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statistical_methods.F
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!--------------------------------------------------------------------------------------------------!
! CP2K: A general program to perform molecular dynamics simulations !
! Copyright 2000-2024 CP2K developers group <https://cp2k.org> !
! !
! SPDX-License-Identifier: GPL-2.0-or-later !
!--------------------------------------------------------------------------------------------------!
! **************************************************************************************************
!> \brief Methods to perform on the fly statistical analysis of data
!> -) Schiferl and Wallace, J. Chem. Phys. 83 (10) 1985
!> \author Teodoro Laino (01.2007) [tlaino]
!> \par History
!> - Teodoro Laino (10.2008) [tlaino] - University of Zurich
!> module made publicly available
! **************************************************************************************************
MODULE statistical_methods
USE cp_log_handling, ONLY: cp_logger_get_default_io_unit
USE global_types, ONLY: global_environment_type
USE kinds, ONLY: dp
USE util, ONLY: sort
#include "./base/base_uses.f90"
IMPLICIT NONE
PRIVATE
CHARACTER(len=*), PARAMETER, PRIVATE :: moduleN = 'statistical_methods'
LOGICAL, PARAMETER :: debug_this_module = .FALSE.
INTEGER, PARAMETER, PUBLIC :: min_sample_size = 20
PUBLIC :: sw_test, &
k_test, &
vn_test
CONTAINS
! **************************************************************************************************
!> \brief Shapiro - Wilk's test or W-statistic to test normality of a distribution
!> R94 APPL. STATIST. (1995) VOL.44, NO.4
!> Calculates the Shapiro-Wilk W test and its significance level
!> \param ix ...
!> \param n ...
!> \param w ...
!> \param pw ...
!> \par History
!> Teodoro Laino (02.2007) [tlaino]
! **************************************************************************************************
SUBROUTINE sw_test(ix, n, w, pw)
REAL(KIND=dp), DIMENSION(:), POINTER :: ix
INTEGER, INTENT(IN) :: n
REAL(KIND=dp), INTENT(OUT) :: w, pw
REAL(KIND=dp), PARAMETER :: c1(6) = (/0.000000_dp, 0.221157_dp, -0.147981_dp, -2.071190_dp, &
4.434685_dp, -2.706056_dp/), c2(6) = (/0.000000_dp, 0.042981_dp, -0.293762_dp, &
-1.752461_dp, 5.682633_dp, -3.582633_dp/), &
c3(4) = (/0.544000_dp, -0.399780_dp, 0.025054_dp, -0.6714E-3_dp/), &
c4(4) = (/1.3822_dp, -0.77857_dp, 0.062767_dp, -0.0020322_dp/), &
c5(4) = (/-1.5861_dp, -0.31082_dp, -0.083751_dp, 0.0038915_dp/), &
c6(3) = (/-0.4803_dp, -0.082676_dp, 0.0030302_dp/), g(2) = (/-2.273_dp, 0.459_dp/), &
one = 1.0_dp, pi6 = 1.909859_dp, qtr = 0.25_dp, small = EPSILON(0.0_dp), &
sqrth = 0.70711_dp
REAL(KIND=dp), PARAMETER :: stqr = 1.047198_dp, th = 0.375_dp, two = 2.0_dp, zero = 0.0_dp
INTEGER :: i, i1, j, n2, output_unit
INTEGER, ALLOCATABLE, DIMENSION(:) :: itmp
LOGICAL :: failure
REAL(KIND=dp) :: a1, a2, an, an25, asa, fac, gamma, m, &
range, rsn, s, sa, sax, ssa, ssassx, &
ssumm2, ssx, summ2, sx, w1, xi, xsx, &
xx, y
REAL(KIND=dp), ALLOCATABLE, DIMENSION(:) :: a, x
failure = .FALSE.
output_unit = cp_logger_get_default_io_unit()
! Check for N < 3
IF (n < 3 .OR. n > 5000) THEN
IF (output_unit > 0) WRITE (output_unit, '(A)') &
"Shapiro Wilk test: number of points less than 3 or greated than 5000."
IF (output_unit > 0) WRITE (output_unit, '(A)') &
"Not able to perform the test!"
END IF
! Sort input array of data in ascending order
IF (MOD(n, 2) == 0) THEN
n2 = n/2
ELSE
n2 = (n - 1)/2
END IF
ALLOCATE (x(n))
ALLOCATE (itmp(n))
ALLOCATE (a(n2))
x(:) = ix
CALL sort(x, n, itmp)
! Check for zero range
range = x(n) - x(1)
IF (range < small) failure = .TRUE.
IF (failure .AND. (output_unit > 0)) THEN
WRITE (output_unit, '(A)') "Shapiro Wilk test: two data points are numerically identical."
WRITE (output_unit, '(A)') "Not able to perform the test!"
