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mo_distributions.f90
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mo_distributions.f90
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!> \file mo_distributions.f90
!> \brief Statistical distributions
!> \details Continuous distribution density functions such as the normal distribution.
!> \author Matthias Cuntz
!> \date Apr 2016
module mo_distributions
! This module provides distribution density functions such as the normal distribution.
! Written Matthias Cuntz, Apr 2016
! License
! -------
! This file is part of the JAMS Fortran package, distributed under the MIT License.
!
! Copyright (c) 2016 Matthias Cuntz - mc (at) macu (dot) de
!
! 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.
implicit none
private
public :: ep ! Exponential Power of Box and Tiao (1992)
public :: ep01 ! Exponential Power of Box and Tiao (1992) with location=0, scale=1
public :: laplace ! Laplace = Double Exponential
public :: laplace01 ! Laplace = Double Exponential with location=0, scale=1
public :: normal ! Normal = Gauss
public :: normal01 ! Normal = Gauss with location=0, scale=1
public :: sep ! Skew Exponential Power
public :: sep01 ! Skew Exponential Power with location=0, scale=1
public :: sep_fs ! Skew Exponential Power after Fernandez and Steel (1998)
public :: sep01_fs ! Skew Exponential Power after Fernandez and Steel (1998) with location=0, scale=1
! public :: sep_fs_mean ! Mean of skew Exponential Power after Fernandez and Steel (1998)
! public :: sep01_fs_mean ! Mean of skew Exponential Power after Fernandez and Steel (1998) with location=0, scale=1
! public :: sep_fs_std ! Standard dev of skew Exponential Power after Fernandez and Steel (1998)
! public :: sep01_fs_std ! Standard dev of skew Exponential Power after Fernandez and Steel (1998) with location=0, scale=1
public :: st ! Skew Student t
public :: st01 ! Skew Student t with location=0, scale=1
public :: st_fs ! Skew Student t after Fernandez and Steel (1998)
public :: st01_fs ! Skew Student t after Fernandez and Steel (1998) with location=0, scale=1
public :: st_fs_mean ! mean of skew Student t after Fernandez and Steel (1998)
public :: st01_fs_mean ! mean of skew Student t after Fernandez and Steel (1998) with location=0, scale=1
public :: st_fs_std ! Standard dev of skew Student t after Fernandez and Steel (1998)
public :: st01_fs_std ! Standard dev of skew Student t after Fernandez and Steel (1998) with location=0, scale=1
public :: t ! Student t
public :: t01 ! Student t with location=0, scale=1
public :: beta_den ! beta probability density function
public :: weibull_pdf ! weibull density function
public :: weibull_cdf ! weibull cumulative distribution function
public :: gamma_pdf ! gamma distribution
! ------------------------------------------------------------------
! NAME
! ep
! PURPOSE
!> \brief Exponential power distribution of Box and Tiao (1992)
!> \details The exponential power distribution of Box and Tiao (1992) with given location \f$ loc \f$,
!> scale \f$ sca \f$ or standard deviation \f$ \sigma \f$ and kurtosis \f$ \beta \f$ parameter:
!> \f[ EP(x;loc,sca,\beta) = EP((x-loc)/sac;0,1,\beta) / sca \f]
!> \f[ EP(x;0,1,\beta) = \omega_\beta
!> \exp \left\{ -c_\beta |x|^\frac{2}{1+\beta} \right\} \f]
!> with
!> \f[ c_\beta = \left( \frac{\Gamma(3[1+\beta]/2)}{\Gamma([1+\beta]/2)} \right)^\frac{1}{1+\beta} \f]
!> \f[ \omega_\beta = \frac{\sqrt(\Gamma(3[1+\beta]/2))}{(1+\beta)(\Gamma([1+\beta]/2))^{3/2}} \f]
! CALLING SEQUENCE
! out = ep(x, loc=loc, sca=sca, beta=beta, sig=sig)
! INTENT(IN)
!> \param[in] "real(sp/dp) :: x input (array) number"
! INTENT(INOUT)
! None
! INTENT(OUT)
! None
! INTENT(IN), OPTIONAL
!> \param[in] "real(sp/dp) :: loc location (default: 0)"
!> \param[in] "real(sp/dp) :: sca scale (default: 1)"
!> \param[in] "real(sp/dp) :: beta kurtosis parameter (default: 0)"
!> \param[in] "real(sp/dp) :: sig standard deviation (default: ???), overwrites scale"
! INTENT(INOUT), OPTIONAL
! None
! INTENT(OUT), OPTIONAL
! None
! RETURN
!> \return real(sp/dp) :: ep — \f$ EP(x;loc,sca,\beta) \f$
! RESTRICTIONS
! Kurtosis factor must be greater than -1 and lower +1: -1 < \f$\beta\f$ < 1.
