pm_distLogNorm::getLogNormCDF Interface Reference

Generate and return the Cumulative Distribution Function (CDF) of the univariate Lognormal distribution. More...

Detailed Description

Generate and return the Cumulative Distribution Function (CDF) of the univariate Lognormal distribution.

Parameters

[in]	x	: The input scalar or array of the same shape as other array-like arguments, of type real of kind any supported by the processor (e.g., RK, RK32, RK64, or RK128), representing the point(s) at which the CDF must be computed.
[in]	mu	: The input scalar or array of the same shape as other array-like arguments of the same type and kind as x representing the location parameter of the distribution. (optional, default = 0)
[in]	sigma	: The input scalar of the same type and kind as x representing the scale parameter of the distribution. (optional, default = 1.)

Returns

cdf : The output scalar or array of the same shape as the input array-like arguments, of the same type and kind as the input x representing the CDF of the Lognormal distribution at the given input x.

Possible calling interfaces ⛓

use pm_distLogNorm, only: getLogNormCDF
 
cdf = getLogNormCDF(x, mu = mu, sigma = sigma)

Warning

The condition 0. < sigma must hold for the corresponding procedure argument.
This condition is 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.

Example usage ⛓

program example
 
    use pm_kind, only: SK
    use pm_kind, only: IK, RK ! all real kinds are supported.
    use pm_distLogNorm, only: getLogNormCDF
    use pm_arraySpace, only: setLinSpace
    use pm_arraySpace, only: setLogSpace
    use pm_io, only: display_type
 
    implicit none
 
    integer(IK), parameter  :: NP = 1000_IK
    real(RK), dimension(NP) :: Point, mu, Sigma, CDF
 
    type(display_type) :: disp
    disp = display_type(file = "main.out.F90")
 
    call setLinSpace(mu, x1 = -5._RK, x2 = +5._RK)
    call setLogSpace(Point, logx1 = -10._RK, logx2 = +3._RK)
    call setLogSpace(Sigma, logx1 = log(0.1_RK), logx2 = log(10._RK))
 
    call disp%skip()
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("! Compute the Cumulative Distribution Function (CDF) of the (Standard) LogNormal distribution.")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%skip()
 
    call disp%skip()
    call disp%show("!%%%%%%%%%%%%%%")
    call disp%show("! Standard CDF.")
    call disp%show("!%%%%%%%%%%%%%%")
    call disp%skip()
 
    call disp%skip()
    call disp%show("Point(NP/2)")
    call disp%show( Point(NP/2) )
    call disp%show("CDF(1) = getLogNormCDF(Point(NP/2))")
                    CDF(1) = getLogNormCDF(Point(NP/2))
    call disp%show("CDF(1)")
    call disp%show( CDF(1) )
    call disp%skip()
 
    call disp%skip()
    call disp%show("!%%%%%%%%%%%%%%%%%")
    call disp%show("! CDF with a mean.")
    call disp%show("!%%%%%%%%%%%%%%%%%")
    call disp%skip()
 
    call disp%skip()
    call disp%show("mu(1)")
    call disp%show( mu(1) )
    call disp%show("Point(1)")
    call disp%show( Point(1) )
    call disp%show("CDF(1) = getLogNormCDF(Point(1), mu(1))")
                    CDF(1) = getLogNormCDF(Point(1), mu(1))
    call disp%show("CDF(1)")
    call disp%show( CDF(1) )
    call disp%skip()
 
    call disp%skip()
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("! CDF with a standard deviation.")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%skip()
 
    call disp%skip()
    call disp%show("Sigma(1)")
    call disp%show( Sigma(1) )
    call disp%show("Point(1)")
    call disp%show( Point(1) )
    call disp%show("CDF(1) = getLogNormCDF(Point(1), Sigma(1))")
                    CDF(1) = getLogNormCDF(Point(1), Sigma(1))
    call disp%show("CDF(1)")
    call disp%show( CDF(1) )
    call disp%skip()
 
    call disp%skip()
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("! CDF with a mean and a standard deviation.")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%skip()
 
