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mo_boxcox.f90
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mo_boxcox.f90
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MODULE mo_boxcox
! This module contains routines to calculate the Box-Cox transformation
! as well as estimating the best exponent for the Box-Cox transformation
! Usage:
! USE mo_boxcox, ONLY: boxcox, get_boxcox
! lmbda = get_boxcox(data)
! new_data = boxcox(data, lmbda)
! data = invboxcox(new_data, lmbda)
! Written March 2011, Matthias Cuntz
! - modified Python code of Travis Oliphant (2002): boxcox, llf_boxcox, get_boxcox
! - modified numerical recipes: brent, mnbrak, swap, shft
! License
! -------
! This file is part of the JAMS Fortran package, distributed under the MIT License.
!
! Copyright (c) 2011-2012 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.
! Note on Numerical Recipes License
! ---------------------------------
! Be aware that some code is under the Numerical Recipes License 3rd
! edition <http://www.nr.com/aboutNR3license.html>
! The Numerical Recipes Personal Single-User License lets you personally
! use Numerical Recipes code ("the code") on any number of computers,
! but only one computer at a time. You are not permitted to allow anyone
! else to access or use the code. You may, under this license, transfer
! precompiled, executable applications incorporating the code to other,
! unlicensed, persons, providing that (i) the application is
! noncommercial (i.e., does not involve the selling or licensing of the
! application for a fee), and (ii) the application was first developed,
! compiled, and successfully run by you, and (iii) the code is bound
! into the application in such a manner that it cannot be accessed as
! individual routines and cannot practicably be unbound and used in
! other programs. That is, under this license, your application user
! must not be able to use Numerical Recipes code as part of a program
! library or "mix and match" workbench.
! Businesses and organizations that purchase the disk or code download,
! and that thus acquire one or more Numerical Recipes Personal
! Single-User Licenses, may permanently assign those licenses, in the
! number acquired, to individual employees. Such an assignment must be
! made before the code is first used and, once made, it is irrevocable
! and can not be transferred.
! If you do not hold a Numerical Recipes License, this code is only for
! informational and educational purposes but cannot be used.
USE mo_kind, ONLY: i4, sp, dp
USE mo_utils, only: eq, le, ge
IMPLICIT NONE
PUBLIC :: boxcox ! Calculate Box-Cox power transformed values given the original values and the exponent lambda
PUBLIC :: get_boxcox ! Find the exponent that maximises the log-likelihood function
PUBLIC :: invboxcox ! Calculate the inverse Box-Cox given the transformed values and the exponent lambda
! ------------------------------------------------------------------
! NAME
! boxcox
! PURPOSE
! Transform a positive dataset with a Box-Cox power transformation.
! boxcox(x) = (x**lambda - 1)/lambda if lambda <> 0
! ln(x) if lambda = 0
!
! If an optinal mask is given, then the Box-Cox transformation is only performed on
! those locations that correspond to true values in the mask.
! x can be single or double precision. The result will have the same numerical precision.
! CALLING SEQUENCE
! out = boxcox(x, lmbda, mask=mask)
! INTENT(IN)
! real(sp/dp) :: x(:) 1D-array with input numbers (>0.)
! real(sp/dp) :: lmbda Exponent power of Box-Cox transform (>= 0.)
! INTENT(INOUT)
! None
! INTENT(OUT)
! real(sp/dp) :: boxcox Power transformed values (at mask=.True.)
! INTENT(IN), OPTIONAL
! logical :: mask(:) 1D-array of logical values with size(x).
! If present, only those locations in vec corresponding to the true values in mask are used.
! INTENT(INOUT), OPTIONAL
! None
! INTENT(OUT), OPTIONAL
! None
! RESTRICTIONS
! Input values must be positive, i.e. > 0.
! EXAMPLE
! lmbda = get_boxcox(data, mask=(data>0.))
! new_data = boxcox(data, lmbda, mask=(data>0.))
! idata = invboxcox(new_data, lmbda, mask=(data>0.))