! EXAMPLE
! vec = (/ 1., 2, 3., -999., 5., 6. /)
! m = ep(vec, loc, sca, beta)
! -> see also example in test directory
! LITERATURE
! Schoups G & Vrugt JA (2010) A formal likelihood function for parameter and
! predictive inference of hydrologic models with correlated, heteroscedastic,
! and non-Gaussian errors. Water Resources Research 46, W10531.
! HISTORY
!> \author Written, Matthias Cuntz
!> \date Apr 2016
interface ep
module procedure ep_sp, ep_dp
end interface ep
! ------------------------------------------------------------------
! NAME
! ep01
! PURPOSE
!> \brief Exponential power distribution of Box and Tiao (1992) with location=0 and scale=1
!> \details The exponential power distribution of Box and Tiao (1992) at location zero and with scale one
!> given kurtosis parameter \f$ \beta \f$:
!> \f[ EP(x;0,1,\beta) = \omega_\beta
!> \exp \left\{ -c_\beta |x|^\frac{2}{1+\beta} \right\} \f]
!> with
!> \f[ c_\beta = \left( \frac{\Gamma(3[1+\beta]/2)}{\Gamma([1+\beta]/2)} \right)^\frac{1}{1+\beta} \f]
!> \f[ \omega_\beta = \frac{\sqrt(\Gamma(3[1+\beta]/2))}{(1+\beta)(\Gamma([1+\beta]/2))^{3/2}} \f]
! CALLING SEQUENCE
! out = ep01(x, beta=beta)
! INTENT(IN)
!> \param[in] "real(sp/dp) :: x input (array) number"
! INTENT(INOUT)
! None
! INTENT(OUT)
! None
! INTENT(IN), OPTIONAL
!> \param[in] "real(sp/dp) :: beta kurtosis parameter (default: 0)"
! INTENT(INOUT), OPTIONAL
! None
! INTENT(OUT), OPTIONAL
! None
! RETURN
!> \return real(sp/dp) :: ep01 — \f$ EP(x;0,1,\xi,\beta) \f$
! RESTRICTIONS
! Kurtosis factor must be greater than -1 and lower +1: -1 < \f$\beta\f$ < 1.
! EXAMPLE
! vec = (/ 1., 2, 3., -999., 5., 6. /)
! m = ep01((vec-loc)/sca, 0.1)/sca
! -> see also example in test directory
! LITERATURE
! Schoups G & Vrugt JA (2010) A formal likelihood function for parameter and
! predictive inference of hydrologic models with correlated, heteroscedastic,
! and non-Gaussian errors. Water Resources Research 46, W10531.
! HISTORY
!> \author Written, Matthias Cuntz
!> \date Apr 2016
interface ep01
module procedure ep01_sp, ep01_dp
end interface ep01
! ------------------------------------------------------------------
! NAME
! laplace
! PURPOSE
!> \brief The Laplace or double exponential distribution
!> \details Samples the probability density function of a the Laplace distribution
!> given location \f$ loc \f$, scale \f$ sca \f$ or standard deviation \f$ \sigma \f$:\n
!> \f[ L(x;loc,sca) = L((x-loc)/sca;0,1) / sca \f]
!> \f[ L(z;0,1) = \frac{1}{2} \exp (-|z|) \f]
! CALLING SEQUENCE
! out = laplace(x, loc=loc, sca=sca, sig=sig)
! INTENT(IN)
!> \param[in] "real(sp/dp) :: x input (array) number"
! INTENT(INOUT)
! None
! INTENT(OUT)
! None
! INTENT(IN), OPTIONAL
!> \param[in] "real(sp/dp) :: loc location (default: 0)"
!> \param[in] "real(sp/dp) :: sca scale (default: 1)"
!> \param[in] "real(sp/dp) :: sig standard deviation (default: ???), overwrites scale"
! INTENT(INOUT), OPTIONAL
! None
! INTENT(OUT), OPTIONAL
! None
! RETURN
!> \return real(sp/dp) :: laplace — \f$ L(x;loc,sca) \f$
! RESTRICTIONS
! Standard deviation must be greater than zero: sca or sig > 0.