    call disp%skip()
    call disp%show("mu(1)")
    call disp%show( mu(1) )
    call disp%show("Sigma(1)")
    call disp%show( Sigma(1) )
    call disp%show("Point(1)")
    call disp%show( Point(1) )
    call disp%show("CDF(1) = getLogNormCDF(Point(1), mu(1), Sigma(1))")
                    CDF(1) = getLogNormCDF(Point(1), mu(1), Sigma(1))
    call disp%show("CDF(1)")
    call disp%show( CDF(1) )
    call disp%skip()
 
    call disp%skip()
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("! A vector of CDF at different points with the same mean and standard deviation.")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%skip()
 
    call disp%skip()
    call disp%show("mu(1)")
    call disp%show( mu(1) )
    call disp%show("Sigma(1)")
    call disp%show( Sigma(1) )
    call disp%show("Point(1:NP:NP/5)")
    call disp%show( Point(1:NP:NP/5) )
    call disp%show("CDF(1:NP:NP/5) = getLogNormCDF(Point(1:NP:NP/5), mu(1), Sigma(1))")
                    CDF(1:NP:NP/5) = getLogNormCDF(Point(1:NP:NP/5), mu(1), Sigma(1))
    call disp%show("CDF(1:NP:NP/5)")
    call disp%show( CDF(1:NP:NP/5) )
    call disp%skip()
 
    call disp%skip()
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("! A vector of CDF at the same point but with different means and standard deviations.")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%skip()
 
    call disp%skip()
    call disp%show("mu(1:NP:NP/5)")
    call disp%show( mu(1:NP:NP/5) )
    call disp%show("Sigma(1:NP:NP/5)")
    call disp%show( Sigma(1:NP:NP/5) )
    call disp%show("Point(1)")
    call disp%show( Point(1) )
    call disp%show("CDF(1:NP:NP/5) = getLogNormCDF(Point(1), mu(1:NP:NP/5), Sigma(1:NP:NP/5))")
                    CDF(1:NP:NP/5) = getLogNormCDF(Point(1), mu(1:NP:NP/5), Sigma(1:NP:NP/5))
    call disp%show("CDF(1:NP:NP/5)")
    call disp%show( CDF(1:NP:NP/5) )
    call disp%skip()
 
    call disp%skip()
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("! A vector of CDF at different points with different means and a standard deviations.")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%skip()
 
    call disp%skip()
    call disp%show("mu(1:NP:NP/5)")
    call disp%show( mu(1:NP:NP/5) )
    call disp%show("Sigma(1:NP:NP/5)")
    call disp%show( Sigma(1:NP:NP/5) )
    call disp%show("Point(1:NP:NP/5)")
    call disp%show( Point(1:NP:NP/5) )
    call disp%show("CDF(1:NP:NP/5) = getLogNormCDF(Point(1:NP:NP/5), mu(1:NP:NP/5), Sigma(1:NP:NP/5))")
                    CDF(1:NP:NP/5) = getLogNormCDF(Point(1:NP:NP/5), mu(1:NP:NP/5), Sigma(1:NP:NP/5))
    call disp%show("CDF(1:NP:NP/5)")
    call disp%show( CDF(1:NP:NP/5) )
    call disp%skip()
 
    !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    ! Output an example array for visualization.
    !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
    block
        integer(IK) :: fileUnit, i
        open(newunit = fileUnit, file = "getLogNormCDF.RK.txt")
        write(fileUnit,"(5(g0,:,' '))") (Point(i), getLogNormCDF(Point(i), [0._RK, 0._RK, 0._RK, -2._RK], sigma = [3.0_RK, 1.0_RK, 0.3_RK, 1.0_RK]), i = 1, NP)
        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 ⛓

 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! Compute the Cumulative Distribution Function (CDF) of the (Standard) LogNormal distribution.
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
!%%%%%%%%%%%%%%
! Standard CDF.
!%%%%%%%%%%%%%%
 