! -> see also example in test directory
! HISTORY
! Written, Matthias Cuntz, March 2011
! - Modified Python code of Travis Oliphant (2002): boxcox, llf_boxcox, get_boxcox
! - Modified numerical recipes: brent, mnbrak, swap, shft
INTERFACE boxcox
MODULE PROCEDURE boxcox_sp, boxcox_dp
END INTERFACE boxcox
! ------------------------------------------------------------------
! NAME
! get_boxcox
! PURPOSE
! Get lambda for Box-Cox transformation.
!
! Transform a positive dataset with a Box-Cox power transformation.
! boxcox(x) = (x**lambda - 1)/lambda if lambda <> 0
! ln(x) if lambda = 0
!
! If an optinal mask is given, then the Box-Cox transformation is only performed on
! those locations that correspond to true values in the mask.
! x can be single or double precision. The result will have the same numerical precision.
! CALLING SEQUENCE
! out = get_boxcox(x, mask=mask)
! INTENT(IN)
! real(sp/dp) :: x(:) 1D-array with input numbers (>0.)
! INTENT(INOUT)
! None
! INTENT(OUT)
! real(sp/dp) :: lambda Lambda for power transformed values (at mask=.True.)
! INTENT(IN), OPTIONAL
! logical :: mask(:) 1D-array of logical values with size(x).
! If present, only those locations in vec corresponding to the true values in mask are used.
! INTENT(INOUT), OPTIONAL
! None
! INTENT(OUT), OPTIONAL
! None
! RESTRICTIONS
! Input values must be positive, i.e. > 0.
! EXAMPLE
! lmbda = get_boxcox(data, mask=(data>0.))
! new_data = boxcox(data, lmbda, mask=(data>0.))
! idata = invboxcox(new_data, lmbda, mask=(data>0.))
! -> see also example in test directory
! HISTORY
! Written, Matthias Cuntz, March 2011
! - Modified Python code of Travis Oliphant (2002): boxcox, llf_boxcox, get_boxcox
! - Modified numerical recipes: brent, mnbrak, swap, shft
INTERFACE get_boxcox
MODULE PROCEDURE get_boxcox_sp, get_boxcox_dp
END INTERFACE get_boxcox
! ------------------------------------------------------------------
! NAME
! invboxcox
! PURPOSE
! Back-transformation of Box-Cox-transformed data.
! boxcox(x) = (x**lambda - 1)/lambda if lambda <> 0
! ln(x) if lambda = 0
! invboxcox(y) = (lambda*y + 1)**(1/lambda) if lambda <> 0
! exp(y) if lambda = 0
!
! If an optinal mask is given, then the inverse Box-Cox transformation is only performed on
! those locations that correspond to true values in the mask.
! x can be single or double precision. The result will have the same numerical precision.
! x can be scalar or vector
! CALLING SEQUENCE
! out = invboxcox(x, lmbda) ! scalar x
! out = invboxcox(x, lmbda, mask=mask) ! vector x
! INTENT(IN)
! real(sp/dp) :: x / x(:) scalar/1D-array with input numbers (>0.)
! real(sp/dp) :: lmbda Exponent power of Box-Cox transform (>= 0.)
! INTENT(INOUT)
! None
! INTENT(OUT)
! real(sp/dp) :: boxcox Back transformed values (at mask=.True.)
! INTENT(IN), OPTIONAL
! logical :: mask(:) 1D-array of logical values with size(x).
! If present, only those locations in vec corresponding to the true values in mask are used.
! Only applicable if x is a 1D-array
! INTENT(INOUT), OPTIONAL
! None
! INTENT(OUT), OPTIONAL
! None
! RESTRICTIONS
! None
! EXAMPLE
! lmbda = get_boxcox(data, mask=(data>0.))
! new_data = boxcox(data, lmbda, mask=(data>0.))
! idata = invboxcox(new_data, lmbda, mask=(data>0.))
