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mo_sobol_index.f90
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mo_sobol_index.f90
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!> \file mo_sobol_index.f90
!> \brief Parameter sensitivity estimation using Sobol index (main and total effect).
!> \details This module contains routines to determine parameter sensitivities using
!> the Sobol index (SI) based on the variance of model output changing only one parameter at a time and
!> all parameter except one at a time.\n
!> Based on sobol_index.py which is part of the JAMS Python library.
!> \authors Juliane Mai
!> \date Jul 2013
MODULE mo_sobol_index
! Written Juliane Mai, Jul 2013
! License
! This file is part of the JAMS Fortran package, distributed under the MIT License.
!
! Copyright (c) 2013 Juliane Mai
!
! 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.
USE mo_kind, ONLY: i4, sp, dp
USE mo_moment, ONLY: variance
IMPLICIT NONE
PUBLIC :: sobol_index ! Sobol index
! ------------------------------------------------------------------
! NAME
! sobol_index
! PURPOSE
!> \brief The Sobol index (SI).
!
!> \details Calculates the Sobol index which is based on the change of variance of
!> model response due to parameter changes.\n
!> Here, the main effect (Si) and the total effect (STi) are implemented. \n
!> The model output has to be provided for:\n
!> - a parameter set A = f(A)\n
!> - a parameter set B which is independent to A = f(B)\n
!> For method #3, #4, #5, #6, #7, #8, #9=Default: \n
!> - n parameter sets Ci where Ci equals parameter set A except parameter i
!> which is the value of parameter set B = f(A_B)\n
!> For method #1, #2, #6: \n
!> - n parameter sets Ci where Ci equals parameter set B except parameter i
!> which is the value of parameter set A = f(B_A)\n
!> This has to be repeated several times. \n
!> The model response can be either a scalar or a vector (e.g. a timeseries).
!> If the model response is a vector, the Si and STi are calculated for each time point
!> independently such that the returned Si and STi are also vectors. \n
!
! INTENT(IN)
!> \param[in] "real(sp/dp) :: ya(:)/ya(:,:)" Model output for parameter sets A \n
!> dim_1 = number of parameter sets \n
!> dim_2 = number of model outputs, e.g. time steps
!> \param[in] "real(sp/dp) :: yb(:)/yb(:,:)" Model output for parameter sets B \n
!> dim_1 = number of parameter sets \n
!> dim_2 = number of model outputs, e.g. time steps
!> \param[in] "real(sp/dp) :: yc(:,:)/yc(:,:,:)" Model output for parameter sets C(i), i=1,npara \n
!> dim_1 = number of parameter sets \n
!> dim_2 = number of parameters \n
!> dim_3 = number of model outputs, e.g. time steps
!
! INTENT(INOUT)
! None
! INTENT(OUT)
!> \param[out] "real(sp/dp) :: si(:)/si(:,:)" Sobol index - Main effect \n
!> (0.0 at masked entries) \n
!> dim_1 = number of parameters \n
!> dim_2 = number of model outputs, e.g. time steps
!> \param[out] "real(sp/dp) :: sti(:)/sti(:,:)" Sobol index - Total effect \n
!> (0.0 at masked entries) \n
!> dim_1 = number of parameters \n
!> dim_2 = number of model outputs, e.g. time steps
!
! INTENT(IN), OPTIONAL
!> \param[in] "logical, optional :: mask(:)" Mask of model output \n
!> (same for all parameter sets) \n
!> (only applicable if ya has at least 2 dims)\n
!> dim_1 = number of model outputs, e.g. time steps
!
! INTENT(INOUT), OPTIONAL
! None
!
! INTENT(OUT), OPTIONAL
!> \param[out] "real(sp/dp), optional :: smean(:,:)" Average index per parameter \n
!> dim_1 = number of parameters \n
!> dim_2 = 2, i.e. SI and STI
!> \param[out] "real(sp/dp), optional :: wmean(:,:)" Variance weighted average index per parameter \n
!> dim_1 = number of parameters \n
!> dim_2 = 2, i.e. SI and STI
!> \param[out] "real(sp/dp), optional :: wmean_frac(:,:)" Nominator and denominator which was used to
!> calculate wmean \n
!> (only be callable, if wmean present)\n
!> dim1 = number of parameters\n
!> dim2 = 4 (1 - nominator Si)\n
!> (2 - denominator Si)\n
!> (3 - nominator STi)\n
!> (4 - denominator STi)\n
!> be aware that it depends on the method how
!> wmean is then calculated, i.e.
!> nominator/denominator or
!> 1.0 - nominator/denominator
!
! RETURN
! None
!
! RESTRICTIONS
!> \note Input values must be floating points.\n
!> Average indexes are only calculated when ya has more than one dimension, i.e. modeloutput is not scalar.
!> Else average indexes are equal si and sti.
!
! EXAMPLE
! ya = (/ 1., 2, 3., -999., 5., 6. /)
! yb = (/ 2., 3, 1., 7., 8., 9. /)
! yc(:,1) = (/ 2., 3, 1., 7., 8., 9. /)
! yc(:,2) = (/ 3., 3, 3., 3., 3., 3. /)
! call sobol_index(ya, yb, yc, si, sti)
! -> see also example in test directory
! LITERATURE
! Homma, T., & Saltelli, A. (1996).
! Importance measures in global sensitivity analysis of nonlinear models.
! Reliability Engineering & System Safety, 52(1), 1-17.
! Saltelli, A. (2002).
! Making best use of model evaluations to compute sensitivity indices.
! Computer Physics Communications, 145(2), 280-297. doi:10.1016/S0010-4655(02)00280-1
! Saltelli, A., Ratto, M., Andres, T., Campolongo, F., Cariboni, J., Gatelli, D., et al. (2008).