 
Point(NP/2)
+0.30001541764020159E-1
CDF(1) = getLogNormCDF(Point(NP/2))
CDF(1)
+0.22701516800954780E-3
 
 
!%%%%%%%%%%%%%%%%%
! CDF with a mean.
!%%%%%%%%%%%%%%%%%
 
 
mu(1)
-5.0000000000000000
Point(1)
+0.45399929762484847E-4
CDF(1) = getLogNormCDF(Point(1), mu(1))
CDF(1)
+0.28665157186802404E-6
 
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! CDF with a standard deviation.
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
Sigma(1)
+0.10000000000000003
Point(1)
+0.45399929762484847E-4
CDF(1) = getLogNormCDF(Point(1), Sigma(1))
CDF(1)
+0.0000000000000000
 
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! CDF with a mean and a standard deviation.
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
mu(1)
-5.0000000000000000
Sigma(1)
+0.10000000000000003
Point(1)
+0.45399929762484847E-4
CDF(1) = getLogNormCDF(Point(1), mu(1), Sigma(1))
CDF(1)
+0.0000000000000000
 
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! A vector of CDF at different points with the same mean and standard deviation.
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
mu(1)
-5.0000000000000000
Sigma(1)
+0.10000000000000003
Point(1:NP:NP/5)
+0.45399929762484847E-4, +0.61284568112625879E-3, +0.82726962539369248E-2, +0.11167167431796345, +1.5074363257375289
CDF(1:NP:NP/5) = getLogNormCDF(Point(1:NP:NP/5), mu(1), Sigma(1))
CDF(1:NP:NP/5)
+0.0000000000000000, +0.0000000000000000, +0.97991769798544137, +1.0000000000000000, +1.0000000000000000
 
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! A vector of CDF at the same point but with different means and standard deviations.
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
mu(1:NP:NP/5)
-5.0000000000000000, -2.9979979979979978, -0.99599599599599564, +1.0060060060060056, +3.0080080080080087
Sigma(1:NP:NP/5)
+0.10000000000000003, +0.25142033481427983, +0.63212184758124557, +1.5892828656229783, +3.9957803018952722
Point(1)
+0.45399929762484847E-4
CDF(1:NP:NP/5) = getLogNormCDF(Point(1), mu(1:NP:NP/5), Sigma(1:NP:NP/5))
CDF(1:NP:NP/5)
+0.0000000000000000, +0.0000000000000000, +0.0000000000000000, +0.21777024628022446E-11, +0.56609157067133431E-3
 
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! A vector of CDF at different points with different means and a standard deviations.
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
mu(1:NP:NP/5)
-5.0000000000000000, -2.9979979979979978, -0.99599599599599564, +1.0060060060060056, +3.0080080080080087
Sigma(1:NP:NP/5)
+0.10000000000000003, +0.25142033481427983, +0.63212184758124557, +1.5892828656229783, +3.9957803018952722
Point(1:NP:NP/5)
+0.45399929762484847E-4, +0.61284568112625879E-3, +0.82726962539369248E-2, +0.11167167431796345, +1.5074363257375289
CDF(1:NP:NP/5) = getLogNormCDF(Point(1:NP:NP/5), mu(1:NP:NP/5), Sigma(1:NP:NP/5))
CDF(1:NP:NP/5)
+0.0000000000000000, +0.0000000000000000, +0.92991403466413658E-9, +0.22091367749118629E-1, +0.25781859748430380
 
 

Postprocessing of the example output ⛓

#!/usr/bin/env python
 
import matplotlib.pyplot as plt
import pandas as pd
import numpy as np
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"]:
 
    pattern = "*." + 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 = 200)
        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("Cumulative Distribution Function (CDF)", 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.savefig(fileList[0].replace(".txt",".png"))
 
    elif len(fileList) > 1:
 
        sys.exit("Ambiguous file list exists.")

Visualization of the example output ⛓

Test:

test_pm_distLogNorm

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, Oct 16, 2009, 11:14 AM, Michigan

Definition at line 602 of file pm_distLogNorm.F90.

---

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

• src/fortran/main/pm_distLogNorm.F90