! -> see also example in test directory
! HISTORY
! Written, Matthias Cuntz, March 2011
! - Modified MC: Python code of Travis Oliphant (2002): boxcox, llf_boxcox, get_boxcox
! - Modified MC: numerical recipes: brent, mnbrak, swap, shft
! - Modified JM: scalar version of invboxcox
INTERFACE invboxcox
MODULE PROCEDURE invboxcox_0d_sp, invboxcox_0d_dp, invboxcox_1d_sp, invboxcox_1d_dp
END INTERFACE invboxcox
! ------------------------------------------------------------------
PRIVATE
! ------------------------------------------------------------------
CONTAINS
! ------------------------------------------------------------------
FUNCTION boxcox_sp(x, lmbda, mask)
IMPLICIT NONE
REAL(sp), DIMENSION(:), INTENT(in) :: x
REAL(sp), INTENT(in) :: lmbda
LOGICAL, DIMENSION(:), INTENT(in), OPTIONAL :: mask
REAL(sp), DIMENSION(size(x)) :: boxcox_sp
LOGICAL, DIMENSION(size(x)) :: maske
REAL(sp) :: lmbda1
maske(:) = .true.
if (present(mask)) then
if (size(mask) /= size(x)) stop 'Error boxcox_sp: size(mask) /= size(x)'
maske = mask
endif
if (any((le(x,0.0_sp)) .and. maske)) stop 'Error boxcox_sp: x <= 0'
if (abs(lmbda) .lt. tiny(0.0_sp)) then
where (maske)
boxcox_sp = log(x)
elsewhere
boxcox_sp = x
end where
else
lmbda1 = 1.0_sp / lmbda
boxcox_sp = merge((exp(lmbda*log(x)) - 1.0_sp) * lmbda1, x, maske)
endif
END FUNCTION boxcox_sp
FUNCTION boxcox_dp(x, lmbda, mask)
IMPLICIT NONE
REAL(dp), DIMENSION(:), INTENT(in) :: x
REAL(dp), INTENT(in) :: lmbda
LOGICAL, DIMENSION(:), INTENT(in), OPTIONAL :: mask
REAL(dp), DIMENSION(size(x)) :: boxcox_dp
LOGICAL, DIMENSION(size(x)) :: maske
REAL(dp) :: lmbda1
maske(:) = .true.
if (present(mask)) then
if (size(mask) /= size(x)) stop 'Error boxcox_dp: size(mask) /= size(x)'
maske = mask
endif
if (any((le(x,0.0_dp)) .and. maske)) then
stop 'Error boxcox_dp: x <= 0'
end if
if (abs(lmbda) .lt. tiny(0.0_dp)) then
where (maske)
boxcox_dp = log(x)
elsewhere
boxcox_dp = x
end where
else
lmbda1 = 1.0_dp / lmbda
boxcox_dp = merge((exp(lmbda*log(x)) - 1.0_dp) * lmbda1, x, maske)
endif
END FUNCTION boxcox_dp
! ------------------------------------------------------------------
! Given a function f, and given a bracketing triplet of abscissas ax, bx, cx (such that bx is
! between ax and cx, and f(bx) is less than both f(ax) and f(cx)), this routine isolates
! the minimum to a fractional precision of about tol using Brent''s method. The abscissa of
! the minimum is returned.
! Parameters: Maximum allowed number of iterations; goldenratio; and a small number that
! protects against trying to achieve fractional accuracy for a minimum that happens to be
! exactly zero.