! Global Sensitivity Analysis. Wiley-Interscience.
! HISTORY
!> \author Juliane Mai
!> \date Jul 2013
! Modified,
! The code is based on sobol_index.py by Matthias Cuntz. sobol_index.py is part of the JAMS Python library.
INTERFACE sobol_index
MODULE PROCEDURE sobol_index_0d_dp, sobol_index_0d_sp, &
sobol_index_1d_dp, sobol_index_1d_sp, &
sobol_index_2d_dp, sobol_index_2d_sp, &
sobol_index_3d_dp, sobol_index_3d_sp
END INTERFACE sobol_index
! ------------------------------------------------------------------
PRIVATE
! ------------------------------------------------------------------
CONTAINS
subroutine sobol_index_0d_dp(ya, yb, yc, si, sti, method, smean, wmean)
real(dp), dimension(:), intent(in) :: ya ! Output running model with parameter sets A
real(dp), dimension(:), intent(in) :: yb ! Output running model with parameter sets B
real(dp), dimension(:,:), intent(in) :: yc ! Output running model with parameter sets C
! ! dim1 = number of parameter sets
! ! = size(ya) = size(yb)
! ! dim2 = number of parameters
real(dp), dimension(size(yc,2)), intent(out) :: si ! Sobol index (main effect)
real(dp), dimension(size(yc,2)), intent(out) :: sti ! Sobol index (total effect)
integer(i4), optional, intent(in) :: method ! (1) 'Saltelli2008' needs Ci = B_A
! ! (2) 'Homma1996' needs Ci = B_A
! ! (3) 'Sobol2007' needs Ci = A_B and B_A --> NA
! ! (4) 'Saltelli2010' needs Ci = A_B
! ! (5) 'Jansen1999' needs Ci = A_B
! ! (6) 'Mai2012' needs Ci = B_A
! ! (7) 'Mai2013' needs Ci = A_B
! ! (8) 'Mai2014' needs Ci = A_B
! ! (9) 'Mai1999' needs Ci = A_B ----> Default
real(dp), dimension(size(yc,2), 2), optional, intent(out) :: smean ! Mean Sobol index (main and total effect)
! ! dim1 = number of parameters
! ! dim2 = 2 (i.e. SI and STI)
! ! == (/ si, sti /) in 0d version !!
real(dp), dimension(size(yc,2), 2), optional, intent(out) :: wmean ! Variance weighted mean Sobol index
! ! (main and total effect)
! ! dim1 = number of parameters
! ! dim2 = 2 (i.e. SI and STI)
! ! == (/ si, sti /) in 0d version !!
! local variables
integer(i4) :: ii
integer(i4) :: nsets, npara
integer(i4) :: meth
real(dp), dimension(:), allocatable :: yab
real(dp) :: var_a, var_b, var_ab
real(dp) :: mean_a, mean_b, mean_ab
if (present(method)) then
meth = method
else
meth = 9
end if
nsets = size(yc,1)
npara = size(yc,2)
select case(meth)
case(1)
! 'Saltelli2008' - The formulation presented in 'The Primer'. (yc=f(B_A))
! Si = (1/n*sum_j(f(A)_j*f(B_A^i)_j) - mean(f(A))^2)/var(f(A))
! STi = (var(f(A))-(1/n*sum_j(f(B)_j*f(B_A^i)_j) - mean(f(A))^2))/var(f(A))
var_a = variance(ya, ddof=1_i4)
mean_a = sum( ya(:) ) / real(nsets,dp)
if ( var_a .gt. tiny(1.0_dp) ) then
! model outputs are different (usual case)
do ii=1, npara
si(ii) = ( dot_product( ya(:) , (yc(:,ii)) ) / real(nsets,dp) - mean_a**2 ) / var_a
sti(ii) = 1.0_dp - ( dot_product( yb(:) , (yc(:,ii)) ) / real(nsets,dp) - mean_a**2 ) / var_a
end do
else
! model outputs are equal (should never happen)
si(:) = 0.0_dp
sti(:) = 0.0_dp
end if
case(2)
! 'Homma1996' - Si takes mixed A, B term for squared mean in nominator. (yc=f(B_A))
! Si = 1/n*sum_j(f(A)_j*(f(B_A^i)_j - f(B)_j))/var(f(A))
! STi as Saltelli2008
var_a = variance(ya, ddof=1_i4)
mean_a = sum( ya(:) ) / real(nsets,dp)
if ( var_a .gt. tiny(1.0_dp) ) then
! model outputs are different (usual case)
do ii=1, npara
si(ii) = ( dot_product( ya(:) , (yc(:,ii)-yb(:)) ) / real(nsets,dp) ) / var_a
sti(ii) = 1.0_dp - ( dot_product( yb(:) , (yc(:,ii)) ) / real(nsets,dp) - mean_a**2 ) / var_a
end do
else
! model outputs are equal (should never happen)
si(:) = 0.0_dp
sti(:) = 0.0_dp
end if
case(3)
stop 'mo_sobol_index: Sobol2007 would need f(A_B) and f(B_A). It is thus not implemented here.'