! EXAMPLE
! vec = (/ 1., 2, 3., -999., 5., 6. /)
! m = laplace(vec, loc=2, sig=1.5)
! -> see also example in test directory
! HISTORY
!> \author Written, Matthias Cuntz
!> \date Apr 2016
interface laplace
module procedure laplace_sp, laplace_dp
end interface laplace
! ------------------------------------------------------------------
! NAME
! laplace01
! PURPOSE
!> \brief The Laplace or double exponential distribution with location=0 and scale=1
!> \details Samples the probability density function of the Laplace distribution
!> at location zero and with scale one:\n
!> \f[ L(z;0,1) = \frac{1}{2} \exp (-|z|) \f]
! CALLING SEQUENCE
! out = laplace01(x)
! INTENT(IN)
!> \param[in] "real(sp/dp) :: x input (array) number"
! INTENT(INOUT)
! None
! INTENT(OUT)
! None
! INTENT(IN), OPTIONAL
! None
! INTENT(INOUT), OPTIONAL
! None
! INTENT(OUT), OPTIONAL
! None
! RETURN
!> \return real(sp/dp) :: laplace — \f$ L(x;0,1) \f$
! RESTRICTIONS
! None
! EXAMPLE
! vec = (/ 1., 2, 3., -999., 5., 6. /)
! m = laplace01((vec-loc)/sca)/sca
! -> see also example in test directory
! HISTORY
!> \author Written, Matthias Cuntz
!> \date Apr 2016
interface laplace01
module procedure laplace01_sp, laplace01_dp
end interface laplace01
! ------------------------------------------------------------------
! NAME
! normal
! PURPOSE
!> \brief The normal (gaussian) distribution
!> \details Samples the probability density function of a the normal distribution
!> given location \f$ loc \f$, scale \f$ sca \f$ or standard deviation \f$ \sigma \f$:\n
!> \f[ N(x;loc,sca) = N((x-loc)/sca;0,1) / sca \f]
!> \f[ N(z;0,1) = \frac{1}{\sqrt{2\pi}} \exp (-z^2/2) \f]
! CALLING SEQUENCE
! out = normal(x, loc=loc, sca=sca, sig=sig)
! INTENT(IN)
!> \param[in] "real(sp/dp) :: x input (array) number"
! INTENT(INOUT)
! None
! INTENT(OUT)
! None
! INTENT(IN), OPTIONAL
!> \param[in] "real(sp/dp) :: loc location (default: 0)"
!> \param[in] "real(sp/dp) :: sca scale (default: 1)"
!> \param[in] "real(sp/dp) :: sig standard deviation (default: ???), overwrites scale"
! INTENT(INOUT), OPTIONAL
! None
! INTENT(OUT), OPTIONAL
! None
! RETURN
!> \return real(sp/dp) :: normal — \f$ N(x;loc,scale) \f$
! RESTRICTIONS
! Standard deviation must be greater than zero: sca or sig > 0.
! EXAMPLE
! vec = (/ 1., 2, 3., -999., 5., 6. /)
! m = normal(vec, loc=2, sig=1.5)
! -> see also example in test directory
! HISTORY
!> \author Written, Matthias Cuntz
!> \date Apr 2016
interface normal
module procedure normal_sp, normal_dp
end interface normal
! ------------------------------------------------------------------
! NAME
! normal01
! PURPOSE
!> \brief The normal (gaussian) distribution with location=0 and scale=1
!> \details Samples the probability density function of the normal distribution
!> at location zero and with scale one:\n
!> \f[ N(z;0,1) = \frac{1}{\sqrt{2\pi}} \exp (-z^2/2) \f]
! CALLING SEQUENCE
! out = normal01(x)
! INTENT(IN)
!> \param[in] "real(sp/dp) :: x input (array) number"
! INTENT(INOUT)
! None
! INTENT(OUT)
! None
! INTENT(IN), OPTIONAL
! None
! INTENT(INOUT), OPTIONAL
! None
! INTENT(OUT), OPTIONAL
! None
! RETURN
!> \return real(sp/dp) :: normal — \f$ N(x;0,1) \f$
! RESTRICTIONS
! None
! EXAMPLE
! vec = (/ 1., 2, 3., -999., 5., 6. /)
! m = normal01((vec-loc)/sca)/sca
! -> see also example in test directory
! HISTORY
!> \author Written, Matthias Cuntz
!> \date Apr 2016
interface normal01
module procedure normal01_sp, normal01_dp
end interface normal01
! ------------------------------------------------------------------
! NAME
! sep
! PURPOSE
!> \brief Skew exponential power distribution
!> \details The skew exponential power distribution with given location \f$ loc \f$,
!> scale \f$ sca \f$ or standard deviation \f$ \sigma \f$, skewness \f$ \xi \f$ and
!> kurtosis \f$ \beta \f$ parameters:
!> \f[ SEP(x;loc,sca,\xi,\beta) = SEP((x-loc)/sca;0,1,\xi,\beta) / sca \f]
! CALLING SEQUENCE
! out = sep(x, loc=loc, sca=sca, xi=xi, beta=beta, sig=sig)
! INTENT(IN)
!> \param[in] "real(sp/dp) :: x input (array) number"
! INTENT(INOUT)
! None
! INTENT(OUT)
! None
! INTENT(IN), OPTIONAL
!> \param[in] "real(sp/dp) :: loc location (default: 0)"
!> \param[in] "real(sp/dp) :: sca scale (default: 1)"
!> \param[in] "real(sp/dp) :: xi skewness parameter (default: 1)"
!> \param[in] "real(sp/dp) :: beta kurtosis parameter (default: 0)"
!> \param[in] "real(sp/dp) :: sig standard deviation (default: ???), owerwrites scale"
! INTENT(INOUT), OPTIONAL
! None
! INTENT(OUT), OPTIONAL
! None
! RETURN
!> \return real(sp/dp) :: sep — \f$ SEP(x;loc,scale,\xi,\beta) \f$
! RESTRICTIONS
! Skewness factor must be greater than zero: \f$\xi\f$ > 0.
