pm_distMultiNorm::getMultiNormLogPDF Interface Reference

Generate and return the natural logarithm of the Probability Density Function (PDF) of the MultiVariate Normal distribution as defined in the description of pm_distMultiNorm. More...

Detailed Description

Generate and return the natural logarithm of the Probability Density Function (PDF) of the MultiVariate Normal distribution as defined in the description of pm_distMultiNorm.

See the documentation of pm_distMultiNorm for the definition of the MVN PDF.

Parameters

[in]	X	: The input array of shape (:) of type real of kind any supported by the processor (e.g., RK, RK32, RK64, or RK128) of arbitrary size (ndim : representing the number of dimensions of the MVN) at which the log(PDF) of the MVN must be computed.
[in]	mean	: The input array of the same type kind, and shape as the input X containing the mean of the MVN distribution. (optional, default = [( 0., integer :: i = 1, size(X) )])
[in]	invCov	: The input square matrix (of shape (ndim,ndim)) of the same type and kind as the input X, containing the inverse of the covariance matrix of the MVN distribution. (optional, default = Identity Matrix : getMatInit([size(X), size(X)], uppLowDia_type(), 0., 0., 1.))
[in]	logPDFNF	: The input scalar of the same type and kind as the input X containing the normalization factor of the MVN distribution. Specifying this input argument can lead to faster runtime when the log(PDF) must be computed for multiple different X values given the same inverse covariance matrix for the MVN PDF. It can be obtained by calling getMultiNormLogPDFNF. (optional, default = getMultiNormLogPDFNF(ndim = size(X), logSqrtDetInvCov))

Returns

logPDF : The output scalar or array of the same shape as the input array-like arguments, of the same type and kind as the input logDetInvCovMat representing the normalization factor of the MVN distribution.

Possible calling interfaces ⛓

use pm_distMultiNorm, only: getMultiNormLogPDF
 
logPDF = getMultiNormLogPDF(X(1:ndim))
logPDF = getMultiNormLogPDF(X(1:ndim), mean(1:ndim))
logPDF = getMultiNormLogPDF(X(1:ndim), invCov(1:ndim, 1:ndim))
logPDF = getMultiNormLogPDF(X(1:ndim), mean(1:ndim), invCov(1:ndim, 1:ndim))
logPDF = getMultiNormLogPDF(X(1:ndim), mean(1:ndim), invCov(1:ndim, 1:ndim), logPDFNF)
logPDF = getMultiNormLogPDF(X(1:ndim), invCov(1:ndim, 1:ndim), logPDFNF)
logPDF = getMultiNormLogPDF(X(1:ndim), mean(1:ndim), logPDFNF)
logPDF = getMultiNormLogPDF(X(1:ndim), logPDFNF)

Warning

The condition all(size(mean) == size(X, 1)) must hold for the corresponding input arguments.
The condition all(shape(invCov) == size(X, 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.

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.

Remarks

The procedures under discussion are elemental.

See also

getMultiNormLogPDFNF

Example usage ⛓

program example
 
    use pm_kind, only: SK, IK
    use pm_kind, only: RKG => RK ! all real kinds are supported.
    use pm_distMultiNorm, only: getMultiNormLogPDF
    use pm_arraySpace, only: getLinSpace
    use pm_arraySpace, only: setLinSpace
    use pm_arraySpace, only: setLogSpace
    use pm_io, only: display_type
 
    implicit none
 
    integer(IK), parameter  :: NP = 5_IK
 
    type(display_type) :: disp
    disp = display_type(file = "main.out.F90")
 
    call disp%skip()
    call disp%show("getMultiNormLogPDF([0._RKG]) ! 1D Norm")
    call disp%show( getMultiNormLogPDF([0._RKG]) )
    call disp%skip()
 
    call disp%skip()
    call disp%show("getMultiNormLogPDF([0._RKG], mean = [10._RKG], invCov = reshape([1._RKG], [1, 1])) ! 1D Norm")
    call disp%show( getMultiNormLogPDF([0._RKG], mean = [10._RKG], invCov = reshape([1._RKG], [1, 1])) )
    call disp%skip()
 
    call disp%skip()
    call disp%show("getMultiNormLogPDF([real(RKG) :: 0, 1], mean = [-1._RKG, 1._RKG], invCov = reshape([+1.3_RKG, -.66_RKG, -.66_RKG, 1.3_RKG], [2, 2])) ! 2D Norm")
    call disp%show( getMultiNormLogPDF([real(RKG) :: 0, 1], mean = [-1._RKG, 1._RKG], invCov = reshape([+1.3_RKG, -.66_RKG, -.66_RKG, 1.3_RKG], [2, 2])) )
    call disp%skip()
 
    !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    ! Output an example density array for visualization.
    !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
    block
        integer(IK) :: fileUnit, i, j
        real(RKG)   , parameter :: signif = 5
        integer(IK) , parameter :: ndim = 2, npnt = 500
        real(RKG) :: grid(ndim, npnt, npnt), mean(ndim), invCov(ndim, ndim)
        mean = [+5._RKG, -5._RKG]
        invCov = reshape([+1.3_RKG, -.66_RKG, -.66_RKG, 1.3_RKG], [ndim, ndim])
        do i = 1, ndim
            grid(i, :, :) = spread  ( getLinSpace( x1 = mean(i) - signif &
                                                    , x2 = mean(i) + signif &
                                                    , count = npnt &
                                                    ) , 3 - i, npnt)
        end do
        open(newunit = fileUnit, file = "getMultiNormLogPDF.D2.RK.txt")
        do i = 1, size(grid, 2)
            do j = 1, size(grid, 3)
                write(fileUnit,"(5(g0,:,','))") grid(:, i, j), exp(getMultiNormLogPDF(grid(:, i, j), mean, invCov))
            end do
        end do
        close(fileUnit)
    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 ⛓