case(4)
! 'Saltelli2010' - Si takes B, A_B instead of A, B_A (yc=f(A_B))
! Si = 1/n*sum_j(f(B)_j*(f(A_B^i)_j - f(A)_j))/var(f(A))
! STi = 1/n*sum_j(f(A)_j*(f(A)_j-f(A_B^i)_j))/var(f(A))
var_a = variance(ya, ddof=1_i4)
if ( var_a .gt. tiny(1.0_dp) ) then
! model outputs are different (usual case)
do ii=1, npara
si(ii) = ( dot_product( yb(:) , (yc(:,ii)-ya(:)) ) / real(nsets,dp) ) / var_a
sti(ii) = ( dot_product( ya(:) , (ya(:)-yc(:,ii)) ) / real(nsets,dp) ) / var_a
end do
else
! model outputs are equal (should never happen)
si(:) = 0.0_dp
sti(:) = 0.0_dp
end if
case(5)
! 'Jansen1999' - Calculate Si and STi by expectation(variance) instead of variance(expectation) (yc=f(A_B))
! Si = (var(f(A)) - 1/2n*sum_j(f(B)_j - f(A_B^i)_j)^2)/var(f(A))
! STi = 1/2n*sum_j(f(A)_j - f(A_B^i)_j)^2/var(f(A))
var_a = variance(ya, ddof=1_i4)
if ( var_a .gt. tiny(1.0_dp) ) then
! model outputs are different (usual case)
do ii=1, npara
si(ii) = 1.0_dp - ( sum( (yb(:)-yc(:,ii))**2 ) / real(2*nsets,dp) ) / var_a
sti(ii) = ( sum( (ya(:)-yc(:,ii))**2 ) / real(2*nsets,dp) ) / var_a
end do
else
! model outputs are equal (should never happen)
si(:) = 0.0_dp
sti(:) = 0.0_dp
end if
case(6)
! 'Mai2012' - Si of Homma 1996 but denominator is variance of A and B (yc=f(B_A)) and
! STi of Homma 1996 but mean and variance of f(A) replaced by f(B)
! Si = 1/n*sum_j(f(A)_j*(f(B_A^i)_j - f(B)_j))/var([f(A),f(B)])
! STi = (var(f(A))-(1/n*sum_j(f(B)_j*f(B_A^i)_j) - mean(f(B)^2)))/var(f(B))
allocate(yab(2*nsets))
yab( 1: nsets) = ya
yab(nsets+1:2*nsets) = yb
var_ab = variance(yab, ddof=1_i4)
var_b = variance(yb, ddof=1_i4)
mean_ab = dot_product(ya,yb) / real(nsets,dp)
mean_b = sum(yb) / real(nsets,dp)
if ( var_ab .gt. tiny(1.0_dp) ) then
! model outputs are different (usual case)
do ii=1, npara
si(ii) = ( dot_product(ya(:),yc(:,ii)) / real(nsets,dp) - mean_ab) / var_ab
sti(ii) = 1.0_dp - ( dot_product(yb(:),yc(:,ii)) / real(nsets,dp) - mean_b**2) / var_b
end do
else
! model outputs are equal (should never happen)
si(:) = 0.0_dp
sti(:) = 0.0_dp
end if
deallocate(yab)
case(7)
! 'Mai2013' - Si of Saltelli2010 but denominator is variance of A and B (yc=f(A_B))
! Si = 1/n*sum_j(f(B)_j*(f(A_B^i)_j - f(A)_j))/var([f(A),f(B)])
! STi = 1/n*sum_j(f(A)_j*(f(A)_j)-f(A_B^i)_j)/var(f(A))
allocate(yab(2*nsets))
yab( 1: nsets) = ya
yab(nsets+1:2*nsets) = yb
var_ab = variance(yab, ddof=1_i4)
var_a = variance(ya, ddof=1_i4)
if ( var_ab .gt. tiny(1.0_dp) ) then
! model outputs are different (usual case)
do ii=1, npara
si(ii) = ( dot_product( yb(:) , (yc(:,ii)-ya(:)) ) / real(nsets,dp) ) / var_ab
sti(ii) = ( dot_product( ya(:) , (ya(:)-yc(:,ii)) ) / real(nsets,dp) ) / var_a
end do
else
! model outputs are equal (should never happen)
si(:) = 0.0_dp
sti(:) = 0.0_dp
end if
deallocate(yab)
case(8)
! 'Mai2014' - SI of Jansen1999 but denominator is variance of A and B (yc=f(A_B))
! Si = (var([f(A),f(B)]) - 1/2n*sum_j(f(B)_j - f(A_B^i)_j)^2)/var([f(A),f(B)])
! STi = 1/2n*sum_j(f(A)_j - f(A_B^i)_j)^2/var(f(A))
allocate(yab(2*nsets))
yab( 1: nsets) = ya
yab(nsets+1:2*nsets) = yb
var_ab = variance(yab, ddof=1_i4)
var_a = variance(ya, ddof=1_i4)
if ( var_ab .gt. tiny(1.0_dp) ) then
! model outputs are different (usual case)
do ii=1, npara
si(ii) = 1.0_dp - ( sum( (yb(:)-yc(:,ii))**2 ) / real(2*nsets,dp) ) / var_ab
sti(ii) = ( sum( (ya(:)-yc(:,ii))**2 ) / real(2*nsets,dp) ) / var_a
end do
else
! model outputs are equal (should never happen)
si(:) = 0.0_dp
sti(:) = 0.0_dp
end if
deallocate(yab)
case(9)
! 'Mai1999' - SI of Mai2013 (Saltelli2010 with var([f(A),f(B)])) and STi of Jansen1999 (yc=f(A_B))
! Si = 1/n*sum_j(f(B)_j*(f(A_B^i)_j - f(A)_j))/var([f(A),f(B)])
! STi = 1/2n*sum_j(f(A)_j - f(A_B^i)_j)^2/var(f(A))
allocate(yab(2*nsets))
yab( 1: nsets) = ya
yab(nsets+1:2*nsets) = yb
var_ab = variance(yab, ddof=1_i4)
var_a = variance(ya, ddof=1_i4)
if ( var_ab .gt. tiny(1.0_dp) ) then
! model outputs are different (usual case)
do ii=1, npara
si(ii) = ( dot_product( yb(:) , (yc(:,ii)-ya(:)) ) / real(nsets,dp) ) / var_ab
sti(ii) = ( sum( (ya(:)-yc(:,ii))**2 ) / real(2*nsets,dp) ) / var_a
end do
else
! model outputs are equal (should never happen)
si(:) = 0.0_dp
sti(:) = 0.0_dp
end if
deallocate(yab)
case default
stop 'mo_sobol_index: This method is not implemented!'
