def exercise_ellipsoid(n_trials=100, n_sub_trials=10):
from gltbx import quadrics
rnd = random.Random(0)
for i in xrange(n_trials):
centre = matrix.col([ rnd.random() for k in xrange(3) ])
half_lengths = matrix.col([ 0.1 + rnd.random() for k in xrange(3) ])
r = scitbx.math.euler_angles_as_matrix(
[ rnd.uniform(0, 360) for i in xrange(3) ], deg=True)
metrics = r * matrix.diag([ x**2 for x in half_lengths ]) * r.transpose()
t = quadrics.ellipsoid_to_sphere_transform(centre, metrics.as_sym_mat3())
assert approx_equal(t.translation_part(), centre)
m = matrix.sqr(t.linear_part())
assert m.determinant() > 0
for j in xrange(n_sub_trials):
y = matrix.col([ rnd.random() for k in xrange(3) ])
c_y = y.transpose() * y
x = m*y
c_x = x.transpose() * metrics.inverse() * x
assert approx_equal(c_x, c_y)
r = scitbx.math.euler_angles_as_matrix((30, 115, 260), deg=True)
centre = matrix.col((-1, 2, 3))
metrics = r * matrix.diag((-1, 0.1, 1)) * r.transpose()
t = quadrics.ellipsoid_to_sphere_transform(centre, metrics.as_sym_mat3())
assert t.non_positive_definite()
x = r * matrix.col((1,0,0))
assert x.transpose() * metrics.inverse() * x > 0
def exercise_ellipsoid(n_trials=100, n_sub_trials=10):
from gltbx import quadrics
rnd = random.Random(0)
for i in range(n_trials):
centre = matrix.col([rnd.random() for k in range(3)])
half_lengths = matrix.col([0.1 + rnd.random() for k in range(3)])
r = scitbx.math.euler_angles_as_matrix(
[rnd.uniform(0, 360) for i in range(3)], deg=True)
metrics = r * matrix.diag([x**2 for x in half_lengths]) * r.transpose()
t = quadrics.ellipsoid_to_sphere_transform(centre,
metrics.as_sym_mat3())
assert approx_equal(t.translation_part(), centre)
m = matrix.sqr(t.linear_part())
assert m.determinant() > 0
for j in range(n_sub_trials):
y = matrix.col([rnd.random() for k in range(3)])
c_y = y.transpose() * y
x = m * y
c_x = x.transpose() * metrics.inverse() * x
assert approx_equal(c_x, c_y)
r = scitbx.math.euler_angles_as_matrix((30, 115, 260), deg=True)
centre = matrix.col((-1, 2, 3))
metrics = r * matrix.diag((-1, 0.1, 1)) * r.transpose()
t = quadrics.ellipsoid_to_sphere_transform(centre, metrics.as_sym_mat3())
assert t.non_positive_definite()
x = r * matrix.col((1, 0, 0))
assert x.transpose() * metrics.inverse() * x > 0
def print_eigen_values_and_vectors_of_observed_covariance(A, s0):
"""
Print the eigen values and vectors of a matrix
"""
# Compute the eigen decomposition of the covariance matrix
eigen_decomposition = linalg.eigensystem.real_symmetric(
A.as_flex_double_matrix())
Q = matrix.sqr(eigen_decomposition.vectors())
L = matrix.diag(eigen_decomposition.values())
# Print the matrix eigen values
logger.info("")
logger.info("Eigen Values:")
logger.info("")
print_matrix(L, indent=2)
logger.info("")
logger.info("Eigen Vectors:")
logger.info("")
print_matrix(Q, indent=2)
logger.info("")
logger.info("Observed covariance in degrees equivalent units")
logger.info("C1: %.5f degrees" % (sqrt(L[0]) * (180.0 / pi) / s0.length()))
logger.info("C2: %.5f degrees" % (sqrt(L[3]) * (180.0 / pi) / s0.length()))
def glm(x, p=None):
from math import exp, sqrt, log, factorial, lgamma
from scitbx import matrix
from scipy.stats import poisson
beta0 = 0
if p is None:
p = [1.0] * len(x)
X = matrix.rec([1] * len(x), (len(x), 1))
c = 1.345
while True:
n = beta0
mu = exp(n)
z = [n + (xx - mu) / mu for xx in x]
w = [pp * mu for pp in p]
r = [(xx - mu) / sqrt(mu) for xx in x]
w2 = [huber(rr, c) for rr in r]
W = matrix.diag(w)
# W2 = matrix.diag(w2)
Z = matrix.col(z)
beta = (X.transpose() * W * X).inverse() * X.transpose() * W * z
print beta
if abs(beta[0] - beta0) < 1e-3:
break
beta0 = beta[0]
# W12 = matrix.diag([sqrt(ww) for ww in w])
# H = W12 * X * (X.transpose() * W * X).inverse() * X.transpose() * W12
mu = exp(n)
return mu
def print_eigen_values_and_vectors(A):
"""
Print the eigen values and vectors of a matrix
"""
# Compute the eigen decomposition of the covariance matrix
eigen_decomposition = linalg.eigensystem.real_symmetric(
A.as_flex_double_matrix())
Q = matrix.sqr(eigen_decomposition.vectors())
L = matrix.diag(eigen_decomposition.values())
# Print the matrix eigen values
logger.info("")
logger.info("Eigen Values:")
logger.info("")
print_matrix(L, indent=2)
logger.info("")
logger.info("Eigen Vectors:")
logger.info("")
print_matrix(Q, indent=2)
logger.info("")
logger.info("Mosaicity in degrees equivalent units")
logger.info("M1: %.5f degrees" % (sqrt(L[0]) * 180.0 / pi))
logger.info("M2: %.5f degrees" % (sqrt(L[4]) * 180.0 / pi))
logger.info("M3: %.5f degrees" % (sqrt(L[8]) * 180.0 / pi))
def glm33(X, Y, B, P):
from math import exp , sqrt, log, factorial, lgamma
from scitbx import matrix
from scipy.stats import poisson
X = matrix.rec(list(X), X.all())
Y = matrix.col(list(Y))
B = matrix.col(list(B))
P = matrix.col(list(P))
while True:
n = X * B
mu = matrix.col([exp(nn) for nn in n])
z = [(yy - mui) / mui for yy, mui in zip(Y, mu)]
w = [pp * mui for pp, mui in zip(P, mu)]
W = matrix.diag(w)
Z = matrix.col(z)
delta = (X.transpose() * W * X).inverse() * X.transpose()*W*z
relE = sqrt(sum([d*d for d in delta])/max(1e-10, sum([d*d for d in B])))
B = B + delta
# print relE
if relE < 1e-3:
break
return B
def compute_mode(mu, sigma):
# Compute the eigen decomposition of the covariance matrix
eigen_decomposition = eigensystem.real_symmetric(
sigma.as_flex_double_matrix())
Q = matrix.sqr(eigen_decomposition.vectors())
L = matrix.diag(eigen_decomposition.values())
# To simplify calculation of the mode, we change basis
mu_prime = Q.transpose() * mu
# Compute the initial value of the lagrange multiplier
l0 = compute_l0(mu_prime.normalize(), mu_prime, L)
# Compute the value via Halley's method
while True:
mu1, mu2, mu3 = mu_prime
s1, s2, s3 = L[0], L[4], L[8]
U = f(l0, mu1, mu2, mu3, s1, s2, s3)
Up = df(l0, mu1, mu2, mu3, s1, s2, s3)
Up2 = d2f(l0, mu1, mu2, mu3, s1, s2, s3)
l = l0 - 2 * U * Up / (2 * Up**2 - U * Up2)
if abs(l - l0) < 1e-7:
break
l0 = l
# Compute the mode
x = matrix.col((
mu_prime[0] / (1 - l * L[0]),
mu_prime[1] / (1 - l * L[4]),
mu_prime[2] / (1 - l * L[8]),
))
# Now transform back into the original basis
return Q * x
def generate_sigma():
"""
Generate a random sigma
"""
# Random eigenvalues
L = matrix.diag((uniform(1e-5, 1e-4), uniform(1e-5,
1e-4), uniform(1e-5, 1e-4)))
# Random angles
ax = uniform(0, 2 * pi)
ay = uniform(0, 2 * pi)
az = uniform(0, 2 * pi)
# X rotation
Rx = matrix.sqr((1, 0, 0, 0, cos(ax), -sin(ax), 0, sin(ax), cos(ax)))
# Y rotation
Ry = matrix.sqr((cos(ay), 0, sin(ay), 0, 1, 0, -sin(ay), 0, cos(ay)))
# Z rotation
Rz = matrix.sqr((cos(az), -sin(az), 0, sin(az), cos(az), 0, 0, 0, 1))
# Eigen vectors
Q = Rz * Ry * Rx
# Compute sigma
sigma = Q * L * Q.inverse()
return sigma
def test_lefebvre():
"""Run the test presented in part 4 of the paper Lefebvre et al. (1999),
http://arxiv.org/abs/hep-ex/9909031."""