END IF
pw = one
w = one
an = n
! Calculates coefficients for the test
IF (n == 3) THEN
a(1) = sqrth
ELSE
an25 = an + qtr
summ2 = zero
DO i = 1, n2
CALL ppnd7((i - th)/an25, a(i))
summ2 = summ2 + a(i)**2
END DO
summ2 = summ2*two
ssumm2 = SQRT(summ2)
rsn = one/SQRT(an)
a1 = poly(c1, 6, rsn) - a(1)/ssumm2
! Normalize coefficients
IF (n > 5) THEN
i1 = 3
a2 = -a(2)/ssumm2 + poly(c2, 6, rsn)
fac = SQRT((summ2 - two*a(1)**2 - two*a(2)**2)/(one - two*a1**2 - two*a2**2))
a(1) = a1
a(2) = a2
ELSE
i1 = 2
fac = SQRT((summ2 - two*a(1)**2)/(one - two*a1**2))
a(1) = a1
END IF
DO i = i1, n2
a(i) = -a(i)/fac
END DO
END IF
! scaled X
xx = x(1)/range
sx = xx
sa = -a(1)
j = n - 1
DO i = 2, n
xi = x(i)/range
sx = sx + xi
IF (i /= j) sa = sa + SIGN(1, i - j)*a(MIN(i, j))
xx = xi
j = j - 1
END DO
! Calculate W statistic as squared correlation
! between data and coefficients
sa = sa/n
sx = sx/n
ssa = zero
ssx = zero
sax = zero
j = n
DO i = 1, n
IF (i /= j) THEN
asa = SIGN(1, i - j)*a(MIN(i, j)) - sa
ELSE
asa = -sa
END IF
xsx = x(i)/range - sx
ssa = ssa + asa*asa
ssx = ssx + xsx*xsx
sax = sax + asa*xsx
j = j - 1
END DO
! W1 equals (1-W) calculated to avoid excessive rounding error
! for W very near 1 (a potential problem in very large samples)
ssassx = SQRT(ssa*ssx)
w1 = (ssassx - sax)*(ssassx + sax)/(ssa*ssx)
w = one - w1
! Calculate significance level for W (exact for N=3)
IF (n == 3) THEN
pw = pi6*(ASIN(SQRT(w)) - stqr)
ELSE
y = LOG(w1)
xx = LOG(an)
m = zero
s = one
IF (n <= 11) THEN
gamma = poly(g, 2, an)
IF (y >= gamma) THEN
pw = small
ELSE
y = -LOG(gamma - y)
m = poly(c3, 4, an)
s = EXP(poly(c4, 4, an))
pw = alnorm((y - m)/s, .TRUE.)
END IF
ELSE
m = poly(c5, 4, xx)
s = EXP(poly(c6, 3, xx))
pw = alnorm((y - m)/s, .TRUE.)
END IF
END IF
DEALLOCATE (x)
DEALLOCATE (itmp)
DEALLOCATE (a)
END SUBROUTINE sw_test
! **************************************************************************************************
!> \brief Produces the normal deviate Z corresponding to a given lower tail area of P
!> Z is accurate to about 1 part in 10**7.
!> AS241 APPL. STATIST. (1988) VOL. 37, NO. 3, 477- 484.
!> \param p ...
!> \param normal_dev ...
!> \par History
!> Original version by Alain J. Miller - 1996
!> Teodoro Laino (02.2007) [tlaino]
! **************************************************************************************************
SUBROUTINE ppnd7(p, normal_dev)
REAL(KIND=dp), INTENT(IN) :: p
REAL(KIND=dp), INTENT(OUT) :: normal_dev
REAL(KIND=dp), PARAMETER :: a0 = 3.3871327179E+00_dp, a1 = 5.0434271938E+01_dp, &
a2 = 1.5929113202E+02_dp, a3 = 5.9109374720E+01_dp, b1 = 1.7895169469E+01_dp, &
b2 = 7.8757757664E+01_dp, b3 = 6.7187563600E+01_dp, c0 = 1.4234372777E+00_dp, &
c1 = 2.7568153900E+00_dp, c2 = 1.3067284816E+00_dp, c3 = 1.7023821103E-01_dp, &
const1 = 0.180625_dp, const2 = 1.6_dp, d1 = 7.3700164250E-01_dp, &
d2 = 1.2021132975E-01_dp, e0 = 6.6579051150E+00_dp, e1 = 3.0812263860E+00_dp, &
e2 = 4.2868294337E-01_dp, e3 = 1.7337203997E-02_dp, f1 = 2.4197894225E-01_dp, &
f2 = 1.2258202635E-02_dp, half = 0.5_dp, one = 1.0_dp
REAL(KIND=dp), PARAMETER :: split1 = 0.425_dp, split2 = 5.0_dp, zero = 0.0_dp
REAL(KIND=dp) :: q, r
q = p - half
IF (ABS(q) <= split1) THEN
r = const1 - q*q
normal_dev = q*(((a3*r + a2)*r + a1)*r + a0)/ &
(((b3*r + b2)*r + b1)*r + one)
RETURN
ELSE
IF (q < zero) THEN
r = p
ELSE
r = one - p
END IF
IF (r <= zero) THEN
normal_dev = zero
RETURN
END IF
r = SQRT(-LOG(r))
IF (r <= split2) THEN
r = r - const2
normal_dev = (((c3*r + c2)*r + c1)*r + c0)/((d2*r + d1)*r + one)
ELSE
r = r - split2
normal_dev = (((e3*r + e2)*r + e1)*r + e0)/((f2*r + f1)*r + one)
END IF
IF (q < zero) normal_dev = -normal_dev
RETURN
END IF
END SUBROUTINE ppnd7
! **************************************************************************************************
!> \brief Evaluates the tail area of the standardised normal curve
!> from x to infinity if upper is .true. or
!> from minus infinity to x if upper is .false.
