This module contains classes and procedures for computing various statistical quantities related to the Exponential distribution. More...

Data Types
type	distExp_type
	This is the derived type for signifying distributions that are of type Exponential as defined in the description of pm_distExp. More...

interface	getExpCDF
	Generate and return the Cumulative Distribution Function (CDF) of the Exponential distribution for an input `x` within the support of the distribution \([\mu, +\infty)\). More...

interface	getExpLogPDF
	Generate and return the natural logarithm of the Probability Density Function (PDF) of the Exponential distribution for an input `x` within the support of the distribution \([\mu, +\infty)\). More...

interface	getExpRand
	Return a scalar (or array of arbitrary rank of) random value(s) from the Exponential distribution, optionally with the specified input location and scale parameters `mu, sigma`. More...

interface	setExpCDF
	Return the Cumulative Distribution Function (CDF) of the Exponential distribution for an input `x` within the support of the distribution \([\mu, +\infty)\). More...

interface	setExpLogPDF
	Return the natural logarithm of the Probability Density Function (PDF) of the Exponential distribution for an input `x` within the support of the distribution \([\mu, +\infty)\). More...

interface	setExpRand
	Return a scalar (or array of arbitrary rank of) random value(s) from the Exponential distribution, optionally with the specified input location and scale parameters `mu, sigma`. More...

Variables
character(*, SK), parameter	MODULE_NAME = "@pm_distExp"

Detailed Description

This module contains classes and procedures for computing various statistical quantities related to the Exponential distribution.

Specifically, this module contains routines for computing the following quantities of the Exponential distribution:

the Probability Density Function (PDF)
the Cumulative Distribution Function (CDF)
the Random Number Generation from the distribution (RNG)
the Inverse Cumulative Distribution Function (ICDF) or the Quantile Function

The PDF of the Exponential distribution with the two location and scale parameters \((\mu, \sigma)\) is defined as,

\begin{equation} \large F(x | \mu, \sigma) = \begin{cases} \frac{1}{\sigma} \exp(-\frac{x - \mu}{\sigma}) &,~ \mu \leq x < +\infty \\ 0 &,~ x < \mu \end{cases} \end{equation}

where \(-\infty < \mu < +\infty\) is the location parameter and \(0 < \sigma < +\infty\) is the scale parameter of the distribution.

The corresponding CDF with the two location and scale parameters \((\mu, \sigma)\) is defined as,

\begin{equation} \large \mathrm{CDF}(x | \mu, \sigma) = \begin{cases} 1 - \exp(-\frac{x - \mu}{\sigma}) &,~ \mu \leq x < +\infty \\ 0 &,~ x < \mu \end{cases} \end{equation}

where \(-\infty < \mu < +\infty\) is the location parameter and \(0 < \sigma < +\infty\) is the scale parameter of the distribution.

Random Number Generation

Assuming \(U \in (0, 1]\) is a uniformly-distributed random variate,

\begin{equation} \large T = -\sigma\log(U) + \mu ~, \end{equation}

is a random variate from the Exponential distribution with the location parameter \(-\infty < \mu < +\infty\) and the scale parameter \(0 < \sigma < +\infty\) such that \(\mu \leq T < +\infty\).

Note: Note that the mean of the Exponential distribution is the scale parameter: \(\mathrm{mean} = \sigma\)).

See also: pm_distUnif
pm_distNegExp

Benchmarks:

Benchmark :: The runtime performance of getExpLogPDF vs. setExpLogPDF ⛓

! Test the performance of `getExpLogPDF()` vs. `setExpLogPDF()`.
program benchmark
 
    use iso_fortran_env, only: error_unit
    use pm_bench, only: bench_type
    use pm_kind, only: IK, RK, SK
 
    implicit none
 
    integer(IK)                         :: i
    integer(IK)                         :: isize
    integer(IK)                         :: fileUnit
    integer(IK)     , parameter         :: NSIZE = 18_IK
    integer(IK)     , parameter         :: NBENCH = 2_IK
    integer(IK)                         :: arraySize(NSIZE)
    real(RK)        , allocatable       :: Array(:), Point(:)
    real(RK)                            :: dummy = 0._RK
    type(bench_type)                    :: bench(NBENCH)
 