# Simple test following the paper. First MC sim
mat = matrix.sqr((0.7, 0.2, 0.4, 0.6))
sig_mat = mat * 0.01
p_mats = [perturb_mat(mat) for i in range(10000)]
inv_p_mats = [m.inverse() for m in p_mats]
# Here use the more exact formula for population covariance as we know the
# true means rather than sample covariance where we calculate the means
# from the data
cov_inv_mat_MC = calc_monte_carlo_population_covariances(
inv_p_mats, mean_matrix=mat.inverse())
# Now analytical formula
cov_mat = matrix.diag([e**2 for e in sig_mat])
cov_inv_mat = matrix_inverse_error_propagation(mat, cov_mat)
cov_inv_mat = cov_inv_mat.as_flex_double_matrix()
# Get fractional differences
frac = (cov_inv_mat_MC - cov_inv_mat) / cov_inv_mat_MC
# assert all fractional errors are within 5%. When the random seed is allowed
# to vary this fails about 7 times in every 100 runs. This is mainly due to
# the random variation in the MC method itself, as with just 10000 samples
# one MC estimate compared to another MC estimate regularly has some
# fractional errors greater than 5%.
assert all([abs(e) < 0.05 for e in frac])
return
def glm33(X, Y, B, P):
from math import exp, sqrt, log, factorial, lgamma
from scitbx import matrix
from scipy.stats import poisson
X = matrix.rec(list(X), X.all())
Y = matrix.col(list(Y))
B = matrix.col(list(B))
P = matrix.col(list(P))
while True:
n = X * B
mu = matrix.col([exp(nn) for nn in n])
z = [(yy - mui) / mui for yy, mui in zip(Y, mu)]
w = [pp * mui for pp, mui in zip(P, mu)]
W = matrix.diag(w)
Z = matrix.col(z)
delta = (X.transpose() * W * X).inverse() * X.transpose() * W * z
relE = sqrt(
sum([d * d for d in delta]) / max(1e-10, sum([d * d for d in B])))
B = B + delta
# print relE
if relE < 1e-3:
break
return B
def glm(x, p=None):
from math import exp, sqrt, log, factorial, lgamma
from scitbx import matrix
from scipy.stats import poisson
beta0 = 0
if p is None:
p = [1.0] * len(x)
X = matrix.rec([1] * len(x), (len(x), 1))
c = 1.345
while True:
n = beta0
mu = exp(n)
z = [n + (xx - mu) / mu for xx in x]
w = [pp * mu for pp in p]
r = [(xx - mu) / sqrt(mu) for xx in x]
w2 = [huber(rr, c) for rr in r]
W = matrix.diag(w)
# W2 = matrix.diag(w2)
Z = matrix.col(z)
beta = (X.transpose() * W * X).inverse() * X.transpose() * W * z
print(beta)
if abs(beta[0] - beta0) < 1e-3:
break
beta0 = beta[0]
# W12 = matrix.diag([sqrt(ww) for ww in w])
# H = W12 * X * (X.transpose() * W * X).inverse() * X.transpose() * W12
mu = exp(n)
return mu
def exercise_impl(svd_impl_name, use_fortran):
import scitbx.linalg
svd_impl = getattr(scitbx.linalg, "lapack_%s" % svd_impl_name)
from scitbx.array_family import flex
from scitbx import matrix
from libtbx.test_utils import approx_equal
#
for diag in [0, 1]:
for n in xrange(1, 11):
a = flex.double(flex.grid(n, n), 0)
for i in xrange(n):
a[(i, i)] = diag
a_inp = a.deep_copy()
svd = svd_impl(a=a, use_fortran=use_fortran)
if svd is None:
if not use_fortran:
print "Skipping tests: lapack_%s not available." % svd_impl_name
return
assert svd.info == 0
assert approx_equal(svd.s, [diag] * n)
assert svd.u.all() == (n, n)
assert svd.vt.all() == (n, n)
mt = flex.mersenne_twister(seed=0)
for m in xrange(1, 11):
for n in xrange(1, 11):
a = matrix.rec(elems=tuple(mt.random_double(m * n) * 4 - 2), n=(m, n))
svd = svd_impl(a=a.transpose().as_flex_double_matrix(), use_fortran=use_fortran)
assert svd.info == 0
sigma = matrix.diag(svd.s) # min(m,n) x min(m,n)
# FORTRAN layout, so transpose
u = matrix.rec(svd.u, svd.u.all()).transpose()
vt = matrix.rec(svd.vt, svd.vt.all()).transpose()
assert approx_equal(u * sigma * vt, a)
#
a = matrix.rec(elems=[0.47, 0.10, -0.21, -0.21, -0.03, 0.35], n=(3, 2))
svd = svd_impl(a=a.transpose().as_flex_double_matrix(), use_fortran=use_fortran)
assert svd.info == 0
assert approx_equal(svd.s, [0.55981345199567534, 0.35931726783538481])
# again remember column-major storage
assert approx_equal(
svd.u,
[
0.81402078804155853,
-0.5136261274467826,
0.27121644094748704,
-0.42424674329757839,
-0.20684171439391938,
0.88160717215094342,
],
)
assert approx_equal(svd.vt, [0.8615633693608673, -0.50765003750177129, 0.50765003750177129, 0.8615633693608673])
def exercise_impl(svd_impl_name, use_fortran):
import scitbx.linalg
svd_impl = getattr(scitbx.linalg, "lapack_%s" % svd_impl_name)
from scitbx.array_family import flex
from scitbx import matrix
from libtbx.test_utils import approx_equal
#
for diag in [0, 1]:
for n in range(1, 11):
a = flex.double(flex.grid(n, n), 0)
for i in range(n):
a[(i, i)] = diag
a_inp = a.deep_copy()
svd = svd_impl(a=a, use_fortran=use_fortran)
if (svd is None):
if (not use_fortran):
print("Skipping tests: lapack_%s not available." %
svd_impl_name)
return
assert svd.info == 0
assert approx_equal(svd.s, [diag] * n)
assert svd.u.all() == (n, n)
assert svd.vt.all() == (n, n)
mt = flex.mersenne_twister(seed=0)
for m in range(1, 11):
for n in range(1, 11):
a = matrix.rec(elems=tuple(mt.random_double(m * n) * 4 - 2),
n=(m, n))
svd = svd_impl(a=a.transpose().as_flex_double_matrix(),
use_fortran=use_fortran)
assert svd.info == 0
sigma = matrix.diag(svd.s) # min(m,n) x min(m,n)
# FORTRAN layout, so transpose
u = matrix.rec(svd.u, svd.u.all()).transpose()