! Kurtosis factor must be greater than -1 and lower +1: -1 < \f$\beta\f$ < 1.
! EXAMPLE
! vec = (/ 1., 2, 3., -999., 5., 6. /)
! m = sep(vec, loc, sca, xi, beta)
! -> see also example in test directory
! LITERATURE
! Schoups G & Vrugt JA (2010) A formal likelihood function for parameter and
! predictive inference of hydrologic models with correlated, heteroscedastic,
! and non-Gaussian errors. Water Resources Research 46, W10531.
! HISTORY
!> \author Written, Matthias Cuntz
!> \date Apr 2016
interface sep
module procedure sep_sp, sep_dp
end interface sep
! ------------------------------------------------------------------
! NAME
! sep01
! PURPOSE
!> \brief Skew exponential power distribution with location=0 and scale=1
!> \details The skew exponential power distribution at location zero and with scale one
!> given skewness and kurtosis parameters \f$ \xi \f$ and \f$ \beta \f$:
!> \f[ SEP(x;0,1,\xi,\beta) = \frac{2\sigma_\xi}{\xi+\xi^{-1}} \omega_\beta
!> \exp \left\{ -c_\beta |x_\xi|^\frac{2}{1+\beta} \right\} \f]
!> with
!> \f[ x_\xi = \xi^{\mathit{sign}(\mu_\xi + \sigma_\xi x)} \f]
!> \f[ c_\beta = \left( \frac{\Gamma(3[1+\beta]/2)}{\Gamma([1+\beta]/2)} \right)^\frac{1}{1+\beta} \f]
!> \f[ \omega_\beta = \frac{\sqrt(\Gamma(3[1+\beta]/2))}{(1+\beta)(\Gamma([1+\beta]/2))^{3/2}} \f]
!> \f[ \sigma_\xi = \sqrt{(M_2-M_1^2)(\xi^2+\xi^{-2})+2M_1^2-M_2} \f]
!> \f[ \mu_\xi = M_1(\xi-\xi^{-1}) \f]
!> \f[ M_1 = \frac{\Gamma(1+\beta)}{\sqrt{\Gamma(3[1+\beta]/2)\Gamma([1+\beta]/2)}} \f]
!> \f[ M_2 = 1 \f]
!> \f[ M_1 = \frac{\Gamma(1+\beta)}{\sqrt{\Gamma(3[1+\beta]/2)\Gamma([1+\beta]/2)}} \f]
! CALLING SEQUENCE
! out = sep01(x, xi=xi, beta=beta)
! INTENT(IN)
!> \param[in] "real(sp/dp) :: x input (array) number"
! INTENT(INOUT)
! None
! INTENT(OUT)
! None
! INTENT(IN), OPTIONAL
!> \param[in] "real(sp/dp) :: xi skewness parameter (default: 1)"
!> \param[in] "real(sp/dp) :: beta kurtosis parameter (default: 0)"
! INTENT(INOUT), OPTIONAL
! None
! INTENT(OUT), OPTIONAL
! None
! RETURN
!> \return real(sp/dp) :: sep01 — \f$ SEP(x;0,1,\xi,\beta) \f$
! RESTRICTIONS
! Skewness factor must be greater than zero: \f$\xi\f$ > 0.
! Kurtosis factor must be greater than -1 and lower +1: -1 < \f$\beta\f$ < 1.
! EXAMPLE
! vec = (/ 1., 2, 3., -999., 5., 6. /)
! m = sep01((vec-loc)/sca, 1.1, 0.1)/sca
! -> see also example in test directory
! LITERATURE
! Schoups G & Vrugt JA (2010) A formal likelihood function for parameter and
! predictive inference of hydrologic models with correlated, heteroscedastic,
! and non-Gaussian errors. Water Resources Research 46, W10531.