!> AS66 Applied Statistics (1973) vol.22, no.3
!> \param x ...
!> \param upper ...
!> \return ...
!> \par History
!> Original version by Alain J. Miller - 1996
!> Teodoro Laino (02.2007) [tlaino]
! **************************************************************************************************
FUNCTION alnorm(x, upper) RESULT(fn_val)
REAL(KIND=dp), INTENT(IN) :: x
LOGICAL, INTENT(IN) :: upper
REAL(KIND=dp) :: fn_val
REAL(KIND=dp), PARAMETER :: a1 = 5.75885480458_dp, a2 = 2.62433121679_dp, &
a3 = 5.92885724438_dp, b1 = -29.8213557807_dp, b2 = 48.6959930692_dp, c1 = -3.8052E-8_dp, &
c2 = 3.98064794E-4_dp, c3 = -0.151679116635_dp, c4 = 4.8385912808_dp, &
c5 = 0.742380924027_dp, c6 = 3.99019417011_dp, con = 1.28_dp, d1 = 1.00000615302_dp, &
d2 = 1.98615381364_dp, d3 = 5.29330324926_dp, d4 = -15.1508972451_dp, &
d5 = 30.789933034_dp, half = 0.5_dp, ltone = 7.0_dp, one = 1.0_dp, p = 0.398942280444_dp, &
q = 0.39990348504_dp, r = 0.398942280385_dp, utzero = 18.66_dp, zero = 0.0_dp
LOGICAL :: up
REAL(KIND=dp) :: y, z
up = upper
z = x
IF (z < zero) THEN
up = .NOT. up
z = -z
END IF
IF (.NOT. (z <= ltone .OR. up .AND. z <= utzero)) THEN
fn_val = zero
IF (.NOT. up) fn_val = one - fn_val
RETURN
END IF
y = half*z*z
IF (z <= con) THEN
fn_val = r*EXP(-y)/(z + c1 + d1/(z + c2 + d2/(z + c3 + d3/(z + c4 + d4/(z + c5 + d5/(z + c6))))))
ELSE
fn_val = half - z*(p - q*y/(y + a1 + b1/(y + a2 + b2/(y + a3))))
END IF
IF (.NOT. up) fn_val = one - fn_val
END FUNCTION alnorm
! **************************************************************************************************
!> \brief Calculates the algebraic polynomial of order nored-1 with
!> array of coefficients c. Zero order coefficient is c(1)
!> AS 181.2 Appl. Statist. (1982) Vol. 31, No. 2
!> \param c ...
!> \param nord ...
!> \param x ...
!> \return ...
!> \par History
!> Original version by Alain J. Miller - 1996
!> Teodoro Laino (02.2007) [tlaino]
! **************************************************************************************************
FUNCTION poly(c, nord, x) RESULT(fn_val)
REAL(KIND=dp), INTENT(IN) :: c(:)
INTEGER, INTENT(IN) :: nord
REAL(KIND=dp), INTENT(IN) :: x
REAL(KIND=dp) :: fn_val
INTEGER :: i, j, n2
REAL(KIND=dp) :: p
fn_val = c(1)
IF (nord == 1) RETURN
p = x*c(nord)
IF (nord == 2) THEN
fn_val = fn_val + p
RETURN
END IF
n2 = nord - 2
j = n2 + 1
DO i = 1, n2
p = (p + c(j))*x
j = j - 1
END DO
fn_val = fn_val + p
END FUNCTION poly
! **************************************************************************************************
!> \brief Kandall's test for correlation
!> \param xdata ...
!> \param istart ...
!> \param n ...
!> \param tau ...
!> \param z ...