end select
if ( present(smean) ) then
smean(:,1) = si(:)
smean(:,2) = sti(:)
end if
if ( present(wmean) ) then
wmean(:,1) = si(:)
wmean(:,2) = sti(:)
end if
end subroutine sobol_index_0d_dp
subroutine sobol_index_0d_sp(ya, yb, yc, si, sti, method, smean, wmean)
real(sp), dimension(:), intent(in) :: ya ! Output running model with parameter sets A
real(sp), dimension(:), intent(in) :: yb ! Output running model with parameter sets B
real(sp), dimension(:,:), intent(in) :: yc ! Output running model with parameter sets C
! ! dim1 = number of parameter sets
! ! = size(ya) = size(yb)
! ! dim2 = number of parameters
real(sp), dimension(size(yc,2)), intent(out) :: si ! Sobol index (main effect)
real(sp), dimension(size(yc,2)), intent(out) :: sti ! Sobol index (total effect)
integer(i4), optional, intent(in) :: method ! (1) 'Saltelli2008' needs Ci = B_A
! ! (2) 'Homma1996' needs Ci = B_A
! ! (3) 'Sobol2007' needs Ci = A_B and B_A --> NA
! ! (4) 'Saltelli2010' needs Ci = A_B
! ! (5) 'Jansen1999' needs Ci = A_B
! ! (6) 'Mai2012' needs Ci = B_A
! ! (7) 'Mai2013' needs Ci = A_B
! ! (8) 'Mai2014' needs Ci = A_B
! ! (9) 'Mai1999' needs Ci = A_B ----> Default
real(sp), dimension(size(yc,2), 2), optional, intent(out) :: smean ! Mean Sobol index (main and total effect)
! ! dim1 = number of parameters
! ! dim2 = 2 (i.e. SI and STI)
! ! == (/ si, sti /) in 0d version !!
real(sp), dimension(size(yc,2), 2), optional, intent(out) :: wmean ! Variance weighted mean Sobol index
! ! (main and total effect)
! ! dim1 = number of parameters
! ! dim2 = 2 (i.e. SI and STI)
! ! == (/ si, sti /) in 0d version !!
! local variables
integer(i4) :: ii
integer(i4) :: nsets, npara
integer(i4) :: meth
real(sp), dimension(:), allocatable :: yab
real(sp) :: var_a, var_b, var_ab
real(sp) :: mean_a, mean_b, mean_ab
if (present(method)) then
meth = method
else
meth = 9
end if
nsets = size(yc,1)
npara = size(yc,2)
select case(meth)
case(1)
! 'Saltelli2008' - The formulation presented in 'The Primer'. (yc=f(B_A))
! Si = (1/n*sum_j(f(A)_j*f(B_A^i)_j) - mean(f(A))^2)/var(f(A))
! STi = (var(f(A))-(1/n*sum_j(f(B)_j*f(B_A^i)_j) - mean(f(A))^2))/var(f(A))
var_a = variance(ya, ddof=1_i4)
mean_a = sum( ya(:) ) / real(nsets,sp)
if ( var_a .gt. tiny(1.0_sp) ) then
! model outputs are different (usual case)
do ii=1, npara
si(ii) = ( dot_product( ya(:) , (yc(:,ii)) ) / real(nsets,sp) - mean_a**2 ) / var_a
sti(ii) = 1.0_sp - ( dot_product( yb(:) , (yc(:,ii)) ) / real(nsets,sp) - mean_a**2 ) / var_a
end do
else
! model outputs are equal (should never happen)
si(:) = 0.0_sp
sti(:) = 0.0_sp
end if
case(2)
! 'Homma1996' - Si takes mixed A, B term for squared mean in nominator. (yc=f(B_A))
! Si = 1/n*sum_j(f(A)_j*(f(B_A^i)_j - f(B)_j))/var(f(A))
! STi as Saltelli2008
var_a = variance(ya, ddof=1_i4)
mean_a = sum( ya(:) ) / real(nsets,sp)
if ( var_a .gt. tiny(1.0_sp) ) then
! model outputs are different (usual case)
do ii=1, npara
si(ii) = ( dot_product( ya(:) , (yc(:,ii)-yb(:)) ) / real(nsets,sp) ) / var_a
sti(ii) = 1.0_sp - ( dot_product( yb(:) , (yc(:,ii)) ) / real(nsets,sp) - mean_a**2 ) / var_a
end do
else
! model outputs are equal (should never happen)
si(:) = 0.0_sp
sti(:) = 0.0_sp
end if
case(3)
stop 'mo_sobol_index: Sobol2007 would need f(A_B) and f(B_A). It is thus not implemented here.'