! HISTORY
!> \author Written, Matthias Cuntz
!> \date Apr 2016
interface sep01
module procedure sep01_sp, sep01_dp
end interface sep01
! ------------------------------------------------------------------
! NAME
! sep_fs
! PURPOSE
!> \brief Skew exponential power distribution after Fernandez and Steel (1998)
!> \details The skew exponential power distribution after Fernandez and Steel (1998)
!> with given location \f$ loc \f$, scale \f$ sca \f$ or standard deviation \f$ \sigma \f$,
!> skewness \f$ \xi \f$, and kurtosis \f$ \beta \f$ parameters:
!> \f[ SEP_{FS}(x;loc,sca,\xi,\beta) = SEP_{FS}((x-loc)/sca;0,1,\xi,\beta) / sca \f]
! CALLING SEQUENCE
! out = sep_fs(x, loc=loc, sca=sca, xi=xi, beta=beta, sig=sig)
! INTENT(IN)
!> \param[in] "real(sp/dp) :: x input (array) number"
! INTENT(INOUT)
! None
! INTENT(OUT)
! None
! INTENT(IN), OPTIONAL
!> \param[in] "real(sp/dp) :: loc location (default: 0)"
!> \param[in] "real(sp/dp) :: sca scale (default: 1)"
!> \param[in] "real(sp/dp) :: xi skewness parameter (default: 1)"
!> \param[in] "real(sp/dp) :: beta kurtosis parameter (default: 0)"
!> \param[in] "real(sp/dp) :: sig standard deviation (default: 1), owerwrites scale"
! INTENT(INOUT), OPTIONAL
! None
! INTENT(OUT), OPTIONAL
! None
! RETURN
!> \return real(sp/dp) :: sep_fs — \f$ SEP_{FS}(x;loc,scale,\xi,\beta) \f$
! RESTRICTIONS
! Skewness factor must be greater than zero: \f$\xi\f$ > 0.
! Kurtosis factor must be greater than -1 and lower +1: -1 < \f$\beta\f$ < 1.
! EXAMPLE
! vec = (/ 1., 2, 3., -999., 5., 6. /)
! m = sep_fs(vec, loc, sca, xi, beta)
! -> see also example in test directory
! LITERATURE
! Schoups G & Vrugt JA (2010) A formal likelihood function for parameter and
! predictive inference of hydrologic models with correlated, heteroscedastic,
! and non-Gaussian errors. Water Resources Research 46, W10531.
! HISTORY
!> \author Written, Matthias Cuntz
!> \date Apr 2016
interface sep_fs
module procedure sep_fs_sp, sep_fs_dp
end interface sep_fs
! ------------------------------------------------------------------
! NAME
! sep01_fs
! PURPOSE
!> \brief Skew exponential power distribution after Fernandez and Steel (1998)
!> with location=0 and scale=1
!> \details The skew exponential power distribution after Fernandez and Steel (1998)
!> at location zero and with scale one, given skewness and kurtosis parameters \f$ \xi \f$
!> and \f$ \beta \f$:
!> \f[ SEP_fs(x;0,1,\xi,\beta) = \frac{2}{\xi+\xi^{-1}}
!> EP(xi^{-sign(x)}x;0,1,\beta) \f]
! CALLING SEQUENCE
! out = sep01_fs(x, xi=xi, beta=beta)
! INTENT(IN)
!> \param[in] "real(sp/dp) :: x input (array) number"
! INTENT(INOUT)
! None
! INTENT(OUT)
! None
! INTENT(IN), OPTIONAL
!> \param[in] "real(sp/dp) :: xi skewness parameter (default: 1)"
!> \param[in] "real(sp/dp) :: beta kurtosis parameter (default: 0)"
! INTENT(INOUT), OPTIONAL
! None
! INTENT(OUT), OPTIONAL
! None
! RETURN
!> \return real(sp/dp) :: sep01_fs — \f$ SEP_fs(x;0,1,\xi,\beta) \f$
! RESTRICTIONS
! Skewness factor must be greater than zero: \f$\xi\f$ > 0.
! Kurtosis factor must be greater than -1 and lower +1: -1 < \f$\beta\f$ < 1.
! EXAMPLE
! vec = (/ 1., 2, 3., -999., 5., 6. /)
! m = sep01_fs((vec-loc)/sca, 1.1, 0.1)/sca
! -> see also example in test directory
! LITERATURE
! Schoups G & Vrugt JA (2010) A formal likelihood function for parameter and
! predictive inference of hydrologic models with correlated, heteroscedastic,
! and non-Gaussian errors. Water Resources Research 46, W10531.