vt = matrix.rec(svd.vt, svd.vt.all()).transpose()
assert approx_equal(u * sigma * vt, a)
#
a = matrix.rec(elems=[0.47, 0.10, -0.21, -0.21, -0.03, 0.35], n=(3, 2))
svd = svd_impl(a=a.transpose().as_flex_double_matrix(),
use_fortran=use_fortran)
assert svd.info == 0
assert approx_equal(svd.s, [0.55981345199567534, 0.35931726783538481])
# again remember column-major storage
assert approx_equal(svd.u, [
0.81402078804155853, -0.5136261274467826, 0.27121644094748704,
-0.42424674329757839, -0.20684171439391938, 0.88160717215094342
])
assert approx_equal(svd.vt, [
0.8615633693608673, -0.50765003750177129, 0.50765003750177129,
0.8615633693608673
])
def exercise_tensor_rank_2_orth_and_frac_linear_maps():
from cctbx import adptbx, sgtbx
p1 = sgtbx.space_group_info('P1')
for i in range(100):
uc = p1.any_compatible_unit_cell(27)
u_star = matrix.col.random(n=6, a=0, b=1)
u_iso_ref = adptbx.u_star_as_u_iso(uc, u_star)
u_iso = matrix.col(uc.u_star_to_u_iso_linear_form()).dot(u_star)
assert approx_equal(u_iso, u_iso_ref, eps=1e-15)
u_cart_ref = adptbx.u_star_as_u_cart(uc, u_star)
u_cart = matrix.sqr(uc.u_star_to_u_cart_linear_map()) * u_star
assert approx_equal(u_cart, u_cart_ref, eps=1e-15)
u_cif_ref = adptbx.u_star_as_u_cif(uc, u_star)
u_cif = matrix.diag(uc.u_star_to_u_cif_linear_map()) * (u_star)
assert approx_equal(u_cif, u_cif_ref)
def exercise_tensor_rank_2_orth_and_frac_linear_maps():
from cctbx import adptbx, sgtbx
p1 = sgtbx.space_group_info('P1')
for i in xrange(100):
uc = p1.any_compatible_unit_cell(27)
u_star = matrix.col.random(n=6, a=0, b=1)
u_iso_ref = adptbx.u_star_as_u_iso(uc, u_star)
u_iso = matrix.col(uc.u_star_to_u_iso_linear_form()).dot(u_star)
assert approx_equal(u_iso, u_iso_ref, eps=1e-15)
u_cart_ref = adptbx.u_star_as_u_cart(uc, u_star)
u_cart = matrix.sqr(uc.u_star_to_u_cart_linear_map()) * u_star
assert approx_equal(u_cart, u_cart_ref, eps=1e-15)
u_cif_ref = adptbx.u_star_as_u_cif(uc, u_star)
u_cif = matrix.diag(uc.u_star_to_u_cif_linear_map())*(u_star)
assert approx_equal(u_cif, u_cif_ref)
def __init__(self, experiment, reflections):
from dials.algorithms.refinement.parameterisation.crystal_parameters import (
CrystalUnitCellParameterisation, )
from dials.algorithms.refinement.parameterisation.crystal_parameters import (
CrystalOrientationParameterisation, )
from dials_scratch.jmp.stills import Model
from dials.array_family import flex
from scitbx import simplex
from math import sqrt
# Store the input
self.experiment = experiment
self.reflections = reflections
self.crystal = self.experiment.crystal
# Get the current values and generate some offsets
values = flex.double((0.001, 0, 0.001, 0, 0, 0.001))
offset = flex.double([0.001 for v in values])
# The optimization history
self.history = []
# Get the initial cell and initial score
M = matrix.sqr((values[0], 0, 0, values[1], values[2], 0, values[3],
values[4], values[5]))
initial_sigma = M * M.transpose()
initial_score = self.target(values)
self.sigma = initial_sigma
# Perform the optimization
optimizer = simple_simplex(values, offset, self, 2000)
result = optimizer.get_solution()
M = matrix.sqr((result[0], 0, 0, result[1], result[2], 0, result[3],
result[4], result[5]))
self.sigma = M * M.transpose()
print("Initial sigma:", initial_sigma)
print("Final sigma: ", self.sigma)
# Compute the eigen decomposition of the covariance matrix
eigen_decomposition = eigensystem.real_symmetric(
self.sigma.as_flex_double_matrix())
Q = matrix.sqr(eigen_decomposition.vectors())
L = matrix.diag(eigen_decomposition.values())
print(Q)
print(L)
def test_B_matrix():
"""Test errors in an inverted B matrix, when the errors in B are known (and
are independent)"""
Bmat = create_Bmat()
inv_Bmat = Bmat.inverse()
# Note that elements in the strict upper triangle of B are zero
# so their perturbed values will be zero too and the errors in these elements
# will also be zero (this is what we want)
# calculate the covariance of elements of inv_B by Monte Carlo simulation,
# assuming that each element of B has an independent normal error given by a
# sigma of 1% of the element value
perturbed_Bmats = [
perturb_mat(Bmat, fraction_sd=0.01) for i in range(10000)
]
invBs = [m.inverse() for m in perturbed_Bmats]
cov_invB_MC = calc_monte_carlo_population_covariances(invBs, inv_Bmat)
# Now calculate using the analytical formula. First need the covariance
# matrix of B itself. This is the diagonal matrix of errors applied in the
# simulation.
n = Bmat.n_rows()
sig_B = Bmat * 0.01
cov_B = matrix.diag([e**2 for e in sig_B])
# Now can use the analytical formula
cov_invB = matrix_inverse_error_propagation(Bmat, cov_B)
cov_invB = cov_invB.as_flex_double_matrix()
# Get fractional differences
frac = flex.double(flex.grid(n**2, n**2), 0.0)
for i in range(frac.all()[0]):
for j in range(frac.all()[1]):
e1 = cov_invB_MC[i, j]
e2 = cov_invB[i, j]
if e1 < 1e-10: continue # avoid divide-by-zero errors below
if e2 < 1e-10:
continue # avoid elements that are supposed to be zero
frac[i, j] = (e1 - e2) / e1
# assert all fractional errors are within 5%
assert all([abs(e) < 0.05 for e in frac])
return
def exercise_eigen_core(diag):
u = adptbx.random_rotate_ellipsoid(diag + [0., 0., 0.])