    bench(1) = bench_type(name = SK_"getExpLogPDF", exec = getExpLogPDF , overhead = setOverhead)
    bench(2) = bench_type(name = SK_"setExpLogPDF", exec = setExpLogPDF , overhead = setOverhead)
 
    arraySize = [( 2_IK**isize, isize = 1_IK, NSIZE )]
 
    write(*,"(*(g0,:,' '))")
    write(*,"(*(g0,:,' vs. '))") (bench(i)%name, i = 1, NBENCH)
    write(*,"(*(g0,:,' '))")
 
    open(newunit = fileUnit, file = "main.out", status = "replace")
 
        write(fileUnit, "(*(g0,:,','))") "arraySize", (bench(i)%name, i = 1, NBENCH)
 
        loopOverArraySize: do isize = 1, NSIZE
 
            write(*,"(*(g0,:,' '))") "Benchmarking with size", arraySize(isize)
 
            allocate(Array(arraySize(isize)), Point(arraySize(isize)))
            do i = 1, NBENCH
                bench(i)%timing = bench(i)%getTiming(minsec = 0.05_RK)
            end do
            deallocate(Array, Point)
 
            write(fileUnit,"(*(g0,:,','))") arraySize(isize), (bench(i)%timing%mean, i = 1, NBENCH)
 
        end do loopOverArraySize
 
        write(*,"(*(g0,:,' '))") dummy
        write(*,"(*(g0,:,' '))")
 
    close(fileUnit)
 
contains
 
    !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    ! procedure wrappers.
    !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
    subroutine setOverhead()
        call initialize()
        call finalize()
    end subroutine
 
    subroutine initialize()
        call random_number(Point)
    end subroutine
 
    subroutine finalize()
        dummy = dummy + Array(1)
    end subroutine
 
    subroutine setExpLogPDF()
        block
            use pm_distExp, only: setExpLogPDF
            call initialize()
            call setExpLogPDF(Array, Point)
            call finalize()
        end block
    end subroutine
 
    subroutine getExpLogPDF()
        block
            use pm_distExp, only: getExpLogPDF
            call initialize()
            Array = getExpLogPDF(Point)
            call finalize()
        end block
    end subroutine
 
end program benchmark

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

Postprocessing of the benchmark output ⛓

#!/usr/bin/env python
 
import matplotlib.pyplot as plt
import pandas as pd
import numpy as np
 
fontsize = 14
 
methods = ["setExpLogPDF", "getExpLogPDF"]
 
df = pd.read_csv("main.out")
 
 
 
ax = plt.figure(figsize = 1.25 * np.array([6.4,4.6]), dpi = 200)
ax = plt.subplot()
 
for method in methods:
    plt.plot( df["arraySize"].values
            , df[method].values
            , linewidth = 2
            )
 
plt.xticks(fontsize = fontsize)
plt.yticks(fontsize = fontsize)
ax.set_xlabel("Array Size", fontsize = fontsize)
ax.set_ylabel("Runtime [ seconds ]", fontsize = fontsize)
ax.set_title("setExpLogPDF() vs. getExpLogPDF()\nLower is better.", fontsize = fontsize)
ax.set_xscale("log")
ax.set_yscale("log")
plt.minorticks_on()
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")
ax.legend   ( methods
           #, loc='center left'
           #, bbox_to_anchor=(1, 0.5)
            , fontsize = fontsize
            )
 
plt.tight_layout()
plt.savefig("benchmark.getExpLogPDF_vs_setExpLogPDF.runtime.png")
 