case(4)
! 'Saltelli2010' - Si takes B, A_B instead of A, B_A (yc=f(A_B))
! Si = 1/n*sum_j(f(B)_j*(f(A_B^i)_j - f(A)_j))/var(f(A))
! STi = 1/n*sum_j(f(A)_j*(f(A)_j-f(A_B^i)_j))/var(f(A))
var_a = variance(ya, ddof=1_i4)
if ( var_a .gt. tiny(1.0_sp) ) then
! model outputs are different (usual case)
do ii=1, npara
si(ii) = ( dot_product( yb(:) , (yc(:,ii)-ya(:)) ) / real(nsets,sp) ) / var_a
sti(ii) = ( dot_product( ya(:) , (ya(:)-yc(:,ii)) ) / real(nsets,sp) ) / var_a
end do
else
! model outputs are equal (should never happen)
si(:) = 0.0_sp
sti(:) = 0.0_sp
end if
case(5)
! 'Jansen1999' - Calculate Si and STi by expectation(variance) instead of variance(expectation) (yc=f(A_B))
! Si = (var(f(A)) - 1/2n*sum_j(f(B)_j - f(A_B^i)_j)^2)/var(f(A))
! STi = 1/2n*sum_j(f(A)_j - f(A_B^i)_j)^2/var(f(A))
var_a = variance(ya, ddof=1_i4)
if ( var_a .gt. tiny(1.0_sp) ) then
! model outputs are different (usual case)
do ii=1, npara
si(ii) = 1.0_sp - ( sum( (yb(:)-yc(:,ii))**2 ) / real(2*nsets,sp) ) / var_a
sti(ii) = ( sum( (ya(:)-yc(:,ii))**2 ) / real(2*nsets,sp) ) / var_a
end do
else
! model outputs are equal (should never happen)
si(:) = 0.0_sp
sti(:) = 0.0_sp
end if
case(6)
! 'Mai2012' - Si of Homma 1996 but denominator is variance of A and B (yc=f(B_A)) and
! STi of Homma 1996 but mean and variance of f(A) replaced by f(B)
! Si = 1/n*sum_j(f(A)_j*(f(B_A^i)_j - f(B)_j))/var([f(A),f(B)])
! STi = (var(f(A))-(1/n*sum_j(f(B)_j*f(B_A^i)_j) - mean(f(B)^2)))/var(f(B))
allocate(yab(2*nsets))
yab( 1: nsets) = ya
yab(nsets+1:2*nsets) = yb
var_ab = variance(yab, ddof=1_i4)
var_b = variance(yb, ddof=1_i4)
mean_ab = dot_product(ya,yb) / real(nsets,sp)
mean_b = sum(yb) / real(nsets,sp)
if ( var_ab .gt. tiny(1.0_sp) ) then
! model outputs are different (usual case)
do ii=1, npara
si(ii) = ( dot_product(ya(:),yc(:,ii)) / real(nsets,sp) - mean_ab) / var_ab
sti(ii) = 1.0_sp - ( dot_product(yb(:),yc(:,ii)) / real(nsets,sp) - mean_b**2) / var_b
end do
else
! model outputs are equal (should never happen)
si(:) = 0.0_sp
sti(:) = 0.0_sp
end if
deallocate(yab)
case(7)
! 'Mai2013' - Si of Saltelli2010 but denominator is variance of A and B (yc=f(A_B))
! Si = 1/n*sum_j(f(B)_j*(f(A_B^i)_j - f(A)_j))/var([f(A),f(B)])
! STi = 1/n*sum_j(f(A)_j*(f(A)_j)-f(A_B^i)_j)/var(f(A))
allocate(yab(2*nsets))
yab( 1: nsets) = ya
yab(nsets+1:2*nsets) = yb
var_ab = variance(yab, ddof=1_i4)
var_a = variance(ya, ddof=1_i4)
if ( var_ab .gt. tiny(1.0_sp) ) then
! model outputs are different (usual case)
do ii=1, npara
si(ii) = ( dot_product( yb(:) , (yc(:,ii)-ya(:)) ) / real(nsets,sp) ) / var_ab
sti(ii) = ( dot_product( ya(:) , (ya(:)-yc(:,ii)) ) / real(nsets,sp) ) / var_a
end do
else
! model outputs are equal (should never happen)
si(:) = 0.0_sp
sti(:) = 0.0_sp
end if
deallocate(yab)
case(8)
! 'Mai2014' - SI of Jansen1999 but denominator is variance of A and B (yc=f(A_B))
! Si = (var([f(A),f(B)]) - 1/2n*sum_j(f(B)_j - f(A_B^i)_j)^2)/var([f(A),f(B)])
! STi = 1/2n*sum_j(f(A)_j - f(A_B^i)_j)^2/var(f(A))
allocate(yab(2*nsets))
yab( 1: nsets) = ya
yab(nsets+1:2*nsets) = yb
var_ab = variance(yab, ddof=1_i4)
var_a = variance(ya, ddof=1_i4)
if ( var_ab .gt. tiny(1.0_sp) ) then
! model outputs are different (usual case)
do ii=1, npara
si(ii) = 1.0_sp - ( sum( (yb(:)-yc(:,ii))**2 ) / real(2*nsets,sp) ) / var_ab
sti(ii) = ( sum( (ya(:)-yc(:,ii))**2 ) / real(2*nsets,sp) ) / var_a
end do
else
! model outputs are equal (should never happen)
si(:) = 0.0_sp
sti(:) = 0.0_sp
end if
deallocate(yab)
case(9)
! 'Mai1999' - SI of Mai2013 (Saltelli2010 with var([f(A),f(B)])) and STi of Jansen1999 (yc=f(A_B))
! Si = 1/n*sum_j(f(B)_j*(f(A_B^i)_j - f(A)_j))/var([f(A),f(B)])
! STi = 1/2n*sum_j(f(A)_j - f(A_B^i)_j)^2/var(f(A))
allocate(yab(2*nsets))
yab( 1: nsets) = ya
yab(nsets+1:2*nsets) = yb
var_ab = variance(yab, ddof=1_i4)
var_a = variance(ya, ddof=1_i4)
if ( var_ab .gt. tiny(1.0_sp) ) then
! model outputs are different (usual case)
do ii=1, npara
si(ii) = ( dot_product( yb(:) , (yc(:,ii)-ya(:)) ) / real(nsets,sp) ) / var_ab
sti(ii) = ( sum( (ya(:)-yc(:,ii))**2 ) / real(2*nsets,sp) ) / var_a
end do
else
! model outputs are equal (should never happen)
si(:) = 0.0_sp
sti(:) = 0.0_sp
end if
deallocate(yab)
case default
stop 'mo_sobol_index: This method is not implemented!'
