! SPDX-FileCopyrightText: 2022-2026 Jason Christopherson ! SPDX-License-Identifier: MIT ! ! MIT License ! ! Copyright (c) 2022-2026 Jason Christopherson ! ! Permission is hereby granted, free of charge, to any person obtaining a copy ! of this software and associated documentation files (the "Software"), to deal ! in the Software without restriction, including without limitation the rights ! to use, copy, modify, merge, publish, distribute, sublicense, and/or sell ! copies of the Software, and to permit persons to whom the Software is ! furnished to do so, subject to the following conditions: ! ! The above copyright notice and this permission notice shall be included in all ! copies or substantial portions of the Software. ! ! THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR ! IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, ! FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE ! AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER ! LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, ! OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE ! SOFTWARE. module fstats_poisson_distribution use iso_fortran_env use ieee_arithmetic use fstats_distributions, only : distribution, probability_log use fstats_special_functions implicit none private public :: poisson_distribution type, extends(distribution) :: poisson_distribution !! Defines a Poisson distribution. real(real64), public :: occrence_rate !! The rate of occurrences. contains procedure, public :: log_pdf => pd_log_pdf procedure, public :: survival => pd_survival procedure, public :: log_cdf => pd_log_cdf procedure, public :: log_survival => pd_log_survival procedure, public :: pdf => pd_pdf procedure, public :: cdf => pd_cdf procedure, public :: mean => pd_mean procedure, public :: median => pd_median procedure, public :: mode => pd_mode procedure, public :: variance => pd_variance procedure, public :: recenter => pd_recenter procedure, public :: defined_range => pd_range end type contains pure elemental function pd_pdf(this, x) result(rst) !! Computes the probability mass function. !! !! The PMF for the Poisson distribution is given as \( f(k) = !! \frac{\lambda^{k} e^{-\lambda}}{k!} \). class(poisson_distribution), intent(in) :: this !! The poisson_distribution object. real(real64), intent(in) :: x !! The number of occurrences (\(k\)). real(real64) :: rst !! The value of the function. ! Local Variables real(real64) :: lambda ! Process lambda = this%occrence_rate rst = exp(this%log_pdf(x)) end function pure elemental function pd_cdf(this, x) result(rst) !! Computes the cumulative distribution function. !! !! The CDF for the Poisson distribution is given as !! $$ F(k) = \frac{\Gamma(floor(k + 1), \lambda)}{floor(\lambda}!)} $$. class(poisson_distribution), intent(in) :: this !! The poisson_distribution object. real(real64), intent(in) :: x !! The number of occurrences (\(k\)). real(real64) :: rst !! The value of the function. ! Local Variables real(real64) :: lambda ! Process lambda = this%occrence_rate rst = ieee_value(0.0d0, ieee_quiet_nan) if (.not.ieee_is_finite(lambda) .or. ieee_is_nan(x)) return if (lambda < 0.0d0) return if (x < 0.0d0) then rst = 0.0d0 else if (.not.ieee_is_finite(x)) then rst = 1.0d0 else rst = regularized_gamma_upper(aint(x) + 1.0d0, lambda) end if end function pure function pd_mean(this) result(rst) !! Computes the mean of the distribution. class(poisson_distribution), intent(in) :: this !! The poisson_distribution object. real(real64) :: rst !! The mean. ! Process rst = this%occrence_rate end function pure function pd_median(this) result(rst) !! Computes the median of the distribution. class(poisson_distribution), intent(in) :: this !! The poisson_distribution object. real(real64) :: rst !! The median. real(real64) :: lambda lambda = this%occrence_rate rst = floor(lambda + 1.0d0 / 3.0d0 - 1.0d0 / (5.0d1 * lambda)) end function pure function pd_mode(this) result(rst) !! Computes the mode of the distribution. class(poisson_distribution), intent(in) :: this !! The poisson_distribution object. real(real64) :: rst !! The mode. rst = max( & real(ceiling(this%occrence_rate) - 1.0d0, real64), & real(floor(this%occrence_rate), real64)) end function pure function pd_variance(this) result(rst) !! Computes the variance. class(poisson_distribution), intent(in) :: this !! The poisson_distribution object. real(real64) :: rst !! The variance. rst = this%occrence_rate end function subroutine pd_recenter(this, x) !! Recenters the distribution about the supplied value. This routine has !! no effect for this distribution as it is always centered about 0. class(poisson_distribution), intent(inout) :: this !! The poisson_distribution object. real(real64), intent(in) :: x !! The value about which to recenter. this%occrence_rate = x end subroutine pure function pd_range(this) result(rst) !! Gets the defined range for the distribution. class(poisson_distribution), intent(in) :: this !! The poisson_distribution object. real(real64), dimension(2) :: rst !! The defined range of the probability distributions [0, infinity). As !! using a value of infinity may cause issue, this routine returns !! huge(0.0d0) instead. rst = [0.0d0, huge(0.0d0)] end function pure elemental function pd_log_pdf(this, x) result(rst) class(poisson_distribution), intent(in) :: this !! Distribution with valid parameters. real(real64), intent(in) :: x !! Evaluation point; discrete masses require integer-valued x. real(real64) :: rst !! Log density; outside support gives -infinity, invalid inputs give NaN. !! An integrable density singularity at a support endpoint gives +infinity. rst = ieee_value(0.0d0, ieee_quiet_nan) if (ieee_is_nan(x)) return if (.not.ieee_is_finite(this%occrence_rate) .or. this%occrence_rate < 0.0d0) return rst = ieee_value(0.0d0, ieee_negative_inf) if (x < 0.0d0 .or. .not.ieee_is_finite(x)) return if (x /= aint(x)) return if (this%occrence_rate == 0.0d0) then if (x == 0.0d0) rst = 0.0d0 else rst = x * log(this%occrence_rate) - this%occrence_rate - log_gamma(x + 1.0d0) end if end function pure elemental function pd_survival(this, x) result(rst) class(poisson_distribution), intent(in) :: this !! Distribution with valid parameters. real(real64), intent(in) :: x !! Evaluation point; discrete tails include masses strictly above x. real(real64) :: rst !! Probability on [0, 1]; invalid parameters or NaN x give NaN. rst = ieee_value(0.0d0, ieee_quiet_nan) if (ieee_is_nan(x)) return if (.not.ieee_is_finite(this%occrence_rate) .or. this%occrence_rate < 0.0d0) return if (x < 0.0d0) then rst = 1.0d0 else if (.not.ieee_is_finite(x)) then rst = 0.0d0 else rst = regularized_gamma_lower(aint(x) + 1.0d0, this%occrence_rate) end if end function pure elemental function pd_log_cdf(this, x) result(rst) class(poisson_distribution), intent(in) :: this !! Distribution with valid parameters. real(real64), intent(in) :: x !! Evaluation point. real(real64) :: rst !! Log probability; zero probability gives -infinity, invalid inputs NaN. rst = probability_log(this%cdf(x)) if (ieee_is_nan(rst)) return if (x >= 0.0d0 .and. ieee_is_finite(x)) & rst = log_regularized_gamma_upper(aint(x) + 1.0d0, this%occrence_rate) end function pure elemental function pd_log_survival(this, x) result(rst) class(poisson_distribution), intent(in) :: this !! Distribution with valid parameters. real(real64), intent(in) :: x !! Evaluation point. real(real64) :: rst !! Log probability; zero probability gives -infinity, invalid inputs NaN. rst = probability_log(this%survival(x)) if (ieee_is_nan(rst)) return if (x >= 0.0d0 .and. ieee_is_finite(x)) & rst = log_regularized_gamma_lower(aint(x) + 1.0d0, this%occrence_rate) end function end module fstats_poisson_distribution