! HISTORY
!> \author Written, Matthias Cuntz
!> \date Apr 2016
interface sep01_fs
module procedure sep01_fs_sp, sep01_fs_dp
end interface sep01_fs
! ------------------------------------------------------------------
! NAME
! sep_fs_mean
! PURPOSE
!> \brief Mean of skew exponential power distribution after Fernandez and Steel (1998)
!> \details The mean of the skew exponential power distribution after Fernandez and Steel (1998)
!> with given location \f$ loc \f$, scale \f$ sca \f$ or standard deviation \f$ \sigma \f$,
!> skewness \f$ \xi \f$, and kurtosis \f$ \beta \f$ parameters:
!> \f[ sep_fs_mean = \int_{-\infty}^{\infty} x SEP_{FS}(x;loc,sca,\xi,\beta) dx \f]
! CALLING SEQUENCE
! out = sep_fs_mean(loc=loc, sca=sca, xi=xi, beta=beta, sig=sig)
! INTENT(IN)
! None
! INTENT(INOUT)
! None
! INTENT(OUT)
! None
! INTENT(IN), OPTIONAL
!> \param[in] "real(sp/dp) :: loc location (default: 0)"
!> \param[in] "real(sp/dp) :: sca scale (default: 1)"
!> \param[in] "real(sp/dp) :: xi skewness parameter (default: 1)"
!> \param[in] "real(sp/dp) :: beta kurtosis parameter (default: 0)"
!> \param[in] "real(sp/dp) :: sig standard deviation (default: 1), owerwrites scale"
! INTENT(INOUT), OPTIONAL
! None
! INTENT(OUT), OPTIONAL
! None
! RETURN
!> \return real(sp/dp) :: sep_fs_mean — \f$ \int_{-\infty}^{\infty} x SEP_{FS}(x;loc,scale,\xi,\beta) dx \f$
! RESTRICTIONS
! Skewness factor must be greater than zero: \f$\xi\f$ > 0.
! Kurtosis factor must be greater than -1 and lower +1: -1 < \f$\beta\f$ < 1.
! EXAMPLE
! vec = (/ 1., 2, 3., -999., 5., 6. /)
! m = sep_fs_mean(loc, sca, xi, beta)
! -> see also example in test directory
! LITERATURE
! Schoups G & Vrugt JA (2010) A formal likelihood function for parameter and
! predictive inference of hydrologic models with correlated, heteroscedastic,
! and non-Gaussian errors. Water Resources Research 46, W10531.
! HISTORY
!> \author Written, Matthias Cuntz
!> \date Apr 2016
! interface sep_fs_mean
! module procedure sep_fs_mean_sp, sep_fs_mean_dp
! end interface sep_fs_mean
! ------------------------------------------------------------------
! NAME
! sep01_fs_mean
! PURPOSE
!> \brief Mean of skew exponential power distribution after Fernandez and Steel (1998)
!> with location=0 and scale=1
!> \details The mean of the skew exponential power distribution after Fernandez and Steel (1998)
!> at location zero and with scale one, given skewness and kurtosis parameters \f$ \xi \f$
!> and \f$ \beta \f$:
!> \f[ sep01_fs_mean = \int_{-\infty}^{\infty} x SEP_{FS}(x;0,1,\xi,\beta) dx \f]
! CALLING SEQUENCE
! out = sep01_fs_mean(xi=xi, beta=beta)
! INTENT(IN)
! None
! INTENT(INOUT)
! None
! INTENT(OUT)
! None
! INTENT(IN), OPTIONAL
!> \param[in] "real(sp/dp) :: xi skewness parameter (default: 1)"
!> \param[in] "real(sp/dp) :: beta kurtosis parameter (default: 0)"
! INTENT(INOUT), OPTIONAL
! None
! INTENT(OUT), OPTIONAL
! None
! RETURN
!> \return real(sp/dp) :: sep01_fs_mean — \f$ \int_{-\infty}^{\infty} x SEP_{FS}(x;0,1,\xi,\beta) dx \f$
! RESTRICTIONS
! Skewness factor must be greater than zero: \f$\xi\f$ > 0.
! Kurtosis factor must be greater than -1 and lower +1: -1 < \f$\beta\f$ < 1.
! EXAMPLE
! vec = (/ 1., 2, 3., -999., 5., 6. /)
! m = sep01_fs_mean(1.1, 0.1)/sca
! -> see also example in test directory
! LITERATURE
! Schoups G & Vrugt JA (2010) A formal likelihood function for parameter and
! predictive inference of hydrologic models with correlated, heteroscedastic,
! and non-Gaussian errors. Water Resources Research 46, W10531.