 
 
ax = plt.figure(figsize = 1.25 * np.array([6.4,4.6]), dpi = 200)
ax = plt.subplot()
 
plt.plot( df["arraySize"].values
        , np.ones(len(df["arraySize"].values))
       #, linestyle = "--"
       #, color = "black"
        , linewidth = 2
        )
plt.plot( df["arraySize"].values
        , df["getExpLogPDF"].values / df["setExpLogPDF"].values
        , linewidth = 2
        )
 
plt.xticks(fontsize = fontsize)
plt.yticks(fontsize = fontsize)
ax.set_xlabel("Array Size", fontsize = fontsize)
ax.set_ylabel("Runtime compared to setExpLogPDF()", fontsize = fontsize)
ax.set_title("getExpLogPDF() / setExpLogPDF()\nLower means faster. Lower than 1 means faster than setExpLogPDF().", fontsize = fontsize)
ax.set_xscale("log")
#ax.set_yscale("log")
plt.minorticks_on()
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")
ax.legend   ( ["setExpLogPDF", "getExpLogPDF"]
           #, bbox_to_anchor = (1, 0.5)
           #, loc = "center left"
            , fontsize = fontsize
            )
 
plt.tight_layout()
plt.savefig("benchmark.getExpLogPDF_vs_setExpLogPDF.runtime.ratio.png")

Visualization of the benchmark output ⛓

Benchmark moral ⛓

The procedures under the generic interface getExpLogPDF are functions while the procedures under the generic interface setExpLogPDF are subroutines.
From the benchmark results, it appears that the functional interface performs slightly less efficiently than the subroutine interface when the input array size is small.
Otherwise, the difference appears to be marginal and insignificant in most practical situations.

Benchmark :: The runtime performance of setExpLogPDF with and without logInvSigma ⛓

program benchmark
 
    use iso_fortran_env, only: error_unit
    use pm_bench, only: bench_type
    use pm_kind, only: IK, RK, SK
 
    implicit none
 
    integer(IK)                         :: i
    integer(IK)                         :: isize
    integer(IK)                         :: fileUnit
    integer(IK)     , parameter         :: NSIZE = 18_IK
    integer(IK)     , parameter         :: NBENCH = 2_IK
    integer(IK)                         :: arraySize(0:NSIZE)
    real(RK)        , allocatable       :: logPDF(:), point(:)
    real(RK)        , allocatable       :: logInvSigma(:), invSigma(:)
    real(RK)                            :: dummy = 0._RK
    type(bench_type)                    :: bench(NBENCH)
 
    bench(1) = bench_type(name = SK_"logInvSigmaMissing", exec = logInvSigmaMissing , overhead = setOverhead)
    bench(2) = bench_type(name = SK_"logInvSigmaPresent", exec = logInvSigmaPresent , overhead = setOverhead)
 
    arraySize = [( 2_IK**isize, isize = 0_IK, NSIZE )]
 
    write(*,"(*(g0,:,' '))")
    write(*,"(*(g0,:,' vs. '))") (bench(i)%name, i = 1, NBENCH)
    write(*,"(*(g0,:,' '))")
 
    open(newunit = fileUnit, file = "main.out", status = "replace")
 
        write(fileUnit, "(*(g0,:,','))") "arraySize", (bench(i)%name, i = 1, NBENCH)
 
        loopOverArraySize: do isize = 0, NSIZE
 
            write(*,"(*(g0,:,' '))") "Benchmarking with size", arraySize(isize)
 
            allocate(logPDF(arraySize(isize)), point(arraySize(isize)), invSigma(arraySize(isize)))
            call random_number(point)
            call random_number(invSigma)
            logInvSigma = log(invSigma)
            do i = 1, NBENCH
                bench(i)%timing = bench(i)%getTiming(minsec = 0.05_RK)
            end do
            deallocate(logPDF, point, invSigma)
 
            write(fileUnit,"(*(g0,:,','))") arraySize(isize), (bench(i)%timing%mean, i = 1, NBENCH)
 
        end do loopOverArraySize
 
        write(*,"(*(g0,:,' '))") dummy
        write(*,"(*(g0,:,' '))")
 
    close(fileUnit)
 
contains
 
    !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    ! procedure wrappers.
    !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
    subroutine setOverhead()
        call initialize()
        call finalize()
    end subroutine
 
    subroutine initialize()
        !call random_number(invSigma)
        !logInvSigma = log(invSigma)
    end subroutine
 