Function brent_sp(ax, bx, cx, func_sp, tol, dat)
IMPLICIT NONE
REAL(sp), INTENT(in) :: ax, bx, cx
INTERFACE
FUNCTION func_sp(x, da)
USE mo_kind, ONLY: sp
IMPLICIT NONE
REAL(sp), INTENT(in) :: x
REAL(sp), DIMENSION(:), INTENT(in) :: da
REAL(sp) :: func_sp
END FUNCTION func_sp
END INTERFACE
REAL(sp), INTENT(in) :: tol
REAL(sp), DIMENSION(:), INTENT(in) :: dat
REAL(sp) :: brent_sp
INTEGER(i4), PARAMETER :: ITMAX = 500
REAL(sp), PARAMETER :: CGOLD = 0.3819660_sp
REAL(sp), PARAMETER :: ZEPS = 1.0e-3_sp*epsilon(ax)
INTEGER(i4) :: iter
REAL(sp) :: a, b, d, e, etemp, fu, fv, fw, fx
REAL(sp) :: p, q, r, tol1, tol2, u, v, w, x, xm
d = 0.0_sp
a = min(ax,cx)
b = max(ax,cx)
v = bx
w = v
x = v
e = 0.0_sp
fx = func_sp(x, dat)
fv = fx
fw = fx
do iter=1, ITMAX
xm = 0.5_sp*(a+b)
tol1 = tol*abs(x) + ZEPS
tol2 = 2.0_sp*tol1
if (le(abs(x-xm),(tol2-0.5_sp*(b-a)))) then
brent_sp = x
return
end if
if (abs(e) > tol1) then
r = (x-w)*(fx-fv)
q = (x-v)*(fx-fw)
p = (x-v)*q-(x-w)*r
q = 2.0_sp*(q-r)
if (q > 0.0_sp) p = -p
q = abs(q)
etemp = e
e = d
if (ge(abs(p),abs(0.5_sp*q*etemp)) .or. &
le(p,q*(a-x)) .or. ge(p,q*(b-x))) then
e = merge(a-x, b-x, ge(x,xm))
d = CGOLD*e
else
d = p/q
u = x+d
if (u-a < tol2 .or. b-u < tol2) d = sign(tol1,xm-x)
end if
else
e = merge(a-x, b-x, ge(x,xm))
d = CGOLD*e
end if
u = merge(x+d, x+sign(tol1,d), ge(abs(d),tol1))
fu = func_sp(u, dat)
if (le(fu,fx)) then
if (ge(u,x)) then
a = x
else
b = x
end if
call shft_sp(v,w,x,u)
call shft_sp(fv,fw,fx,fu)
else
if (u < x) then
a = u
else
b = u
end if
if (le(fu,fw) .or. eq(w,x)) then
v = w
fv = fw
w = u
fw = fu
else if (le(fu,fv) .or. eq(v,x) .or. eq(v,w)) then
v = u
fv = fu
end if
end if
end do
! If it comes to here, there were too much iterations
STOP 'brent_sp: exceed maximum iterations'
END FUNCTION brent_sp
FUNCTION brent_dp(ax, bx, cx, func_dp, tol, dat)
IMPLICIT NONE
REAL(dp), INTENT(in) :: ax, bx, cx
INTERFACE
FUNCTION func_dp(x, da)
USE mo_kind, ONLY: dp
IMPLICIT NONE
REAL(dp), INTENT(in) :: x
REAL(dp), DIMENSION(:), INTENT(in) :: da
REAL(dp) :: func_dp
END FUNCTION func_dp
END INTERFACE
REAL(dp), INTENT(in) :: tol
REAL(dp), DIMENSION(:), INTENT(in) :: dat
REAL(dp) :: brent_dp
INTEGER(i4), PARAMETER :: ITMAX=500
REAL(dp), PARAMETER :: CGOLD=0.3819660_dp
REAL(dp), PARAMETER :: ZEPS=1.0e-3_dp*epsilon(ax)
INTEGER(i4) :: iter
REAL(dp) :: a, b, d, e, etemp, fu, fv, fw, fx
REAL(dp) :: p, q, r, tol1, tol2, u, v, w, x, xm
d = 0.0_dp
a = min(ax,cx)
b = max(ax,cx)
v = bx
w = v
x = v
e = 0.0_dp
fx = func_dp(x, dat)
fv = fx
fw = fx
do iter=1, ITMAX
xm = 0.5_dp*(a+b)
tol1 = tol*abs(x) + ZEPS
tol2 = 2.0_dp*tol1
if (le(abs(x-xm),(tol2-0.5_dp*(b-a)))) then
brent_dp = x
return
end if
if (abs(e) > tol1) then
r = (x-w)*(fx-fv)
q = (x-v)*(fx-fw)
p = (x-v)*q-(x-w)*r
q = 2.0_dp*(q-r)
if (q > 0.0_dp) p = -p
q = abs(q)
etemp = e
e = d