 
getMultiNormLogPDF([0._RKG]) ! 1D Norm
-0.91893853320467267
 
 
getMultiNormLogPDF([0._RKG], mean = [10._RKG], invCov = reshape([1._RKG], [1, 1])) ! 1D Norm
-50.918938533204674
 
 
getMultiNormLogPDF([real(RKG) :: 0, 1], mean = [-1._RKG, 1._RKG], invCov = reshape([+1.3_RKG, -.66_RKG, -.66_RKG, 1.3_RKG], [2, 2])) ! 2D Norm
-2.3745483811023420
 
 

Postprocessing of the example output ⛓

#!/usr/bin/env python
 
import matplotlib.pyplot as plt
import pandas as pd
import numpy as np
import math
import glob
import sys
 
linewidth = 2
fontsize = 17
 
marker ={ "CK" : "-"
        , "IK" : "."
        , "RK" : "-"
        }
xlab =  { "CK" : "X ( real/imaginary components )"
        , "IK" : "X ( integer-valued )"
        , "RK" : "X ( real-valued )"
        }
legends =   [ r"$\mu = 0.0,~\sigma = 3.0$"
            , r"$\mu = 0.0,~\sigma = 1.0$"
            , r"$\mu = 0.0,~\sigma = 0.3$"
            , r"$\mu = -2.,~\sigma = 1.0$"
            ]
 
for kind in ["IK", "CK", "RK"]:
 
        # Make 1d plot.
 
    pattern = "*.D1."+kind+".txt"
    fileList = glob.glob(pattern)
    if len(fileList) == 1:
 
        df = pd.read_csv(fileList[0], delimiter = ",")
 
        fig = plt.figure(figsize = 1.25 * np.array([6.4, 4.8]), dpi = 300)
        ax = plt.subplot()
 
        if kind == "CK":
            plt.plot( df.values[:, 0]
                    , df.values[:,1:5]
                    , marker[kind]
                    , linewidth = linewidth
                   #, color = "r"
                    )
            plt.plot( df.values[:, 1]
                    , df.values[:,1:5]
                    , marker[kind]
                    , linewidth = linewidth
                   #, color = "blue"
                    )
        else:
            plt.plot( df.values[:, 0]
                    , df.values[:,1:5]
                    , marker[kind]
                    , linewidth = linewidth
                   #, color = "r"
                    )
            ax.legend   ( legends
                        , fontsize = fontsize
                        )
 
            plt.xticks(fontsize = fontsize - 2)
            plt.yticks(fontsize = fontsize - 2)
            ax.set_xlabel(xlab[kind], fontsize = 17)
            ax.set_ylabel("Probability Density Function (PDF)", fontsize = 17)
 
            plt.grid(visible = True, which = "both", axis = "both", color = "0.85", linestyle = "-")
            ax.tick_params(axis = "y", which = "minor")
            ax.tick_params(axis = "x", which = "minor")
 
            plt.tight_layout()
            plt.savefig(fileList[0].replace(".txt",".png"))
 
    elif len(fileList) > 1:
 
        sys.exit("Ambiguous file list exists.")
    # Make 2d plot.
 
    pattern = "*.D2."+kind+".txt"
    fileList = glob.glob(pattern)
    if len(fileList) == 1:
 
        df = pd.read_csv(fileList[0], delimiter = ",", header = None)
        df = df.values
 
        npnt = math.isqrt(len(df[:, 0]))
        gridx = np.reshape(df[:, 0], newshape = (npnt, npnt), order = 'F')
        gridy = np.reshape(df[:,1], newshape = (npnt, npnt), order = 'F')
        gridz = np.reshape(df[:,2], newshape = (npnt, npnt), order = 'C')
 
        fig, ax = plt.subplots(subplot_kw = {"projection": "3d"})
        fig = plt.figure(figsize = 1.25 * np.array([6.4, 4.8]), dpi = 300)
        ax = fig.add_subplot(1, 1, 1, projection = '3d')
        
        ax.plot_surface(gridx, gridy, gridz, cmap = 'GnBu', linewidth = 0)
        ax.set_xlabel('X axis')
        ax.set_ylabel('Y axis')
        ax.set_zlabel("PDF", fontsize = 17)
        
        plt.xticks(fontsize = fontsize - 2)
        plt.yticks(fontsize = fontsize - 2)
        
        plt.grid(visible = True, which = "both", axis = "both", color = "0.85", linestyle = "-")
        ax.tick_params(axis = "y", which = "minor")
        ax.tick_params(axis = "x", which = "minor")
        
        plt.tight_layout()
        plt.savefig(fileList[0].replace(".txt",".png"))
 
    elif len(fileList) > 1:
 
        sys.exit("Ambiguous file list exists.")

Visualization of the example output ⛓

Test:

test_pm_distMultiNorm

Todo:

Normal Priority: When the input argument logPDFNF is missing and invCov is present, there is a possibility that the Cholesky Factorization in the computation of logSqrtDetInvCov fails.
In such cases, the procedure will halt the program by calling error stop.
An optional info output argument must be added in the future to handle such runtime failures gracefully.

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.

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 449 of file pm_distMultiNorm.F90.

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

• src/fortran/main/pm_distMultiNorm.F90