! HISTORY
!> \author Written, Matthias Cuntz
!> \date Apr 2016
! interface sep01_fs_mean
! module procedure sep01_fs_mean_sp, sep01_fs_mean_dp
! end interface sep01_fs_mean
! ------------------------------------------------------------------
! NAME
! sep_fs_std
! PURPOSE
!> \brief Standard deviation of the skew exponential power distribution after Fernandez and Steel (1998)
!> \details The standard deviation of the skew exponential power distribution
!> after Fernandez and Steel (1998)
!> with given location \f$ loc \f$, scale \f$ sca \f$ or standard deviation \f$ \sigma \f$,
!> skewness \f$ \xi \f$, and kurtosis \f$ \beta \f$ parameters:
!> \f[ sep_fs_std^2 = \int_{-\infty}^{\infty} x^2 SEP_{FS}(x;loc,sca,\xi,\beta) dx \f]
! CALLING SEQUENCE
! out = sep_fs_std(sca=sca, xi=xi, beta=beta, sig=sig)
! INTENT(IN)
! None
! INTENT(INOUT)
! None
! INTENT(OUT)
! None
! INTENT(IN), OPTIONAL
!> \param[in] "real(sp/dp) :: sca scale (default: 1)"
!> \param[in] "real(sp/dp) :: xi skewness parameter (default: 1)"
!> \param[in] "real(sp/dp) :: beta kurtosis parameter (default: 0)"
!> \param[in] "real(sp/dp) :: sig standard deviation (default: 1), owerwrites scale"
! INTENT(INOUT), OPTIONAL
! None
! INTENT(OUT), OPTIONAL
! None
! RETURN
!> \return real(sp/dp) :: sep_fs_std — \f$ \sqrt \int_{-\infty}^{\infty} x^2 SEP_{FS}(x;loc,sca,\xi,\beta) dx \f$
! RESTRICTIONS
! Skewness factor must be greater than zero: \f$\xi\f$ > 0.
! Kurtosis factor must be greater than -1 and lower +1: -1 < \f$\beta\f$ < 1.
! EXAMPLE
! vec = (/ 1., 2, 3., -999., 5., 6. /)
! m = sep_fs_std(sca, xi, beta)
! -> see also example in test directory
! LITERATURE
! Schoups G & Vrugt JA (2010) A formal likelihood function for parameter and
! predictive inference of hydrologic models with correlated, heteroscedastic,
! and non-Gaussian errors. Water Resources Research 46, W10531.
! HISTORY
!> \author Written, Matthias Cuntz
!> \date Apr 2016
! interface sep_fs_std
! module procedure sep_fs_std_sp, sep_fs_std_dp
! end interface sep_fs_std
! ------------------------------------------------------------------
! NAME
! sep01_fs_std
! PURPOSE
!> \brief Standard deviation of the skew exponential power distribution after Fernandez and Steel (1998)
!> with location=0 and scale=1
!> \details The standard deviation of the skew exponential power distribution
!> after Fernandez and Steel (1998)
!> at location zero and with scale one, given skewness and kurtosis parameters \f$ \xi \f$
!> and \f$ \beta \f$:
!> \f[ sep01_fs_std^2 = \int_{-\infty}^{\infty} x^2 SEP_{FS}(x;0,1,\xi,\beta) dx \f]
! CALLING SEQUENCE
! out = sep01_fs_std(xi=xi, beta=beta)
! INTENT(IN)
! None
! INTENT(INOUT)
! None
! INTENT(OUT)
! None
! INTENT(IN), OPTIONAL
!> \param[in] "real(sp/dp) :: xi skewness parameter (default: 1)"
!> \param[in] "real(sp/dp) :: beta kurtosis parameter (default: 0)"
! INTENT(INOUT), OPTIONAL
! None
! INTENT(OUT), OPTIONAL
! None
! RETURN
!> \return real(sp/dp) :: sep01_fs_std — \f$ \sqrt \int_{-\infty}^{\infty} x^2 SEP_{FS}(x;0,1,\xi,\beta) dx \f$
! RESTRICTIONS
! Skewness factor must be greater than zero: \f$\xi\f$ > 0.
! Kurtosis factor must be greater than -1 and lower +1: -1 < \f$\beta\f$ < 1.
! EXAMPLE
! vec = (/ 1., 2, 3., -999., 5., 6. /)
! m = sep01_fs_std(1.1, 0.1)
! -> see also example in test directory
! LITERATURE
! Schoups G & Vrugt JA (2010) A formal likelihood function for parameter and
! predictive inference of hydrologic models with correlated, heteroscedastic,
! and non-Gaussian errors. Water Resources Research 46, W10531.