ev = list(adptbx.eigenvalues(u))
diag.sort()
ev.sort()
for i in xrange(3):
check_eigenvalue(u, ev[i])
for i in xrange(3):
assert abs(diag[i] - ev[i]) < 1.e-4
if (adptbx.is_positive_definite(ev)):
es = adptbx.eigensystem(u)
ev = list(es.values())
ev.sort()
for i in xrange(3):
check_eigenvalue(u, ev[i])
for i in xrange(3):
assert abs(diag[i] - ev[i]) < 1.e-4
evec = []
for i in xrange(3):
check_eigenvector(u, es.values()[i], es.vectors(i))
evec.extend(es.vectors(i))
return # XXX following tests disabled for the moment
# sometimes fail if eigenvalues are very similar but not identical
sqrt_eval = matrix.diag(flex.sqrt(flex.double(es.values())))
evec = matrix.sqr(evec).transpose()
sqrt_u = evec * sqrt_eval * evec.transpose()
u_full = matrix.sym(sym_mat3=u).elems
assert approx_equal(u_full, (sqrt_u.transpose() * sqrt_u).elems,
eps=1.e-3)
assert approx_equal(u_full, (sqrt_u * sqrt_u.transpose()).elems,
eps=1.e-3)
assert approx_equal(u_full, (sqrt_u * sqrt_u).elems, eps=1.e-3)
sqrt_u_plus_shifts = matrix.sym(sym_mat3=[
x + 10 * (random.random() - .5) for x in sqrt_u.as_sym_mat3()
])
sts = (sqrt_u_plus_shifts.transpose() *
sqrt_u_plus_shifts).as_sym_mat3()
ev = adptbx.eigenvalues(sts)
assert min(ev) >= 0
sts = (sqrt_u_plus_shifts *
sqrt_u_plus_shifts.transpose()).as_sym_mat3()
ev = adptbx.eigenvalues(sts)
assert min(ev) >= 0
sts = (sqrt_u_plus_shifts * sqrt_u_plus_shifts).as_sym_mat3()
ev = adptbx.eigenvalues(sts)
assert min(ev) >= 0
def __init__(self, nb, bf=1, skew=0, taper=1):
"""
% autoTree Create System Models of Kinematic Trees
% autoTree(nb,bf,skew,taper) creates system models of kinematic trees
% having revolute joints. nb and bf specify the number of bodies in the
% tree, and the branching factor, respectively. The latter is the average
% number of children of a nonterminal node, and must be >=1. bf=1 produces
% an unbranched tree; bf=2 produces a binary tree; and non-integer values
% produce trees in which the number of children alternates between
% floor(bf) and ceil(bf) in such a way that the average is bf. Trees are
% constructed (and numbered) breadth-first. Link i is a thin-walled
% cylindrical tube of length l(i), radius l(i)/20, and mass m(i), lying
% between 0 and l(i) on the x axis of its local coordinate system. The
% values of l(i) and m(i) are determined by the tapering coefficient:
% l(i)=taper^(i-1) and m(i)=taper^(3*(i-1)). Thus, if taper=1 then
% m(i)=l(i)=1 for all i. The inboard joint axis of link i lies on the
% local z axis, and its outboard axis passes through the point (l(i),0,0)
% and is rotated about the x axis by an angle of skew radians relative to
% the inboard axis. If the link has more than one outboard joint then they
% all have the same axis. If skew=0 then the mechanism is planar. The
% final one, two or three arguments can be omitted, in which case they
% assume default values of taper=1, skew=0 and bf=1.
"""
self.NB = nb
self.pitch = [0] * nb
self.parent = [None] * nb
self.Xtree = []
self.I = []
len_ = []
for i in xrange(nb):
self.parent[i] = ifloor((i - 1 + math.ceil(bf)) / bf) - 1
if (self.parent[i] == -1):
self.Xtree.append(Xtrans([0, 0, 0]))
else:
self.Xtree.append(
Xrotx(skew) * Xtrans([len_[self.parent[i]], 0, 0]))
len_.append(taper**i)
mass = taper**(3 * i)
CoM = len_[i] * matrix.col([0.5, 0, 0])
Icm = mass * len_[i]**2 * matrix.diag(
[0.0025, 1.015 / 12, 1.015 / 12])
self.I.append(mcI(mass, CoM, Icm))
def __init__(self, nb, bf=1, skew=0, taper=1):
"""
% autoTree Create System Models of Kinematic Trees
% autoTree(nb,bf,skew,taper) creates system models of kinematic trees
% having revolute joints. nb and bf specify the number of bodies in the
% tree, and the branching factor, respectively. The latter is the average
% number of children of a nonterminal node, and must be >=1. bf=1 produces
% an unbranched tree; bf=2 produces a binary tree; and non-integer values
% produce trees in which the number of children alternates between
% floor(bf) and ceil(bf) in such a way that the average is bf. Trees are
% constructed (and numbered) breadth-first. Link i is a thin-walled
% cylindrical tube of length l(i), radius l(i)/20, and mass m(i), lying
% between 0 and l(i) on the x axis of its local coordinate system. The
% values of l(i) and m(i) are determined by the tapering coefficient:
% l(i)=taper^(i-1) and m(i)=taper^(3*(i-1)). Thus, if taper=1 then
% m(i)=l(i)=1 for all i. The inboard joint axis of link i lies on the
% local z axis, and its outboard axis passes through the point (l(i),0,0)
% and is rotated about the x axis by an angle of skew radians relative to
% the inboard axis. If the link has more than one outboard joint then they
% all have the same axis. If skew=0 then the mechanism is planar. The
% final one, two or three arguments can be omitted, in which case they
% assume default values of taper=1, skew=0 and bf=1.