    subroutine finalize()
        dummy = dummy + logPDF(1)
    end subroutine
 
    subroutine logInvSigmaPresent()
        use pm_distExp, only: setExpLogPDF
        call initialize()
        if (arraySize(isize) > 1_IK) then
            call setExpLogPDF(logPDF, point, invSigma, logInvSigma)
        else
            call setExpLogPDF(logPDF(1), point(1), invSigma(1), logInvSigma(1))
        end if
        call finalize()
    end subroutine
 
    subroutine logInvSigmaMissing()
        use pm_distExp, only: getExpLogPDF
        call initialize()
        if (arraySize(isize) > 1_IK) then
            logPDF = getExpLogPDF(point, invSigma = invSigma)
        else
            logPDF(1) = getExpLogPDF(point(1), invSigma = invSigma(1))
        end if
        call finalize()
    end subroutine
 
end program benchmark

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

Postprocessing of the benchmark output ⛓

#!/usr/bin/env python
 
import matplotlib.pyplot as plt
import pandas as pd
import numpy as np
 
fontsize = 14
 
methods = ["logInvSigmaPresent", "logInvSigmaMissing"]
 
df = pd.read_csv("main.out")
 
 
 
ax = plt.figure(figsize = 1.25 * np.array([6.4,4.6]), dpi = 200)
ax = plt.subplot()
 
for method in methods:
    plt.plot( df["arraySize"].values
            , df[method].values
            , linewidth = 2
            )
 
plt.xticks(fontsize = fontsize)
plt.yticks(fontsize = fontsize)
ax.set_xlabel("Array Size", fontsize = fontsize)
ax.set_ylabel("Runtime [ seconds ]", fontsize = fontsize)
ax.set_title("logInvSigmaPresent() vs. logInvSigmaMissing()\nLower is better.", fontsize = fontsize)
ax.set_xscale("log")
ax.set_yscale("log")
plt.minorticks_on()
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")
ax.legend   ( methods
           #, loc='center left'
           #, bbox_to_anchor=(1, 0.5)
            , fontsize = fontsize
            )
 
plt.tight_layout()
plt.savefig("benchmark.setExpLogPDF-logInvSigma-missing_vs_present.runtime.png")
 
 
 
ax = plt.figure(figsize = 1.25 * np.array([6.4,4.6]), dpi = 200)
ax = plt.subplot()
 
plt.plot( df["arraySize"].values
        , np.ones(len(df["arraySize"].values))
       #, linestyle = "--"
       #, color = "black"
        , linewidth = 2
        )
plt.plot( df["arraySize"].values
        , df["logInvSigmaMissing"].values / df["logInvSigmaPresent"].values
        , linewidth = 2
        )
 
plt.xticks(fontsize = fontsize)
plt.yticks(fontsize = fontsize)
ax.set_xlabel("Array Size", fontsize = fontsize)
ax.set_ylabel("Runtime compared to logInvSigmaPresent()", fontsize = fontsize)
ax.set_title("logInvSigmaMissing / logInvSigmaPresent\nLower means faster. Lower than 1 means faster than logInvSigmaPresent.", fontsize = fontsize)
ax.set_xscale("log")
#ax.set_yscale("log")
plt.minorticks_on()
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")
ax.legend   ( ["logInvSigmaPresent", "logInvSigmaMissing"]
           #, bbox_to_anchor = (1, 0.5)
           #, loc = "center left"
            , fontsize = fontsize
            )
 
plt.tight_layout()
plt.savefig("benchmark.setExpLogPDF-logInvSigma-missing_vs_present.runtime.ratio.png")

Visualization of the benchmark output ⛓

Benchmark moral ⛓

The procedures under the generic interface setExpLogPDF accept an extra argument logInvSigma = log(invSigma) while the procedures under the generic interface getExpLogPDF compute this term internally with every procedure call.
In the presence of this argument, the logarithmic computation log(invSigma) will be avoided.
As such, the presence of logInvSigma is expected to lead to faster computations.

Test:: test_pm_distExp

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

Variable Documentation

◆ MODULE_NAME

character(*, SK), parameter pm_distExp::MODULE_NAME = "@pm_distExp"

Definition at line 118 of file pm_distExp.F90.

Data Types

Variables

Detailed Description

Variable Documentation

◆ MODULE_NAME