if (ge(abs(p),abs(0.5_dp*q*etemp)) .or. &
le(p,q*(a-x)) .or. ge(p,q*(b-x))) then
e = merge(a-x, b-x, ge(x,xm))
d = CGOLD*e
else
d = p/q
u = x+d
if (u-a < tol2 .or. b-u < tol2) d = sign(tol1,xm-x)
end if
else
e = merge(a-x, b-x, ge(x,xm))
d = CGOLD*e
end if
u = merge(x+d, x+sign(tol1,d), ge(abs(d),tol1))
fu = func_dp(u, dat)
if (le(fu,fx)) then
if (ge(u,x)) then
a = x
else
b = x
end if
call shft_dp(v,w,x,u)
call shft_dp(fv,fw,fx,fu)
else
if (u < x) then
a = u
else
b = u
end if
if (le(fu,fw) .or. eq(w,x)) then
v = w
fv = fw
w = u
fw = fu
else if (le(fu,fv) .or. eq(v,x) .or. eq(v,w)) then
v = u
fv = fu
end if
end if
end do
! If it comes to here, there were too much iterations
STOP 'brent_dp: exceed maximum iterations'
END FUNCTION brent_dp
! ------------------------------------------------------------------
! Find the lambda that maximizes the log-likelihood function
FUNCTION get_boxcox_sp(x, mask)
IMPLICIT NONE
REAL(sp), DIMENSION(:), INTENT(in) :: x
LOGICAL, DIMENSION(:), INTENT(in), OPTIONAL :: mask
REAL(sp) :: get_boxcox_sp
REAL(sp), PARAMETER :: tol = 1.48e-08
REAL(sp) :: ax, bx, cx
REAL(sp) :: fa, fb, fc
LOGICAL, DIMENSION(size(x)) :: maske
REAL(sp), DIMENSION(:), ALLOCATABLE :: xx
INTEGER(i4) :: nn
maske(:) = .true.
if (present(mask)) then
if (size(mask) /= size(x)) stop 'Error get_boxcox_sp: size(mask) /= size(x)'
maske = mask
endif
nn = count(maske)
if (allocated(xx)) deallocate(xx)
allocate(xx(nn))
xx = pack(x, mask=maske)
ax = -2.0_sp
bx = 2.0_sp
call mnbrak_sp(ax, bx, cx, fa, fb, fc, llf_boxcox_sp, xx)
get_boxcox_sp = brent_sp(ax, bx, cx, llf_boxcox_sp, tol, xx)
END FUNCTION get_boxcox_sp
FUNCTION get_boxcox_dp(x, mask)
IMPLICIT NONE
REAL(dp), DIMENSION(:), INTENT(in) :: x
LOGICAL, DIMENSION(:), INTENT(in), OPTIONAL :: mask
REAL(dp) :: get_boxcox_dp
REAL(dp), PARAMETER :: tol = 1.48e-08
REAL(dp) :: ax, bx, cx
REAL(dp) :: fa, fb, fc
LOGICAL, DIMENSION(size(x)) :: maske
REAL(dp), DIMENSION(:), ALLOCATABLE :: xx
INTEGER(i4) :: nn
maske(:) = .true.
if (present(mask)) then
if (size(mask) /= size(x)) stop 'Error get_boxcox_dp: size(mask) /= size(x)'
maske = mask
endif
nn = count(maske)
if (allocated(xx)) deallocate(xx)
allocate(xx(nn))
xx = pack(x, mask=maske)
if (any(le(xx,0.0_dp))) then
stop 'Error get_boxcox_dp: x <= 0'
end if
ax = -5.0_dp
bx = 5.0_dp
call mnbrak_dp(ax, bx, cx, fa, fb, fc, llf_boxcox_dp, xx)
get_boxcox_dp = brent_dp(ax, bx, cx, llf_boxcox_dp, tol, xx)
END FUNCTION get_boxcox_dp
! ------------------------------------------------------------------
FUNCTION invboxcox_1d_sp(x, lmbda, mask)
IMPLICIT NONE
REAL(sp), DIMENSION(:), INTENT(in) :: x
REAL(sp), INTENT(in) :: lmbda
LOGICAL, DIMENSION(:), INTENT(in), OPTIONAL :: mask
REAL(sp), DIMENSION(size(x)) :: invboxcox_1d_sp
LOGICAL, DIMENSION(size(x)) :: maske
REAL(sp) :: lmbda1
REAL(sp), DIMENSION(size(x)) :: temp
maske(:) = .true.