"""
self.NB = nb
self.pitch = [0] * nb
self.parent = [None] * nb
self.Xtree = []
self.I = []
len_ = []
for i in xrange(nb):
self.parent[i] = ifloor((i-1+math.ceil(bf))/bf)-1
if (self.parent[i] == -1):
self.Xtree.append(Xtrans([0,0,0]))
else:
self.Xtree.append(Xrotx(skew) * Xtrans([len_[self.parent[i]],0,0]))
len_.append(taper**i)
mass = taper**(3*i)
CoM = len_[i] * matrix.col([0.5,0,0])
Icm = mass * len_[i]**2 * matrix.diag([0.0025,1.015/12,1.015/12])
self.I.append(mcI(mass, CoM, Icm))
def exercise_eigen_core(diag):
u = adptbx.random_rotate_ellipsoid(diag + [0.0, 0.0, 0.0])
ev = list(adptbx.eigenvalues(u))
diag.sort()
ev.sort()
for i in xrange(3):
check_eigenvalue(u, ev[i])
for i in xrange(3):
assert abs(diag[i] - ev[i]) < 1.0e-4
if adptbx.is_positive_definite(ev):
es = adptbx.eigensystem(u)
ev = list(es.values())
ev.sort()
for i in xrange(3):
check_eigenvalue(u, ev[i])
for i in xrange(3):
assert abs(diag[i] - ev[i]) < 1.0e-4
evec = []
for i in xrange(3):
check_eigenvector(u, es.values()[i], es.vectors(i))
evec.extend(es.vectors(i))
return # XXX following tests disabled for the moment
# sometimes fail if eigenvalues are very similar but not identical
sqrt_eval = matrix.diag(flex.sqrt(flex.double(es.values())))
evec = matrix.sqr(evec).transpose()
sqrt_u = evec * sqrt_eval * evec.transpose()
u_full = matrix.sym(sym_mat3=u).elems
assert approx_equal(u_full, (sqrt_u.transpose() * sqrt_u).elems, eps=1.0e-3)
assert approx_equal(u_full, (sqrt_u * sqrt_u.transpose()).elems, eps=1.0e-3)
assert approx_equal(u_full, (sqrt_u * sqrt_u).elems, eps=1.0e-3)
sqrt_u_plus_shifts = matrix.sym(sym_mat3=[x + 10 * (random.random() - 0.5) for x in sqrt_u.as_sym_mat3()])
sts = (sqrt_u_plus_shifts.transpose() * sqrt_u_plus_shifts).as_sym_mat3()
ev = adptbx.eigenvalues(sts)
assert min(ev) >= 0
sts = (sqrt_u_plus_shifts * sqrt_u_plus_shifts.transpose()).as_sym_mat3()
ev = adptbx.eigenvalues(sts)
assert min(ev) >= 0
sts = (sqrt_u_plus_shifts * sqrt_u_plus_shifts).as_sym_mat3()
ev = adptbx.eigenvalues(sts)
assert min(ev) >= 0
def glm33(x, p):
from math import exp, sqrt, log, factorial, lgamma
from scitbx import matrix
from scipy.stats import poisson
beta0 = 0
while True:
n = beta0
mu = exp(n)
z = [(xx - mu) / mu for xx in x]
w = [pp * mu for pp in p]
W = matrix.diag(w)
Z = matrix.col(z)
X = matrix.rec([1] * len(z), (len(z), 1))
H = X * (X.transpose() * W * X).inverse() * X.transpose() * W
H = [H[i + i * len(z)] for i in range(len(z))]
MSE = sum((xx - mu)**2 for xx in x) / len(x)
r = [xx - mu for xx in x]
D = [rr * rr / (1 * MSE) * (hh / (1 - hh)**2) for rr, hh in zip(r, H)]
N = sum(1 for d in D if d > 4 / len(x))
# print X.transpose()*W*z
# print (W*X)[0]
# print (X.transpose()*W*X).inverse()[0]
delta = (X.transpose() * W * X).inverse() * X.transpose() * W * z
# print delta
relE = sqrt(
sum([d * d
for d in delta]) / max(1e-10, sum([d * d for d in [beta0]])))
beta0 = beta0 + delta[0]
# print relE
if relE < 1e-3:
break
return exp(beta0)
def core(a):
c = flex.double(a)
c.resize(flex.grid(a.n))
u = c.matrix_upper_triangle_as_packed_u()
gwm = scitbx.linalg.gill_murray_wright_cholesky_decomposition_in_place(
u, epsilon=1.e-8)
assert gwm.epsilon == 1.e-8
u = c.matrix_upper_triangle_as_packed_u()
gwm = scitbx.linalg.gill_murray_wright_cholesky_decomposition_in_place(
u)
assert gwm.epsilon == scitbx.math.floating_point_epsilon_double_get()
assert gwm.packed_u.id() == u.id()
p, e = gwm.pivots, gwm.e
r = matrix.sqr(u.matrix_packed_u_as_upper_triangle())
if a.n != (0, 0):
rtr = r.transpose() * r
pm = p_as_mx(p)
papt = pm * a * pm.transpose()
paept = papt + matrix.diag(e)
delta_decomposition = scitbx.linalg.matrix_equality_ratio(
paept, rtr)
assert delta_decomposition < 10, delta_decomposition
b = flex.random_double(size=a.n[0], factor=2) - 1
x = gwm.solve(b=b)
px = pm * matrix.col(x)
pb = pm * matrix.col(b)
if 0:
eigen = scitbx.linalg.eigensystem.real_symmetric(
paept.as_flex_double_matrix())
lambda_ = eigen.values()
print "condition number: ", lambda_[0] / lambda_[-1]
delta_solve = scitbx.linalg.matrix_cholesky_test_ratio(
a=paept.as_flex_double_matrix(),
x=flex.double(px),
b=flex.double(pb),
epsilon=gwm.epsilon)
assert delta_solve < 10, delta_solve
return p, e, r
def core(a):
c = flex.double(a)
c.resize(flex.grid(a.n))
u = c.matrix_upper_triangle_as_packed_u()
gwm = scitbx.linalg.gill_murray_wright_cholesky_decomposition_in_place(
u,
epsilon=1.e-8)
assert gwm.epsilon == 1.e-8
u = c.matrix_upper_triangle_as_packed_u()
gwm = scitbx.linalg.gill_murray_wright_cholesky_decomposition_in_place(u)
assert gwm.epsilon == scitbx.math.floating_point_epsilon_double_get()
assert gwm.packed_u.id() == u.id()
p, e = gwm.pivots, gwm.e
r = matrix.sqr(u.matrix_packed_u_as_upper_triangle())
if a.n != (0,0):
rtr = r.transpose() * r
pm = p_as_mx(p)
papt = pm * a * pm.transpose()
paept = papt + matrix.diag(e)
delta_decomposition = scitbx.linalg.matrix_equality_ratio(paept, rtr)
assert delta_decomposition < 10, delta_decomposition
b = flex.random_double(size=a.n[0], factor=2)-1
x = gwm.solve(b=b)
px = pm * matrix.col(x)
pb = pm * matrix.col(b)
if 0:
eigen = scitbx.linalg.eigensystem.real_symmetric(
paept.as_flex_double_matrix())
lambda_ = eigen.values()
print "condition number: ", lambda_[0]/lambda_[-1]
delta_solve = scitbx.linalg.matrix_cholesky_test_ratio(
a=paept.as_flex_double_matrix(),
x=flex.double(px),
b=flex.double(pb),
epsilon=gwm.epsilon)
assert delta_solve < 10, delta_solve
return p, e, r
def glm33(x, p):
from math import exp , sqrt, log, factorial, lgamma
from scitbx import matrix
from scipy.stats import poisson
beta0 = 0
while True:
n = beta0
mu = exp(n)
z = [(xx - mu) / mu for xx in x]
