pm_distPois Module Reference

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

Data Types
type	distPois_type
	This is the derived type for signifying distributions that are of type Poisson as defined in the description of pm_distPois. More...
interface	getPoisCDF
	Generate and return the Cumulative Distribution Function (CDF) of the Poisson distribution for an input count within the discrete integer support of the distribution \([0, +\infty)\). More...
interface	getPoisLogPMF
	Generate and return the natural logarithm of the Probability Mass Function (PMF) of the Poisson distribution for an input count within the discrete integer support of the distribution \([0, +\infty)\). More...
interface	getPoisRand
	Generate and return a scalar (or array of arbitrary rank of) random value(s) from the Poisson distribution. More...
interface	setPoisCDF
	Return the Cumulative Distribution Function (CDF) of the Poisson distribution. More...
interface	setPoisLogPMF
	Return the natural logarithm of the Probability Mass Function (PMF) of the Poisson distribution for an input count within the discrete integer support of the distribution \([0, +\infty)\). More...
interface	setPoisRand
	Return a scalar (or array of arbitrary rank of) random value(s) from the Poisson distribution. More...

Variables
character(*, SK), parameter	MODULE_NAME = "@pm_distPois"
real(RKB), parameter	LAMBDA_LIMIT = 10._RKB
	The constant scalar of type real of kind RKB, representing the value of the parameter of the Poisson distribution above which the rejection method of Hormann, 1993, The transformed rejection method for generating Poisson random variables for generating Poisson-distributed random values is valid. More...

Detailed Description

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

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

1. the Probability Mass Function (PMF)
2. the Cumulative Distribution Function (CDF)
3. the Random Number Generation from the distribution (RNG)
4. the Inverse Cumulative Distribution Function (ICDF) or the Quantile Function

The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event.
It is named after French mathematician Siméon Denis Poisson.
The Poisson distribution can also be used for the number of events in other specified interval types such as distance, area, or volume.
A discrete random variable \(X\) is said to have a Poisson distribution, with parameter \(\lambda > 0\) if it has a probability mass function given by,

\begin{equation} f(k; \lambda) = \pi(X = k) = \frac {\lambda^k \exp\left(-\lambda\right)}{k!} ~, \end{equation}

where

1. \(k\) is the number of occurrences ( \(k = 0, 1, 2, \ldots\)), and
2. \(\ms{!}\) is the factorial function.

The positive real number \(\lambda\) is equal to the expected value of \(X\) and also to its variance.

\begin{equation} \lambda = \up{E}(X) = \up{Var}(X) ~. \end{equation}

The CDF of the Poisson distribution with parameter \(\lambda\) is defined as,

\begin{eqnarray} \ms{CDF}(k | \lambda) &=& \frac{\Gamma(\lfloor k + 1 \rfloor, \lambda)}{\lfloor k \rfloor !} ~, \nonumber \\ &=& \exp\left(-\lambda\right) \sum _{j=0}^{\lfloor k \rfloor}{\frac{\lambda^{j}}{j!}} ~, \end{eqnarray}

where

1. \(k\) is the number of occurrences ( \(k = 0, 1, 2, \ldots\)), and
2. \(\ms{!}\) is the factorial function, and
3. \(\Gamma(x, y) / \lfloor k \rfloor !\) is the regularized upper incomplete gamma function, and
4. \(\lfloor k \rfloor\) is the floor function.

Random Number Generation

The RNG generic interfaces of this module use two different approaches for Poisson RNG for different ranges of \(\lambda\) parameter values of the Poisson distribution.

1. When \(\lambda <\) LAMBDA_LIMIT, a RNG algorithm due to Donald Ervin Knuth is used.
   
2. When LAMBDA_LIMIT \(\leq \lambda\), a rejection-based RNG algorithm due to Hormann, 1993, The transformed rejection method for generating Poisson random variables is used.
   This rejection method has an efficiency slight less than \(90\%\).
   

See also

pm_distPois
pm_distPois
Poisson distribution

Benchmarks:

Benchmark :: The runtime performance of getPoisLogPMF vs. setPoisLogPMF ⛓

! Test the performance of `getPoisLogPMF()` vs. `setPoisLogPMF()`.
program benchmark
 
    use iso_fortran_env, only: error_unit
    use pm_kind, only: SK, IK, RK, RKG => RK
    use pm_distUnif, only: xoshiro256ssw_type
    use pm_bench, only: bench_type
 
    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)
    integer(IK)     , allocatable       :: count(:)
    real(RKG)       , allocatable       :: array(:)
    real(RKG)                           :: dummy = 0._RKG
    type(bench_type)                    :: bench(NBENCH)
    type(xoshiro256ssw_type)            :: rng
 
    rng = xoshiro256ssw_type()
 
    bench(1) = bench_type(name = SK_"getPoisLogPMF", exec = getPoisLogPMF , overhead = setOverhead)
    bench(2) = bench_type(name = SK_"setPoisLogPMF", exec = setPoisLogPMF , 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)), count(arraySize(isize)))
            do i = 1, NBENCH
                bench(i)%timing = bench(i)%getTiming(minsec = 0.05_RK)
            end do
            deallocate(array, count)
 
            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()
        use pm_distUnif, only: setUnifRand
        call setUnifRand(rng, count, 0_IK, 1023_IK)
    end subroutine
 
    subroutine finalize()
        dummy = dummy + array(1)
    end subroutine
 
    subroutine setPoisLogPMF()
        block
            use pm_distPois, only: setPoisLogPMF
            call initialize()
            call setPoisLogPMF(array, count, lambda = 1._RKG)
            call finalize()
        end block
    end subroutine
 
    subroutine getPoisLogPMF()
        block
            use pm_distPois, only: getPoisLogPMF
            call initialize()
            array = getPoisLogPMF(count, lambda = 1._RKG)
            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 = ["setPoisLogPMF", "getPoisLogPMF"]
 