if (present(mask)) then
if (size(mask) /= size(x)) stop 'Error invboxcox_1d_sp: size(mask) /= size(x)'
maske = mask
endif
if (abs(lmbda) .lt. tiny(0.0_sp)) then
where (maske)
invboxcox_1d_sp = exp(x)
elsewhere
invboxcox_1d_sp = x
end where
else
lmbda1 = 1.0_sp / lmbda
temp = lmbda*x+1.0_sp
where (temp .gt. 0.0_sp)
temp = exp(lmbda1*log(temp))
elsewhere
temp = 0.0_sp
end where
invboxcox_1d_sp = merge(temp, x, maske)
endif
END FUNCTION invboxcox_1d_sp
FUNCTION invboxcox_1d_dp(x, lmbda, mask)
IMPLICIT NONE
REAL(dp), DIMENSION(:), INTENT(in) :: x
REAL(dp), INTENT(in) :: lmbda
LOGICAL, DIMENSION(:), INTENT(in), OPTIONAL :: mask
REAL(dp), DIMENSION(size(x)) :: invboxcox_1d_dp
LOGICAL, DIMENSION(size(x)) :: maske
REAL(dp) :: lmbda1
REAL(dp), DIMENSION(size(x)) :: temp
maske(:) = .true.
if (present(mask)) then
if (size(mask) /= size(x)) stop 'Error invboxcox_1d_dp: size(mask) /= size(x)'
maske = mask
endif
if (abs(lmbda) .lt. tiny(0.0_dp)) then
where (maske)
invboxcox_1d_dp = exp(x)
elsewhere
invboxcox_1d_dp = x
end where
else
lmbda1 = 1.0_dp / lmbda
temp = lmbda*x+1.0_dp
where (temp .gt. 0.0_dp)
temp = exp(lmbda1*log(temp))
elsewhere
temp = 0.0_dp
end where
invboxcox_1d_dp = merge(temp, x, maske)
endif
END FUNCTION invboxcox_1d_dp
FUNCTION invboxcox_0d_sp(x, lmbda)
IMPLICIT NONE
REAL(sp), INTENT(in) :: x
REAL(sp), INTENT(in) :: lmbda
REAL(sp) :: invboxcox_0d_sp
REAL(sp) :: lmbda1
REAL(sp) :: temp
if (abs(lmbda) .lt. tiny(0.0_sp)) then
invboxcox_0d_sp = exp(x)
else
lmbda1 = 1.0_sp / lmbda
temp = lmbda*x+1.0_sp
if (temp .gt. 0.0_sp) then
temp = exp(lmbda1*log(temp))
else
temp = 0.0_sp
end if
invboxcox_0d_sp = temp
endif
END FUNCTION invboxcox_0d_sp
FUNCTION invboxcox_0d_dp(x, lmbda)
IMPLICIT NONE
REAL(dp), INTENT(in) :: x
REAL(dp), INTENT(in) :: lmbda
REAL(dp) :: invboxcox_0d_dp
REAL(dp) :: lmbda1
REAL(dp) :: temp
if (abs(lmbda) .lt. tiny(0.0_dp)) then
invboxcox_0d_dp = exp(x)
else
lmbda1 = 1.0_dp / lmbda
temp = lmbda*x+1.0_dp
if (temp .gt. 0.0_dp) then
temp = exp(lmbda1*log(temp))
else
temp = 0.0_dp
end if
invboxcox_0d_dp = temp
endif
END FUNCTION invboxcox_0d_dp
! ------------------------------------------------------------------
! The boxcox (negative) log-likelihood function.