end select
if ( present(smean) ) then
smean(:,1) = si(:)
smean(:,2) = sti(:)
end if
if ( present(wmean) ) then
wmean(:,1) = si(:)
wmean(:,2) = sti(:)
end if
end subroutine sobol_index_0d_sp
subroutine sobol_index_1d_dp(ya, yb, yc, si, sti, method, mask, smean, wmean, wmean_frac)
real(dp), dimension(:,:), intent(in) :: ya ! Output running model with parameter sets A
! ! dim1 = number of parameter sets
! ! dim2 = number of modeloutputs
! ! (e.g. number of timesteps)
real(dp), dimension(:,:), intent(in) :: yb ! Output running model with parameter sets B
! ! dim1 = number of parameter sets
! ! dim2 = number of modeloutputs
! ! (e.g. number of timesteps)
real(dp), dimension(:,:,:), intent(in) :: yc ! Output running model with parameter sets C
! ! dim1 = number of parameter sets
! ! = size(ya) = size(yb)
! ! dim2 = number of parameters
! ! dim3 = number of modeloutputs
! ! (e.g. number of timesteps)
logical, dimension(:), optional, intent(in) :: mask ! Mask of model output
! ! has to be the same for each parameter set
! ! dim1 = number of modeloutputs
! ! (e.g. number of timesteps)
real(dp), dimension(size(yc,2),size(yc,3)), intent(out) :: si ! Sobol index (main effect)
! ! (0.0 at masked entries)
! ! dim1 = number of parameters
! ! dim2 = number of modeloutputs
! ! (e.g. number of timesteps)
real(dp), dimension(size(yc,2),size(yc,3)), intent(out) :: sti ! Sobol index (total effect)
! ! (0.0 at masked entries)
! ! dim1 = number of parameters
! ! dim2 = number of modeloutputs
! ! (e.g. number of timesteps)
integer(i4), optional, intent(in) :: method ! (1) 'Saltelli2008' needs Ci = B_A
! ! (2) 'Homma1996' needs Ci = B_A
! ! (3) 'Sobol2007' needs Ci = A_B and B_A --> NA
! ! (4) 'Saltelli2010' needs Ci = A_B
! ! (5) 'Jansen1999' needs Ci = A_B
! ! (6) 'Mai2012' needs Ci = B_A
! ! (7) 'Mai2013' needs Ci = A_B
! ! (8) 'Mai2014' needs Ci = A_B
! ! (9) 'Mai1999' needs Ci = A_B ----> Default
real(dp), dimension(size(yc,2), 2), optional, intent(out) :: smean ! Mean Sobol index (main and total effect)
! ! dim1 = number of parameters
! ! dim2 = 2 (i.e. SI and STI)
real(dp), dimension(size(yc,2), 2), optional, intent(out) :: wmean ! Variance weighted mean Sobol index
! ! (main and total effect)
! ! dim1 = number of parameters
! ! dim2 = 2 (i.e. SI and STI)
real(dp), dimension(size(yc,2), 4), optional, intent(out) :: wmean_frac ! Numerator and denominator which was used to
! ! calculate wmean
! ! (only be callable, if wmean present)
! ! dim1 = number of parameters
! ! dim2 = 4 (1 - numerator Si)
! ! (2 - denominator Si)
! ! (3 - numerator STi)
! ! (4 - denominator STi)
! ! be aware that it depends on the method how
! ! wmean is then calculated, i.e.
! ! numerator/denominator or
! ! 1.0 - numerator/denominator
! local variables
integer(i4) :: ii, iout
integer(i4) :: nsets, npara, nout
integer(i4) :: meth
real(dp), dimension(:), allocatable :: yab
real(dp) :: var_a, var_b, var_ab
real(dp) :: mean_a, mean_b, mean_ab
logical, dimension(size(ya,2)) :: imask
! for averaging
real(dp), dimension(:), allocatable :: si_num, sti_num
real(dp) :: si_denom, sti_denom
real(dp) :: si_num_ii, sti_num_ii
if (present(method)) then
meth = method
else
meth = 9
end if
if (present(mask)) then
imask = mask
else
imask = .true.