! HISTORY
!> \author Written, Matthias Cuntz
!> \date Apr 2016
! interface sep01_fs_std
! module procedure sep01_fs_std_sp, sep01_fs_std_dp
! end interface sep01_fs_std
! ------------------------------------------------------------------
! NAME
! st
! PURPOSE
!> \brief Skewed Student's t distribution
!> \details The Skewed Student t-distribution given location \f$ loc \f$,
!> scale \f$ sca \f$ or standard deviation \f$ \sigma \f$ and degrees of freedom \f$ \nu \f$:
!> \f[ st(x;\nu,loc,sca,\xi) = st((x-loc)/sca;\nu,0,1,\xi) / sca \f]
!> \f[ st(x;\nu,0,1,\xi) = \frac{2\sigma_\xi}{\xi+\xi^{-1}}
!> \frac{\Gamma \left( \frac{\nu+1}{2} \right) \sqrt{\frac{\nu}{\nu-2}}}{
!> \Gamma \left( \frac{\nu}{2}\right) \sqrt{\pi \nu} }
!> \left( 1 + \frac{x_\xi^2}{\nu-2} \right) ^{-\frac{\nu+1}{2}} \f]
!> with
!> \f[ x_\xi = \xi^{\mathit{sign}(\mu_\xi + \sigma_\xi x)} \f]
!> \f[ \mu_\xi = \frac{2(\xi^2-\xi^{-2})}{\xi+\xi^{-1}}
!> \frac{\Gamma \left( \frac{\nu+1}{2} \right) \sqrt{\frac{\nu}{\nu-2}}}{
!> \Gamma \left( \frac{\nu}{2}\right) \sqrt{\pi \nu} } \frac{\nu-2}{\nu-1} \f]
!> \f[ \sigma_\xi = \sqrt{-\mu_\xi^2 + \frac{\xi^3+\xi^{-3}}{\xi+\xi^{-1}}} \f]
! CALLING SEQUENCE
! out = st(x, nu, loc=loc, sca=sca, xi=xi, sig=sig)
! INTENT(IN)
!> \param[in] "real(sp/dp) :: x input (array) number"
!> \param[in] "real(sp/dp) :: nu degrees of freedom"
! INTENT(INOUT)
! None
! INTENT(OUT)
! None
! INTENT(IN), OPTIONAL
!> \param[in] "real(sp/dp) :: loc location (default: 0)"
!> \param[in] "real(sp/dp) :: sca scale (default: 1)"
!> \param[in] "real(sp/dp) :: xi skewness parameter (default: 1)"
!> \param[in] "real(sp/dp) :: sig standard deviation (default: ???), overwrites scale"
! INTENT(INOUT), OPTIONAL
! None
! INTENT(OUT), OPTIONAL
! None
! RETURN
!> \return real(sp/dp) :: st — \f$ st(x;\nu,loc,sca,\xi) \f$
! RESTRICTIONS
! Degrees of freedom must be greater than one: \f$\nu\f$ > 1.
! Skewness factor must be greater than zero: \f$\xi\f$ > 0.
! EXAMPLE
! vec = (/ 1., 2, 3., -999., 5., 6. /)
! m = st(vec, 3., loc, sca)
! -> see also example in test directory
! HISTORY
!> \author Written, Matthias Cuntz
!> \date Apr 2016
interface st
module procedure st_sp, st_dp
end interface st
! ------------------------------------------------------------------
! NAME
! st01
! PURPOSE
!> \brief Skewed Student's t distribution at location zero and scale 1.
!> \details The Skewed Student t-distribution at location zero and scale one,
!> given degrees of freedom \f$ \nu \f$:
!> \f[ st(x;\nu,0,1,\xi) = \frac{2\sigma_\xi}{\xi+\xi^{-1}}
!> \frac{\Gamma \left( \frac{\nu+1}{2} \right) \sqrt{\frac{\nu}{\nu-2}}}{
!> \Gamma \left( \frac{\nu}{2}\right) \sqrt{\pi \nu} }
!> \left( 1 + \frac{x_\xi^2}{\nu-2} \right) ^{-\frac{\nu+1}{2}} \f]
!> with
!> \f[ x_\xi = \xi^{\mathit{sign}(\mu_\xi + \sigma_\xi x)} \f]
!> \f[ \mu_\xi = \frac{2(\xi^2-\xi^{-2})}{\xi+\xi^{-1}}
!> \frac{\Gamma \left( \frac{\nu+1}{2} \right) \sqrt{\frac{\nu}{\nu-2}}}{
!> \Gamma \left( \frac{\nu}{2}\right) \sqrt{\pi \nu} } \frac{\nu-2}{\nu-1} \f]
!> \f[ \sigma_\xi = \sqrt{-\mu_\xi^2 + \frac{\xi^3+\xi^{-3}}{\xi+\xi^{-1}}} \f]
! CALLING SEQUENCE
! out = st01(x, nu, xi=xi)
! INTENT(IN)
!> \param[in] "real(sp/dp) :: x input (array) number"
!> \param[in] "real(sp/dp) :: nu degrees of freedom"
! INTENT(INOUT)
! None