FUNCTION llf_boxcox_sp(lmbda, x)
IMPLICIT NONE
REAL(sp), INTENT(in) :: lmbda
REAL(sp), DIMENSION(:), INTENT(in) :: x
REAL(sp) :: llf_boxcox_sp
REAL(sp), DIMENSION(size(x)) :: y
REAL(sp) :: N, my, f, s
N = real(size(x),sp)
y = boxcox_sp(x, lmbda)
my = sum(y) / N
f = (lmbda-1.0_sp) * sum(log(x))
! f = f - 0.5_dp*N * log(sum((y-my)*(y-my))/N)
s = Min( huge(1.0_sp)/N , Max( N*tiny(1.0_sp) , sum((y-my)*(y-my)) ) )
f = f - 0.5_sp*N * log(s/N)
llf_boxcox_sp = -f
END FUNCTION llf_boxcox_sp
FUNCTION llf_boxcox_dp(lmbda, x)
IMPLICIT NONE
REAL(dp), INTENT(in) :: lmbda
REAL(dp), DIMENSION(:), INTENT(in) :: x
REAL(dp) :: llf_boxcox_dp
REAL(dp), DIMENSION(size(x)) :: y
REAL(dp) :: N, my, f, s
N = real(size(x),dp)
y = boxcox_dp(x, lmbda)
my = sum(y) / N
f = (lmbda-1.0_dp) * sum(log(x))
! f = f - 0.5_dp*N * log(sum((y-my)*(y-my))/N)
s = Min( huge(1.0_dp)/N , Max( N*tiny(1.0_dp) , sum((y-my)*(y-my)) ) )
f = f - 0.5_dp*N * log(s/N)
llf_boxcox_dp = -f
END FUNCTION llf_boxcox_dp
! ------------------------------------------------------------------
! Given a function func, and given distinct initial points ax and bx, this routine searches
! in the downhill direction (defined by the functionas evaluated at the initial points) and
! returns new points ax, bx, cx that bracket a minimum of the function.
! Parameters: GOLD is the default ratio by which successive intervals are magnified; GLIMIT
! is the maximum magnification allowed for a parabolic-fit step.
SUBROUTINE mnbrak_sp(ax, bx, cx, fa, fb, fc, func_sp, dat)
IMPLICIT NONE
REAL(sp), INTENT(inout) :: ax, bx
REAL(sp), INTENT(out) :: cx
REAL(sp), INTENT(out) :: fa, fb, fc
INTERFACE
FUNCTION func_sp(x, da)
USE mo_kind, ONLY: sp
IMPLICIT NONE
REAL(sp), INTENT(in) :: x
REAL(sp), DIMENSION(:), INTENT(in) :: da
REAL(sp) :: func_sp
END FUNCTION func_sp
END INTERFACE
REAL(sp), DIMENSION(:), INTENT(in) :: dat
REAL(sp), PARAMETER :: GOLD = 1.618034_sp
REAL(sp), PARAMETER :: GLIMIT = 100.0_sp
REAL(sp), PARAMETER :: TINY = 1.0e-20_sp
REAL(sp) :: fu, q, r, u, ulim
fa = func_sp(ax, dat)
fb = func_sp(bx, dat)
if (fb > fa) then
call swap_sp(ax,bx)
call swap_sp(fa,fb)
end if
cx = bx + GOLD*(bx-ax)
fc = func_sp(cx, dat)
do
if (fb < fc) return
r = (bx-ax)*(fb-fc)
q = (bx-cx)*(fb-fa)
u = bx-((bx-cx)*q-(bx-ax)*r)/(2.0_sp*sign(max(abs(q-r),TINY),q-r))
ulim = bx + GLIMIT*(cx-bx)
if ((bx-u)*(u-cx) > 0.0_sp) then
fu = func_sp(u, dat)
if (fu < fc) then
ax = bx
fa = fb
bx = u
fb = fu
return
else if (fu > fb) then
cx = u
fc = fu
return
end if
u = cx + GOLD*(cx-bx)
fu = func_sp(u, dat)
else if ((cx-u)*(u-ulim) > 0.0_sp) then