end if
if (present(wmean_frac) .and. .not. present(wmean)) then
stop 'mo_sobol_index: wmean_frac is only callable when wmean is present'
end if
nsets = size(yc,1)
npara = size(yc,2)
nout = size(yc,3)
si(:,:) = 0.0_dp
sti(:,:) = 0.0_dp
allocate(si_num(npara))
allocate(sti_num(npara))
si_num(:) = 0.0_dp
sti_num(:) = 0.0_dp
si_denom = 0.0_dp
sti_denom = 0.0_dp
select case(meth)
case(1)
! 'Saltelli2008' - The formulation presented in 'The Primer'. (yc=f(B_A))
! Si = (1/n*sum_j(f(A)_j*f(B_A^i)_j) - mean(f(A))^2)/var(f(A))
! STi = (var(f(A))-(1/n*sum_j(f(B)_j*f(B_A^i)_j) - mean(f(A))^2))/var(f(A))
do iout=1, nout
if (imask(iout)) then
var_a = variance(ya(:,iout), ddof=1_i4)
mean_a = sum(ya(:,iout)) / real(nsets,dp)
si_denom = si_denom + var_a
sti_denom = sti_denom + var_a
if ( var_a .gt. tiny(1.0_dp) ) then
! model outputs are different (usual case)
do ii=1, npara
si_num_ii = ( dot_product( ya(:,iout) , (yc(:,ii,iout)) ) / real(nsets,dp) - mean_a**2 )
sti_num_ii = ( dot_product( yb(:,iout) , (yc(:,ii,iout)) ) / real(nsets,dp) - mean_a**2 )
si_num(ii) = si_num(ii) + si_num_ii
sti_num(ii) = sti_num(ii) + sti_num_ii
si(ii,iout) = si_num_ii / var_a
sti(ii,iout) = 1.0_dp - sti_num_ii / var_a
end do
else
! model outputs are equal (should never happen)
si(:,iout) = 0.0_dp
sti(:,iout) = 0.0_dp
end if
end if
end do
case(2)
! 'Homma1996' - Si takes mixed A, B term for squared mean in nominator. (yc=f(B_A))
! Si = 1/n*sum_j(f(A)_j*(f(B_A^i)_j - f(B)_j))/var(f(A))
! STi as Saltelli2008
do iout=1, nout
if (imask(iout)) then
var_a = variance(ya(:,iout), ddof=1_i4)
mean_a = sum(ya(:,iout)) / real(nsets,dp)
si_denom = si_denom + var_a
sti_denom = sti_denom + var_a
if ( var_a .gt. tiny(1.0_dp) ) then
! model outputs are different (usual case)
do ii=1, npara
si_num_ii = ( dot_product( ya(:,iout) , (yc(:,ii,iout)-yb(:,iout)) ) / real(nsets,dp) )
sti_num_ii = ( dot_product( yb(:,iout) , (yc(:,ii,iout)) ) / real(nsets,dp) - mean_a**2 )
si_num(ii) = si_num(ii) + si_num_ii
sti_num(ii) = sti_num(ii) + sti_num_ii
si(ii,iout) = si_num_ii / var_a
sti(ii,iout) = 1.0_dp - sti_num_ii / var_a
end do
else
! model outputs are equal (should never happen)
si(:,iout) = 0.0_dp
sti(:,iout) = 0.0_dp
end if
end if
end do
case(3)
stop 'mo_sobol_index: Sobol2007 would need f(A_B) and f(B_A). It is thus not implemented here.'
case(4)
! 'Saltelli2010' - Si takes B, A_B instead of A, B_A (yc=f(A_B))
! Si = 1/n*sum_j(f(B)_j*(f(A_B^i)_j - f(A)_j))/var(f(A))
! STi = 1/n*sum_j(f(A)_j*(f(A)_j-f(A_B^i)_j))/var(f(A))
do iout=1, nout
if (imask(iout)) then
var_a = variance(ya(:,iout), ddof=1_i4)
si_denom = si_denom + var_a
sti_denom = sti_denom + var_a
if ( var_a .gt. tiny(1.0_dp) ) then
! model outputs are different (usual case)
do ii=1, npara
si_num_ii = ( dot_product( yb(:,iout) , (yc(:,ii,iout)-ya(:,iout)) ) / real(nsets,dp) )
sti_num_ii = ( dot_product( ya(:,iout) , (ya(:,iout)-yc(:,ii,iout)) ) / real(nsets,dp) )
si_num(ii) = si_num(ii) + si_num_ii
sti_num(ii) = sti_num(ii) + sti_num_ii
si(ii,iout) = si_num_ii / var_a
sti(ii,iout) = sti_num_ii / var_a
end do
else
! model outputs are equal (should never happen)
si(:,iout) = 0.0_dp
sti(:,iout) = 0.0_dp
end if
end if
end do
case(5)
! 'Jansen1999' - Calculate Si and STi by expectation(variance) instead of variance(expectation) (yc=f(A_B))
! Si = (var(f(A)) - 1/2n*sum_j(f(B)_j - f(A_B^i)_j)^2)/var(f(A))
! STi = 1/2n*sum_j(f(A)_j - f(A_B^i)_j)^2/var(f(A))
do iout=1, nout
if (imask(iout)) then
var_a = variance(ya(:,iout), ddof=1_i4)
si_denom = si_denom + var_a
sti_denom = sti_denom + var_a
if ( var_a .gt. tiny(1.0_dp) ) then
! model outputs are different (usual case)
do ii=1, npara
si_num_ii = ( sum( (yb(:,iout)-yc(:,ii,iout))**2 ) / real(2*nsets,dp) )
sti_num_ii = ( sum( (ya(:,iout)-yc(:,ii,iout))**2 ) / real(2*nsets,dp) )
si_num(ii) = si_num(ii) + si_num_ii
sti_num(ii) = sti_num(ii) + sti_num_ii
si(ii,iout) = 1.0_dp - si_num_ii / var_a
sti(ii,iout) = sti_num_ii / var_a
end do
else
! model outputs are equal (should never happen)
si(:,iout) = 0.0_dp
sti(:,iout) = 0.0_dp
end if
end if
end do
case(6)
! 'Mai2012' - Si of Homma 1996 but denominator is variance of A and B (yc=f(B_A)) and
! STi of Homma 1996 but mean and variance of f(A) replaced by f(B)
! Si = 1/n*sum_j(f(A)_j*(f(B_A^i)_j - f(B)_j))/var([f(A),f(B)])