w = [pp * mu for pp in p]
W = matrix.diag(w)
Z = matrix.col(z)
X = matrix.rec([1]*len(z), (len(z),1))
H = X*(X.transpose() *W* X).inverse() * X.transpose()*W
H = [H[i+i*len(z)] for i in range(len(z))]
MSE = sum((xx-mu)**2 for xx in x) / len(x)
r = [xx - mu for xx in x]
D = [rr*rr / (1 * MSE) * (hh / (1-hh)**2) for rr, hh in zip(r, H)]
N = sum(1 for d in D if d > 4 / len(x))
# print X.transpose()*W*z
# print (W*X)[0]
# print (X.transpose()*W*X).inverse()[0]
delta = (X.transpose() * W * X).inverse() * X.transpose()*W*z
# print delta
relE = sqrt(sum([d*d for d in delta])/max(1e-10, sum([d*d for d in [beta0]])))
beta0 = beta0 + delta[0]
# print relE
if relE < 1e-3:
break
return exp(beta0)
def glm2(y):
from math import sqrt, exp, floor, log
from scipy.stats import poisson
from scitbx import matrix
from dials.array_family import flex
y = flex.double(y)
x = flex.double([1.0 for yy in y])
w = flex.double([1.0 for yy in y])
X = matrix.rec(x, (len(x), 1))
c = 1.345
beta = matrix.col([0])
maxiter = 10
accuracy = 1e-3
for iter in range(maxiter):
ni = flex.double([1.0 for xx in x])
sni = flex.sqrt(ni)
eta = flex.double(X * beta)
mu = flex.exp(eta)
dmu_deta = flex.exp(eta)
Vmu = mu
sVF = flex.sqrt(Vmu)
residP = (y - mu) * sni / sVF
phi = 1
sV = sVF * sqrt(phi)
residPS = residP / sqrt(phi)
H = flex.floor(mu * ni - c * sni * sV)
K = flex.floor(mu * ni + c * sni * sV)
# print min(H)
dpH = flex.double([poisson(mui).pmf(Hi) for mui, Hi in zip(mu, H)])
dpH1 = flex.double([poisson(mui).pmf(Hi - 1) for mui, Hi in zip(mu, H)])
dpK = flex.double([poisson(mui).pmf(Ki) for mui, Ki in zip(mu, K)])
dpK1 = flex.double([poisson(mui).pmf(Ki - 1) for mui, Ki in zip(mu, K)])
pHm1 = flex.double([poisson(mui).cdf(Hi - 1) for mui, Hi in zip(mu, H)])
pKm1 = flex.double([poisson(mui).cdf(Ki - 1) for mui, Ki in zip(mu, K)])
pH = pHm1 + dpH # = ppois(H,*)
pK = pKm1 + dpK # = ppois(K,*)
E2f = mu * (dpH1 - dpH - dpK1 + dpK) + pKm1 - pHm1
Epsi = c * (1.0 - pK - pH) + (mu / sV) * (dpH - dpK)
Epsi2 = c * c * (pH + 1.0 - pK) + E2f
EpsiS = c * (dpH + dpK) + E2f / sV
psi = flex.double([huber(rr, c) for rr in residPS])
cpsi = psi - Epsi
temp = cpsi * w * sni / sV * dmu_deta
EEqMat = [0] * len(X)
for j in range(X.n_rows()):
for i in range(X.n_columns()):
k = i + j * X.n_columns()
EEqMat[k] = X[k] * temp[j]
EEqMat = matrix.rec(EEqMat, (X.n_rows(), X.n_columns()))
EEq = []
for i in range(EEqMat.n_columns()):
col = []
for j in range(EEqMat.n_rows()):
k = i + j * EEqMat.n_columns()
col.append(EEqMat[k])
EEq.append(sum(col) / len(col))
EEq = matrix.col(EEq)
DiagB = EpsiS / (sni * sV) * w * (ni * dmu_deta) ** 2
B = matrix.diag(DiagB)
H = (X.transpose() * B * X) / len(y)
dbeta = H.inverse() * EEq
beta_new = beta + dbeta
relE = sqrt(sum([d * d for d in dbeta]) / max(1e-10, sum([d * d for d in beta])))
beta = beta_new
# print relE
if relE < accuracy:
break
weights = [min(1, c / abs(r)) for r in residPS]
eta = flex.double(X * beta)
mu = flex.exp(eta)
return beta
def glm(x, p=None):
from math import exp, sqrt, log, factorial, lgamma
from scitbx import matrix
from scipy.stats import poisson
beta0 = 0
if p is None:
p = [1.0] * len(x)
while True:
n = beta0
mu = exp(n)
z = [n + (xx - mu) / mu for xx in x]
w = [pp * mu for pp in p]
W = matrix.diag(w)
Z = matrix.col(z)
X = matrix.rec([1] * len(z), (len(z), 1))
beta = (X.transpose() * W * X).inverse() * X.transpose() * W * z
if abs(beta[0] - beta0) < 1e-3:
break
beta0 = beta[0]
n = beta[0]
mu = exp(n)
z = [n + (xx - mu) / mu for xx in x]
w = [pp * mu for pp in p]
W = matrix.diag(w)
Z = matrix.col(z)
X = matrix.rec([1] * len(z), (len(z), 1))
W12 = matrix.diag([sqrt(ww) for ww in w])
H = X * (X.transpose() * W * X).inverse() * X.transpose()
r = [(xx - mu) / mu for xx in x]
sum1 = 0
sum2 = 0
sum3 = 0
D = []
for xx in x:
d = 0
if xx != 0:
d += xx * log(xx)
d -= xx * log(mu)
d -= xx - mu
D.append(2 * d)
# r = []
# for xx in x:
# if xx == 0:
# r.append(0)
# else:
# r.append(2*(log(xx) - log(mu)))
# D = []
# for i in range(len(r)):
# h = H[i+i*H.n[0]]
# d = (r[i]/(1 - h))**2 * h/1.0
# D.append(d)
# print min(x), max(x)
print(min(D), max(D), F(1, len(x)))
return exp(beta[0]), max(D) > F(1, len(x))
def exercise_cif_from_cctbx():
quartz = xray.structure(crystal_symmetry=crystal.symmetry(
(5.01, 5.01, 5.47, 90, 90, 120), "P6222"),
scatterers=flex.xray_scatterer([
xray.scatterer("Si", (1 / 2., 1 / 2., 1 / 3.)),
xray.scatterer("O", (0.197, -0.197, 0.83333))
]))
for sc in quartz.scatterers():
sc.flags.set_grad_site(True)
s = StringIO()
loop = geometry.distances_as_cif_loop(
quartz.pair_asu_table(distance_cutoff=2),
site_labels=quartz.scatterers().extract_labels(),
sites_frac=quartz.sites_frac()).loop
print >> s, loop
assert not show_diff(
s.getvalue(), """\
loop_
_geom_bond_atom_site_label_1
_geom_bond_atom_site_label_2
_geom_bond_distance
_geom_bond_site_symmetry_2
Si O 1.6160 4_554
Si O 1.6160 2_554
Si O 1.6160 3_664
Si O 1.6160 5_664
""")
s = StringIO()
loop = geometry.angles_as_cif_loop(
quartz.pair_asu_table(distance_cutoff=2),
site_labels=quartz.scatterers().extract_labels(),
sites_frac=quartz.sites_frac()).loop
print >> s, loop
assert not show_diff(
s.getvalue(), """\
loop_
_geom_angle_atom_site_label_1
_geom_angle_atom_site_label_2
_geom_angle_atom_site_label_3
_geom_angle
_geom_angle_site_symmetry_1
_geom_angle_site_symmetry_3
O Si O 101.3 2_554 4_554
O Si O 111.3 3_664 4_554
O Si O 116.1 3_664 2_554
O Si O 116.1 5_664 4_554
O Si O 111.3 5_664 2_554
O Si O 101.3 5_664 3_664
Si O Si 146.9 3 5
""")
# with a covariance matrix
flex.set_random_seed(1)
vcv_matrix = matrix.diag(
flex.random_double(size=quartz.n_parameters(), factor=1e-5))\
.as_flex_double_matrix().matrix_symmetric_as_packed_u()