fu = func_sp(u, dat)
if (fu < fc) then
bx = cx
cx = u
u = cx + GOLD*(cx-bx)
call shft_sp(fb,fc,fu,func_sp(u, dat))
end if
else if (ge((u-ulim)*(ulim-cx),0.0_sp)) then
u = ulim
fu = func_sp(u, dat)
else
u = cx + GOLD*(cx-bx)
fu = func_sp(u, dat)
end if
call shft_sp(ax,bx,cx,u)
call shft_sp(fa,fb,fc,fu)
end do
END SUBROUTINE mnbrak_sp
SUBROUTINE mnbrak_dp(ax, bx, cx, fa, fb, fc, func_dp, dat)
IMPLICIT NONE
REAL(dp), INTENT(inout) :: ax, bx
REAL(dp), INTENT(out) :: cx
REAL(dp), INTENT(out) :: fa, fb, fc
INTERFACE
FUNCTION func_dp(x, da)
USE mo_kind, ONLY: dp
IMPLICIT NONE
REAL(dp), INTENT(in) :: x
REAL(dp), DIMENSION(:), INTENT(in) :: da
REAL(dp) :: func_dp
END FUNCTION func_dp
END INTERFACE
REAL(dp), DIMENSION(:), INTENT(in) :: dat
REAL(dp), PARAMETER :: GOLD = 1.618034_dp
REAL(dp), PARAMETER :: GLIMIT = 100.0_dp
REAL(dp), PARAMETER :: TINY = 1.0e-20_dp
REAL(dp) :: fu, q, r, u, ulim
fa = func_dp(ax, dat)
fb = func_dp(bx, dat)
if (fb > fa) then
call swap_dp(ax,bx)
call swap_dp(fa,fb)
end if
cx = bx + GOLD*(bx-ax)
fc = func_dp(cx, dat)
do
if (fb < fc) return
r = (bx-ax)*(fb-fc)
q = (bx-cx)*(fb-fa)
u = bx-((bx-cx)*q-(bx-ax)*r)/(2.0_dp*sign(max(abs(q-r),TINY),q-r))
ulim = bx + GLIMIT*(cx-bx)
if ((bx-u)*(u-cx) > 0.0_dp) then
fu = func_dp(u, dat)
if (fu < fc) then
ax = bx
fa = fb
bx = u
fb = fu
return
else if (fu > fb) then
cx = u
fc = fu
return
end if
u = cx + GOLD*(cx-bx)
fu = func_dp(u, dat)
else if ((cx-u)*(u-ulim) > 0.0_dp) then
fu = func_dp(u, dat)
if (fu < fc) then
bx = cx
cx = u
u = cx + GOLD*(cx-bx)
call shft_dp(fb,fc,fu,func_dp(u, dat))
end if
else if (ge((u-ulim)*(ulim-cx),0.0_dp)) then
u = ulim
fu = func_dp(u, dat)
else
u = cx + GOLD*(cx-bx)
fu = func_dp(u, dat)
end if
call shft_dp(ax,bx,cx,u)
call shft_dp(fa,fb,fc,fu)
end do
END SUBROUTINE mnbrak_dp
! ------------------------------------------------------------------
! Helper for brent and mnbrak
SUBROUTINE shft_sp(a,b,c,d)
IMPLICIT NONE
REAL(sp), INTENT(out) :: a
REAL(sp), INTENT(inout) :: b,c
REAL(sp), INTENT(in) :: d
a = b
b = c
c = d
END SUBROUTINE shft_sp
SUBROUTINE shft_dp(a,b,c,d)
IMPLICIT NONE
REAL(dp), INTENT(out) :: a
REAL(dp), INTENT(inout) :: b,c
REAL(dp), INTENT(in) :: d
a = b
b = c
c = d
END SUBROUTINE shft_dp
! ------------------------------------------------------------------
! Helper for mnbrak
SUBROUTINE swap_sp(a,b)
IMPLICIT NONE
REAL(sp), INTENT(inout) :: a, b
REAL(sp) :: dum
dum = a
a = b
b = dum
END SUBROUTINE swap_sp
SUBROUTINE swap_dp(a,b)
IMPLICIT NONE
REAL(dp), INTENT(inout) :: a, b
REAL(dp) :: dum
dum = a
a = b
b = dum
END SUBROUTINE swap_dp
END MODULE mo_boxcox