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("setPoisLogPMF() vs. getPoisLogPMF()\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.getPoisLogPMF_vs_setPoisLogPMF.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["getPoisLogPMF"].values / df["setPoisLogPMF"].values
        , linewidth = 2
        )
 
plt.xticks(fontsize = fontsize)
plt.yticks(fontsize = fontsize)
ax.set_xlabel("Array Size", fontsize = fontsize)
ax.set_ylabel("Runtime compared to setPoisLogPMF()", fontsize = fontsize)
ax.set_title("getPoisLogPMF() / setPoisLogPMF()\nLower means faster. Lower than 1 means faster than setPoisLogPMF().", 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   ( ["setPoisLogPMF", "getPoisLogPMF"]
           #, bbox_to_anchor = (1, 0.5)
           #, loc = "center left"
            , fontsize = fontsize
            )
 
plt.tight_layout()
plt.savefig("benchmark.getPoisLogPMF_vs_setPoisLogPMF.runtime.ratio.png")

Visualization of the benchmark output ⛓

Benchmark moral ⛓

1. The procedures under the generic interface getPoisLogPMF are functions while the procedures under the generic interface setPoisLogPMF 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 setPoisLogPMF with and without logLambda ⛓

program benchmark
 
    use iso_fortran_env, only: error_unit
    use pm_bench, only: bench_type
    use pm_distUnif, only: xoshiro256ssw_type
    use pm_distUnif, only: setUnifRand
    use pm_kind, only: SK, IK, RK, RKG => RK
 
    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)
    integer(IK)     , allocatable       :: count(:)
    real(RKG)       , allocatable       :: logPMF(:)
    real(RKG)       , allocatable       :: logLambda(:), lambda(:)
    real(RKG)                           :: dummy = 0._RKG
    type(bench_type)                    :: bench(NBENCH)
    type(xoshiro256ssw_type)            :: rng
 
    rng = xoshiro256ssw_type()
 
    bench(1) = bench_type(name = SK_"logLambdaMissing", exec = logLambdaMissing , overhead = setOverhead)
    bench(2) = bench_type(name = SK_"logLambdaPresent", exec = logLambdaPresent , 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(logPMF(arraySize(isize)), count(arraySize(isize)), lambda(arraySize(isize)))
            call setUnifRand(rng, count, 0_IK, 1023_IK)
            call random_number(lambda)
            lambda = 1._RKG - lambda
            logLambda = log(lambda)
            do i = 1, NBENCH
                bench(i)%timing = bench(i)%getTiming(minsec = 0.05_RK)
            end do
            deallocate(logPMF, count, lambda)
 
            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(lambda)
        !logLambda = log(lambda)
    end subroutine
 
    subroutine finalize()
        dummy = dummy + logPMF(1)
    end subroutine
 
    subroutine logLambdaPresent()
        use pm_distPois, only: setPoisLogPMF
        call initialize()
        if (arraySize(isize) > 1_IK) then
            call setPoisLogPMF(logPMF, count, lambda, logLambda)
        else
            call setPoisLogPMF(logPMF(1), count(1), lambda(1), logLambda(1))
        end if
        call finalize()
    end subroutine
 
    subroutine logLambdaMissing()
        use pm_distPois, only: getPoisLogPMF
        call initialize()
        if (arraySize(isize) > 1_IK) then
            logPMF = getPoisLogPMF(count, lambda = lambda)
        else
            logPMF(1) = getPoisLogPMF(count(1), lambda = lambda(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 = ["logLambdaPresent", "logLambdaMissing"]
 
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("logLambdaPresent() vs. logLambdaMissing()\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.setPoisLogPMF-logLambda-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["logLambdaMissing"].values / df["logLambdaPresent"].values
        , linewidth = 2
        )
 
plt.xticks(fontsize = fontsize)
plt.yticks(fontsize = fontsize)
ax.set_xlabel("Array Size", fontsize = fontsize)
ax.set_ylabel("Runtime compared to logLambdaPresent()", fontsize = fontsize)
ax.set_title("logLambdaMissing / logLambdaPresent\nLower means faster. Lower than 1 means faster than logLambdaPresent.", 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   ( ["logLambdaPresent", "logLambdaMissing"]
           #, bbox_to_anchor = (1, 0.5)
           #, loc = "center left"
            , fontsize = fontsize
            )
 
plt.tight_layout()
plt.savefig("benchmark.setPoisLogPMF-logLambda-missing_vs_present.runtime.ratio.png")

Visualization of the benchmark output ⛓

Benchmark moral ⛓

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

Test:

test_pm_distPois

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

Variable Documentation

LAMBDA_LIMIT

real(RKB), parameter pm_distPois::LAMBDA_LIMIT = 10._RKB

The constant scalar of type real of kind RKB, representing the value of the parameter of the Poisson distribution above which the rejection method of Hormann, 1993, The transformed rejection method for generating Poisson random variables for generating Poisson-distributed random values is valid.

This constant exists merely as a reference with which decision can be made about the proper setPoisRand interface usage.

Test:

test_pm_distPois

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 762 of file pm_distPois.F90.

MODULE_NAME

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

Definition at line 129 of file pm_distPois.F90.