! STi = (var(f(A))-(1/n*sum_j(f(B)_j*f(B_A^i)_j) - mean(f(B)^2)))/var(f(B))
allocate(yab(2*nsets))
do iout=1, nout
if (imask(iout)) then
yab( 1: nsets) = ya(:,iout)
yab(nsets+1:2*nsets) = yb(:,iout)
var_ab = variance(yab, ddof=1_i4)
var_b = variance(yb(:,iout), ddof=1_i4)
mean_ab = dot_product(ya(:,iout),yb(:,iout)) / real(nsets,dp)
mean_b = sum(yb(:,iout)) / real(nsets,dp)
si_denom = si_denom + var_ab
sti_denom = sti_denom + var_b
if ( var_ab .gt. tiny(1.0_dp) ) then
! model outputs are different (usual case)
do ii=1, npara
si_num_ii = ( dot_product(ya(:,iout),yc(:,ii,iout)) / real(nsets,dp) - mean_ab )
sti_num_ii = ( dot_product(yb(:,iout),yc(:,ii,iout)) / real(nsets,dp) - mean_b**2 )
si_num(ii) = si_num(ii) + si_num_ii
sti_num(ii) = sti_num(ii) + sti_num_ii
si(ii,iout) = si_num_ii / var_ab
sti(ii,iout) = 1.0_dp - sti_num_ii / var_b
end do
else
! model outputs are equal (should never happen)
si(:,iout) = 0.0_dp
sti(:,iout) = 0.0_dp
end if
end if
end do
deallocate(yab)
case(7)
! 'Mai2013' - Si of Saltelli2010 but denominator is variance of A and B (yc=f(A_B))
! Si = 1/n*sum_j(f(B)_j*(f(A_B^i)_j - f(A)_j))/var([f(A),f(B)])
! STi = 1/n*sum_j(f(A)_j*(f(A)_j)-f(A_B^i)_j)/var(f(A))
allocate(yab(2*nsets))
do iout=1, nout
if (imask(iout)) then
yab( 1: nsets) = ya(:,iout)
yab(nsets+1:2*nsets) = yb(:,iout)
var_ab = variance(yab, ddof=1_i4)
var_a = variance(ya(:,iout), ddof=1_i4)
si_denom = si_denom + var_ab
sti_denom = sti_denom + var_a
if ( var_ab .gt. tiny(1.0_dp) ) then
! model outputs are different (usual case)
do ii=1, npara
si_num_ii = ( dot_product( yb(:,iout) , (yc(:,ii,iout)-ya(:,iout)) ) / real(nsets,dp) )
sti_num_ii = ( dot_product( ya(:,iout) , (ya(:,iout)-yc(:,ii,iout)) ) / real(nsets,dp) )
si_num(ii) = si_num(ii) + si_num_ii
sti_num(ii) = sti_num(ii) + sti_num_ii
si(ii,iout) = si_num_ii / var_ab
sti(ii,iout) = sti_num_ii / var_a
end do
else
! model outputs are equal (should never happen)
si(:,iout) = 0.0_dp
sti(:,iout) = 0.0_dp
end if
end if
end do
deallocate(yab)
case(8)
! 'Mai2014' - SI of Jansen1999 but denominator is variance of A and B (yc=f(A_B))
! Si = (var([f(A),f(B)]) - 1/2n*sum_j(f(B)_j - f(A_B^i)_j)^2)/var([f(A),f(B)])
! STi = 1/2n*sum_j(f(A)_j - f(A_B^i)_j)^2/var(f(A))
allocate(yab(2*nsets))
do iout=1, nout
if (imask(iout)) then
yab( 1: nsets) = ya(:,iout)
yab(nsets+1:2*nsets) = yb(:,iout)
var_ab = variance(yab, ddof=1_i4)
var_a = variance(ya(:,iout), ddof=1_i4)
si_denom = si_denom + var_ab
sti_denom = sti_denom + var_a
if ( var_ab .gt. tiny(1.0_dp) ) then
! model outputs are different (usual case)
do ii=1, npara
si_num_ii = ( sum( (yb(:,iout)-yc(:,ii,iout))**2 ) / real(2*nsets,dp) )
sti_num_ii = ( sum( (ya(:,iout)-yc(:,ii,iout))**2 ) / real(2*nsets,dp) )
si_num(ii) = si_num(ii) + si_num_ii
sti_num(ii) = sti_num(ii) + sti_num_ii
si(ii,iout) = 1.0_dp - si_num_ii / var_ab
sti(ii,iout) = sti_num_ii / var_a
end do
else
! model outputs are equal (should never happen)
si(:,iout) = 0.0_dp
sti(:,iout) = 0.0_dp
end if
end if
end do
deallocate(yab)
case(9)
! 'Mai1999' - SI of Mai2013 (Saltelli2010 with var([f(A),f(B)])) and STi of Jansen1999 (yc=f(A_B))
! Si = 1/n*sum_j(f(B)_j*(f(A_B^i)_j - f(A)_j))/var([f(A),f(B)])
! STi = 1/2n*sum_j(f(A)_j - f(A_B^i)_j)^2/var(f(A))
allocate(yab(2*nsets))
do iout=1, nout
if (imask(iout)) then
yab( 1: nsets) = ya(:,iout)
yab(nsets+1:2*nsets) = yb(:,iout)
var_ab = variance(yab, ddof=1_i4)
var_a = variance(ya(:,iout), ddof=1_i4)
si_denom = si_denom + var_ab
sti_denom = sti_denom + var_a
if ( var_ab .gt. tiny(1.0_dp) ) then
! model outputs are different (usual case)
do ii=1, npara
si_num_ii = ( dot_product( yb(:,iout) , (yc(:,ii,iout)-ya(:,iout)) ) / real(nsets,dp) )
sti_num_ii = ( sum( (ya(:,iout)-yc(:,ii,iout))**2 ) / real(2*nsets,dp) )
si_num(ii) = si_num(ii) + si_num_ii
sti_num(ii) = sti_num(ii) + sti_num_ii
si(ii,iout) = si_num_ii / var_ab
sti(ii,iout) = sti_num_ii / var_a
end do
else
! model outputs are equal (should never happen)
si(:,iout) = 0.0_dp
sti(:,iout) = 0.0_dp
end if
end if
end do
deallocate(yab)
case default
stop 'mo_sobol_index: This method is not implemented!'
end select
if (present(smean)) then
smean = 0.0_dp
smean(:,1) = sum(si, dim=2) / real(count(imask),dp)
smean(:,2) = sum(sti,dim=2) / real(count(imask),dp)
end if