s = StringIO()
loop = geometry.distances_as_cif_loop(
quartz.pair_asu_table(distance_cutoff=2),
site_labels=quartz.scatterers().extract_labels(),
sites_frac=quartz.sites_frac(),
covariance_matrix=vcv_matrix,
parameter_map=quartz.parameter_map()).loop
print >> s, loop
assert not show_diff(
s.getvalue(), """\
loop_
_geom_bond_atom_site_label_1
_geom_bond_atom_site_label_2
_geom_bond_distance
_geom_bond_site_symmetry_2
Si O 1.616(14) 4_554
Si O 1.616(12) 2_554
Si O 1.616(14) 3_664
Si O 1.616(12) 5_664
""")
s = StringIO()
loop = geometry.angles_as_cif_loop(
quartz.pair_asu_table(distance_cutoff=2),
site_labels=quartz.scatterers().extract_labels(),
sites_frac=quartz.sites_frac(),
covariance_matrix=vcv_matrix,
parameter_map=quartz.parameter_map()).loop
print >> s, loop
assert not show_diff(
s.getvalue(), """\
loop_
_geom_angle_atom_site_label_1
_geom_angle_atom_site_label_2
_geom_angle_atom_site_label_3
_geom_angle
_geom_angle_site_symmetry_1
_geom_angle_site_symmetry_3
O Si O 101.3(8) 2_554 4_554
O Si O 111.3(10) 3_664 4_554
O Si O 116.1(9) 3_664 2_554
O Si O 116.1(9) 5_664 4_554
O Si O 111.3(10) 5_664 2_554
O Si O 101.3(8) 5_664 3_664
Si O Si 146.9(9) 3 5
""")
cell_vcv = flex.pow2(matrix.diag(flex.random_double(size=6,factor=1e-1))\
.as_flex_double_matrix().matrix_symmetric_as_packed_u())
s = StringIO()
loop = geometry.distances_as_cif_loop(
quartz.pair_asu_table(distance_cutoff=2),
site_labels=quartz.scatterers().extract_labels(),
sites_frac=quartz.sites_frac(),
covariance_matrix=vcv_matrix,
cell_covariance_matrix=cell_vcv,
parameter_map=quartz.parameter_map()).loop
print >> s, loop
assert not show_diff(
s.getvalue(), """\
loop_
_geom_bond_atom_site_label_1
_geom_bond_atom_site_label_2
_geom_bond_distance
_geom_bond_site_symmetry_2
Si O 1.616(15) 4_554
Si O 1.616(19) 2_554
Si O 1.616(15) 3_664
Si O 1.616(19) 5_664
""")
def compute(self, reflections):
print("Computing bbox for %d reflections" % len(reflections))
# Compute quantile
quantile = chisq_quantile(2, 0.997)
D = sqrt(quantile) * 2
bbox = flex.int6()
print("ML: %f" % D)
for i in range(len(reflections)):
s1 = matrix.col(reflections[i]["s1"])
s2 = matrix.col(reflections[i]["s2"])
s0 = matrix.col(self.experiment.beam.get_s0())
# Ensure our values are ok
assert s1.length() > 0
sigma = MosaicityParameterisation(self.parameters).sigma()
R = compute_change_of_basis_operation(s0, s2)
S = R * sigma * R.transpose()
mu = R * s2
assert abs(1 - mu.normalize().dot(matrix.col((0, 0, 1)))) < 1e-7
S11 = matrix.sqr((S[0], S[1], S[3], S[4]))
S12 = matrix.col((S[2], S[5]))
S21 = matrix.col((S[6], S[7])).transpose()
S22 = S[8]
mu1 = matrix.col((mu[0], mu[1]))
mu2 = mu[2]
mubar = mu1 + S12 * (1 / S22) * (s0.length() - mu2)
Sbar = S11 - S12 * (1 / S22) * S21
eigen_decomposition = eigensystem.real_symmetric(
Sbar.as_flex_double_matrix())
Q = matrix.sqr(eigen_decomposition.vectors())
L = matrix.diag(eigen_decomposition.values())
max_L = max(L)
delta = sqrt(max_L) * D
p1 = mubar + matrix.col((-delta, -delta))
p2 = mubar + matrix.col((-delta, +delta))
p3 = mubar + matrix.col((+delta, -delta))
p4 = mubar + matrix.col((+delta, +delta))
p1 = matrix.col((p1[0], p1[1], s0.length())).normalize()
p2 = matrix.col((p2[0], p2[1], s0.length())).normalize()
p3 = matrix.col((p3[0], p3[1], s0.length())).normalize()
p4 = matrix.col((p4[0], p4[1], s0.length())).normalize()
x1 = R.transpose() * p1
x2 = R.transpose() * p2
x3 = R.transpose() * p3
x4 = R.transpose() * p4
xy1 = self.experiment.detector[0].get_ray_intersection_px(x1)
xy2 = self.experiment.detector[0].get_ray_intersection_px(x2)
xy3 = self.experiment.detector[0].get_ray_intersection_px(x3)
xy4 = self.experiment.detector[0].get_ray_intersection_px(x4)
xx = (xy1[0], xy2[0], xy3[0], xy4[0])
yy = (xy1[1], xy2[1], xy3[1], xy4[1])
x0, x1 = int(floor(min(xx))) - 1, int(ceil(max(xx))) + 1
y0, y1 = int(floor(min(yy))) - 1, int(ceil(max(yy))) + 1
assert x1 > x0
assert y1 > y0
bbox.append((x0, x1, y0, y1, 0, 1))
reflections["bbox"] = bbox
x0, x1, y0, y1, _, _ = bbox.parts()
xsize, ysize = self.experiment.detector[0].get_image_size()
selection = (x1 > 0) & (y1 > 0) & (x0 < xsize) & (y0 < ysize)
reflections = reflections.select(selection)
print("Filtered reflecions with bbox outside image range")
print("Kept %d reflections" % len(reflections))
return reflections
def exercise_hbond_as_cif_loop():
xs = sucrose()
for sc in xs.scatterers():
sc.flags.set_grad_site(True)
radii = [
covalent_radii.table(elt).radius()
for elt in xs.scattering_type_registry().type_index_pairs_as_dict()
]
asu_mappings = xs.asu_mappings(buffer_thickness=2 * max(radii) + 0.5)
pair_asu_table = crystal.pair_asu_table(asu_mappings)
pair_asu_table.add_covalent_pairs(xs.scattering_types(), tolerance=0.5)
hbonds = [
geometry.hbond(1, 5, sgtbx.rt_mx('-X,0.5+Y,2-Z')),
geometry.hbond(5, 14, sgtbx.rt_mx('-X,-0.5+Y,1-Z')),
geometry.hbond(7, 10, sgtbx.rt_mx('1+X,+Y,+Z')),
geometry.hbond(10, 0),
geometry.hbond(12, 14, sgtbx.rt_mx('-1-X,0.5+Y,1-Z')),
geometry.hbond(14, 12, sgtbx.rt_mx('-1-X,-0.5+Y,1-Z')),
geometry.hbond(16, 7)
]
loop = geometry.hbonds_as_cif_loop(hbonds,
pair_asu_table,
xs.scatterers().extract_labels(),
sites_frac=xs.sites_frac()).loop
s = StringIO()
print >> s, loop
assert not show_diff(
s.getvalue(), """\
loop_
_geom_hbond_atom_site_label_D
_geom_hbond_atom_site_label_H
_geom_hbond_atom_site_label_A
_geom_hbond_distance_DH
_geom_hbond_distance_HA
_geom_hbond_distance_DA
_geom_hbond_angle_DHA
_geom_hbond_site_symmetry_A
O2 H2 O4 0.8200 2.0636 2.8635 165.0 2_557
O4 H4 O9 0.8200 2.0559 2.8736 174.9 2_546
O5 H5 O7 0.8200 2.0496 2.8589 169.0 1_655
O7 H7 O1 0.8200 2.0573 2.8617 166.8 .