!> \param prob ...
!> \par History
!> Teodoro Laino (02.2007) [tlaino]
!> \note
!> tau: Kendall's Tau
!> z: number of std devs from 0 of tau
!> prob: tau's probability
! **************************************************************************************************
SUBROUTINE k_test(xdata, istart, n, tau, z, prob)
REAL(KIND=dp), DIMENSION(:), POINTER :: xdata
INTEGER, INTENT(IN) :: istart, n
REAL(KIND=dp) :: tau, z, prob
INTEGER :: is, j, k, nt
REAL(KIND=dp) :: a1, var
nt = n - istart + 1
IF (nt .GE. min_sample_size) THEN
is = 0
DO j = istart, n - 1
DO k = j + 1, n
a1 = xdata(j) - xdata(k)
IF (a1 .GT. 0.0_dp) THEN
is = is + 1
ELSE IF (a1 .LT. 0.0_dp) THEN
is = is - 1
END IF
END DO
END DO
tau = REAL(is, KIND=dp)
var = REAL(nt, KIND=dp)*REAL(nt - 1, KIND=dp)*REAL(2*nt + 5, KIND=dp)/18.0_dp
z = tau/SQRT(var)
prob = erf(ABS(z)/SQRT(2.0_dp))
ELSE
tau = 0.0_dp
z = 0.0_dp
prob = 1.0_dp
END IF
END SUBROUTINE k_test
! **************************************************************************************************
!> \brief Von Neumann test for serial correlation
!> \param xdata ...
!> \param n ...
!> \param r ...
!> \param u ...
!> \param prob ...
!> \par History
!> Teodoro Laino (02.2007) [tlaino]
! **************************************************************************************************
SUBROUTINE vn_test(xdata, n, r, u, prob)
REAL(KIND=dp), DIMENSION(:), POINTER :: xdata
INTEGER, INTENT(IN) :: n
REAL(KIND=dp) :: r, u, prob
INTEGER :: i
REAL(KIND=dp) :: q, s, var, x
IF (n .GE. min_sample_size) THEN
x = 0.0_dp
q = 0.0_dp
s = 0.0_dp
DO i = 1, n - 1
x = x + xdata(i)
q = q + (xdata(i + 1) - xdata(i))**2
END DO
x = x + xdata(n)
x = x/REAL(n, KIND=dp)
DO i = 1, n
s = s + (xdata(i) - x)**2
END DO
s = s/REAL(n - 1, KIND=dp)
q = q/REAL(2*(n - 1), KIND=dp)
r = q/s
var = SQRT(1.0_dp/REAL(n + 1, KIND=dp)*(1.0_dp + 1.0_dp/REAL(n - 1, KIND=dp)))
u = (r - 1.0_dp)/var
prob = erf(ABS(u)/SQRT(2.0_dp))
ELSE
r = 0.0_dp
u = 0.0_dp
prob = 1.0_dp
END IF
END SUBROUTINE vn_test
! **************************************************************************************************
!> \brief Performs tests on statistical methods
!> Debug use only
!> \param xdata ...
!> \param globenv ...
!> \par History
!> Teodoro Laino (02.2007) [tlaino]
! **************************************************************************************************
SUBROUTINE tests(xdata, globenv)
REAL(KIND=dp), DIMENSION(:), POINTER :: xdata
TYPE(global_environment_type), POINTER :: globenv
INTEGER :: i, n
REAL(KINd=dp) :: prob, pw, r, tau, u, w, z
REAL(KIND=dp), DIMENSION(:), POINTER :: ydata
IF (debug_this_module) THEN
n = 50 ! original sample size
NULLIFY (xdata)
ALLOCATE (xdata(n))
DO i = 1, 10
xdata(i) = 5.0_dp - REAL(i, KIND=dp)/2.0_dp + 0.1*globenv%gaussian_rng_stream%next()
WRITE (3, *) xdata(i)
END DO
DO i = 11, n
xdata(i) = 0.1*globenv%gaussian_rng_stream%next()
END DO
! Test for trend
DO i = 1, n
CALL k_test(xdata, i, n, tau, z, prob)
IF (prob <= 0.2_dp) EXIT
END DO
WRITE (*, *) "Mann-Kendall test", i
! Test for normality distribution and for serial correlation
DO i = 1, n
ALLOCATE (ydata(n - i + 1))
ydata = xdata(i:n)
CALL sw_test(ydata, n - i + 1, w, pw)
CALL vn_test(ydata, n - i + 1, r, u, prob)
WRITE (*, *) "Shapiro Wilks test", i, w, pw
WRITE (*, *) "Von Neu", i, r, u, prob
DEALLOCATE (ydata)
END DO
DEALLOCATE (xdata)
END IF
END SUBROUTINE tests
END MODULE statistical_methods