in]	chol	: The input contiguous array of shape (1:ndim, 1:ndim, 1:nell) whose specified triangular subset contains the Cholesky Factorization of the Gramian matrix of the MMVUE distribution.
[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.
[in]	invGram	: The input array of shape (1:ndim, 1:ndim, 1:nell) of the same type and kind as the input argument point, containing the collection of square matrices of full inverse representative Gramian matrix of the \(\ndim\)-dimensional ellipsoids in the distribution. This argument is needed to determine the membership of the generated random vectors for estimating the effective volumes the ellipsoids.
[in]	nsim	: The input positive scalar of type integer of default kind IK, containing the number of random number simulations in approximating the PDF of the distribution. The larger this integer, the more accurate the logPDF estimate will be. (optional, default = 10000.)
[in]	normed	: The input positive scalar of type logical of default kind LK. If set to .false., the output logPDF will contain the usual normalizing factor corresponding to the volume of a unit-ball in the corresponding number of dimensions of the domain of the distribution. If set to .true., the output logPDF will not contain the factor corresponding to the volume of a unit-ball in the corresponding number of dimensions of the domain of the distribution. This factor is returned by the generic interface getLogVolUnitBall. In such a case, the output logPDF is not simply the natural logarithm of the inverse of effective volumes of the ellipsoids, but instead, their effective sum of the square-roots of the multiplicative traces of their Cholesky factors. This normalized value is occasionally useful in clustering algorithms or when the normalization factor is irrelevant. Beware that in this case, the output logPDF does not represent the actual log(PDF) of the distribution, but is a constant-added equivalent to it. (optional, default = .false..)

Returns

logPDF : The output scalar of,

1. type real of kind any supported by the processor (e.g., RK, RK32, RK64, or RK128),

containing the natural logarithm of the PDF of the distribution.

Possible calling interfaces ⛓

use pm_distUnifElls, only: getUnifEllsLogPDF
 
logPDF = getUnifEllsLogPDF(rng, mean(1:ndim, 1:nell), nsim = nsim, normed = normed) ! all ellipsoids are unit-balls centered on `mean` slices.
logPDF = getUnifEllsLogPDF(rng, mean(1:ndim, 1:nell), chol(1:ndim, 1:ndim, 1:nell), subset, invGram(1:ndim, 1:ndim, 1:nell), nsim = nsim, normed = normed)
!

Warning

The condition 0 < nsim must hold for the corresponding input arguments.
The condition all(shape(invGram) == [size(mean, 1), size(mean, 1), size(mean, 2)) must hold for the corresponding input arguments.
The condition size(chol, 1) <= size(chol, 2) must hold for the corresponding input arguments (to ensure that arguments can be passed without data copy).
The condition all([size(chol, 1), size(chol, 3)] == shape(mean)) 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.

Warning

The condition 0 < nell must hold for the corresponding input arguments.
The condition all(invmul <= 1) must hold for the corresponding input arguments.
The condition size(chol, 1) == size(chol, 2) 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

pm_ellipsoid
setUnifEllsRand

Example usage ⛓

program example
 
    use pm_kind, only: SK
    use pm_kind, only: IK, LK
    use pm_io, only: display_type
    use pm_arrayResize, only: setResized
    use pm_distUnif, only: getUnifRand, rngf_type
    use pm_matrixChol, only: getMatChol, uppDia, lowDia
    use pm_distUnifElls, only: getUnifEllsLogPDF
    use pm_distChol, only: getCholRand, uppDia
    use pm_matrixInv, only: getMatInv, choUpp
    use pm_io, only: getErrTableWrite, trans
 
    implicit none
 
    integer(IK) :: ndim, iell, nell, nsim
 
    type(display_type) :: disp
    disp = display_type(file = "main.out.F90")
 
    block
        use pm_kind, only: TKG => RKD
        real(TKG), allocatable :: logPDF, mean(:,:), chol(:, :, :), invGram(:, :, :)
 
        call disp%skip()
        call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
        call disp%show("! Random vectors from Multiple Multivariate Uniform Balls with particular means.")
        call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
        call disp%skip()
 
        call disp%skip()
        call disp%show("ndim = 2; nell = getUnifRand(5, 10); nsim = 2000")
                        ndim = 2; nell = getUnifRand(5, 10); nsim = 2000
        call disp%show("[ndim, nell, nsim]")
        call disp%show( [ndim, nell, nsim] )
        call disp%show("mean = getUnifRand(-5._TKG, +5._TKG, ndim, nell)")
                        mean = getUnifRand(-5._TKG, +5._TKG, ndim, nell)
        call disp%show("mean")
        call disp%show( mean )
        call disp%show("logPDF = getUnifEllsLogPDF(rngf_type(), mean)")
                        logPDF = getUnifEllsLogPDF(rngf_type(), mean)
        call disp%show("logPDF")
        call disp%show( logPDF )
        call disp%skip()
 
        call disp%skip()
        call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
        call disp%show("! Random vectors from Multiple Multivariate Uniform Ellipsoids with particular means.")
        call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
        call disp%skip()
 
        call disp%show("call setResized(chol, [ndim, ndim, nell])")
                        call setResized(chol, [ndim, ndim, nell])
        call disp%show("call setResized(invGram, [ndim, ndim, nell])")
                        call setResized(invGram, [ndim, ndim, nell])
        call disp%show("do iell = 1, nell; chol(:, :, iell) = getCholRand(0._TKG, ndim, uppDia); invGram(:, :, iell) = getMatInv(chol(:, :, iell), choUpp); end do")
                        do iell = 1, nell; chol(:, :, iell) = getCholRand(0._TKG, ndim, uppDia); invGram(:, :, iell) = getMatInv(chol(:, :, iell), choUpp); end do
        call disp%show("logPDF = getUnifEllsLogPDF(rngf_type(), mean, chol, uppDia, invGram)")
                        logPDF = getUnifEllsLogPDF(rngf_type(), mean, chol, uppDia, invGram)
        call disp%show("logPDF")
        call disp%show( logPDF )
        call disp%skip()
 
    end block
 
end program example

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

Example output ⛓

 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! Random vectors from Multiple Multivariate Uniform Balls with particular means.
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
ndim = 2; nell = getUnifRand(5, 10); nsim = 2000
[ndim, nell, nsim]
+2, +8, +2000
mean = getUnifRand(-5._TKG, +5._TKG, ndim, nell)
mean
+4.6721949008481705, +3.8405483717163516, -4.2567678643350497, -1.0083596223611488, -4.1744397360523369, +4.3619606622955018, -0.50427219453448657, -4.1315468790760939
-1.3486744413132055, +3.5202613861617835, -2.7405235151367071, +4.3625928882822134, +3.5714870112763095, +0.34434694199731997, +2.0582991034008256, +0.84244335856979013
logPDF = getUnifEllsLogPDF(rngf_type(), mean)
logPDF
+2.0758350460857242
 
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! Random vectors from Multiple Multivariate Uniform Ellipsoids with particular means.
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
call setResized(chol, [ndim, ndim, nell])
call setResized(invGram, [ndim, ndim, nell])
do iell = 1, nell; chol(:, :, iell) = getCholRand(0._TKG, ndim, uppDia); invGram(:, :, iell) = getMatInv(chol(:, :, iell), choUpp); end do
logPDF = getUnifEllsLogPDF(rngf_type(), mean, chol, uppDia, invGram)
logPDF
+2.6555241789069921
 
 

Test:

test_pm_distUnifElls

Final Remarks ⛓

---

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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.
   
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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.

Definition at line 310 of file pm_distUnifElls.F90.

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

• src/fortran/main/pm_distUnifElls.F90