O8 H8 O9 0.8200 2.1407 2.8943 152.8 2_456
O9 H9 O8 0.8200 2.1031 2.8943 162.1 2_446
O10 H10 O5 0.8200 2.0167 2.7979 159.1 .
""")
# with a covariance matrix
flex.set_random_seed(1)
vcv_matrix = matrix.diag(
flex.random_double(size=xs.n_parameters(), factor=1e-5))\
.as_flex_double_matrix().matrix_symmetric_as_packed_u()
loop = geometry.hbonds_as_cif_loop(hbonds,
pair_asu_table,
xs.scatterers().extract_labels(),
sites_frac=xs.sites_frac(),
covariance_matrix=vcv_matrix,
parameter_map=xs.parameter_map()).loop
s = StringIO()
print >> s, loop
assert not show_diff(
s.getvalue(), """\
loop_
_geom_hbond_atom_site_label_D
_geom_hbond_atom_site_label_H
_geom_hbond_atom_site_label_A
_geom_hbond_distance_DH
_geom_hbond_distance_HA
_geom_hbond_distance_DA
_geom_hbond_angle_DHA
_geom_hbond_site_symmetry_A
O2 H2 O4 0.82(3) 2.06(3) 2.86(3) 165.0(18) 2_557
O4 H4 O9 0.82(4) 2.06(4) 2.87(4) 175(2) 2_546
O5 H5 O7 0.82(2) 2.05(2) 2.859(19) 169.0(18) 1_655
O7 H7 O1 0.82(2) 2.06(2) 2.86(2) 167(2) .
O8 H8 O9 0.82(3) 2.14(3) 2.89(3) 153(3) 2_456
O9 H9 O8 0.82(3) 2.10(3) 2.89(3) 162(2) 2_446
O10 H10 O5 0.82(3) 2.02(3) 2.80(3) 159(3) .
""")
cell_vcv = flex.pow2(matrix.diag(flex.random_double(size=6,factor=1e-1))\
.as_flex_double_matrix().matrix_symmetric_as_packed_u())
loop = geometry.hbonds_as_cif_loop(hbonds,
pair_asu_table,
xs.scatterers().extract_labels(),
sites_frac=xs.sites_frac(),
covariance_matrix=vcv_matrix,
cell_covariance_matrix=cell_vcv,
parameter_map=xs.parameter_map()).loop
s = StringIO()
print >> s, loop
assert not show_diff(
s.getvalue(), """\
loop_
_geom_hbond_atom_site_label_D
_geom_hbond_atom_site_label_H
_geom_hbond_atom_site_label_A
_geom_hbond_distance_DH
_geom_hbond_distance_HA
_geom_hbond_distance_DA
_geom_hbond_angle_DHA
_geom_hbond_site_symmetry_A
O2 H2 O4 0.82(3) 2.06(4) 2.86(4) 165.0(18) 2_557
O4 H4 O9 0.82(4) 2.06(4) 2.87(4) 175(2) 2_546
O5 H5 O7 0.82(2) 2.05(2) 2.86(2) 169.0(18) 1_655
O7 H7 O1 0.82(2) 2.06(3) 2.86(3) 167(2) .
O8 H8 O9 0.82(3) 2.14(4) 2.89(4) 153(3) 2_456
O9 H9 O8 0.82(3) 2.10(3) 2.89(4) 162(2) 2_446
O10 H10 O5 0.82(3) 2.02(3) 2.80(3) 159(3) .
""")
def glm2(y):
from math import sqrt, exp, floor, log
from scipy.stats import poisson
from scitbx import matrix
from dials.array_family import flex
y = flex.double(y)
x = flex.double([1.0 for yy in y])
w = flex.double([1.0 for yy in y])
X = matrix.rec(x, (len(x), 1))
c = 1.345
beta = matrix.col([0])
maxiter = 10
accuracy = 1e-3
for iter in range(maxiter):
ni = flex.double([1.0 for xx in x])
sni = flex.sqrt(ni)
eta = flex.double(X * beta)
mu = flex.exp(eta)
dmu_deta = flex.exp(eta)
Vmu = mu
sVF = flex.sqrt(Vmu)
residP = (y - mu) * sni / sVF
phi = 1
sV = sVF * sqrt(phi)
residPS = residP / sqrt(phi)
H = flex.floor(mu * ni - c * sni * sV)
K = flex.floor(mu * ni + c * sni * sV)
# print min(H)
dpH = flex.double([poisson(mui).pmf(Hi) for mui, Hi in zip(mu, H)])
dpH1 = flex.double(
[poisson(mui).pmf(Hi - 1) for mui, Hi in zip(mu, H)])
dpK = flex.double([poisson(mui).pmf(Ki) for mui, Ki in zip(mu, K)])
dpK1 = flex.double(
[poisson(mui).pmf(Ki - 1) for mui, Ki in zip(mu, K)])
pHm1 = flex.double(
[poisson(mui).cdf(Hi - 1) for mui, Hi in zip(mu, H)])
pKm1 = flex.double(
[poisson(mui).cdf(Ki - 1) for mui, Ki in zip(mu, K)])
pH = pHm1 + dpH # = ppois(H,*)
pK = pKm1 + dpK # = ppois(K,*)
E2f = mu * (dpH1 - dpH - dpK1 + dpK) + pKm1 - pHm1
Epsi = c * (1.0 - pK - pH) + (mu / sV) * (dpH - dpK)
Epsi2 = c * c * (pH + 1.0 - pK) + E2f
EpsiS = c * (dpH + dpK) + E2f / sV
psi = flex.double([huber(rr, c) for rr in residPS])
cpsi = psi - Epsi
temp = cpsi * w * sni / sV * dmu_deta
EEqMat = [0] * len(X)
for j in range(X.n_rows()):
for i in range(X.n_columns()):
k = i + j * X.n_columns()
EEqMat[k] = X[k] * temp[j]
EEqMat = matrix.rec(EEqMat, (X.n_rows(), X.n_columns()))
EEq = []
for i in range(EEqMat.n_columns()):
col = []
for j in range(EEqMat.n_rows()):
k = i + j * EEqMat.n_columns()
col.append(EEqMat[k])
EEq.append(sum(col) / len(col))
EEq = matrix.col(EEq)
DiagB = EpsiS / (sni * sV) * w * (ni * dmu_deta)**2
B = matrix.diag(DiagB)
H = (X.transpose() * B * X) / len(y)
dbeta = H.inverse() * EEq
beta_new = beta + dbeta
relE = sqrt(
sum([d * d for d in dbeta]) / max(1e-10, sum([d * d
for d in beta])))
beta = beta_new
# print relE
if relE < accuracy:
break
weights = [min(1, c / abs(r)) for r in residPS]
eta = flex.double(X * beta)
mu = flex.exp(eta)
return beta
def exercise_gill_murray_wright_cholesky_decomposition():
def p_as_mx(p):
n = len(p)
m = flex.double(flex.grid(n,n))
m.matrix_diagonal_set_in_place(1)
for i in xrange(n):
if p[i] != i:
m.matrix_swap_rows_in_place(i, p[i])
return matrix.sqr(m)
def core(a):
c = flex.double(a)
c.resize(flex.grid(a.n))
u = c.matrix_upper_triangle_as_packed_u()
gwm = scitbx.linalg.gill_murray_wright_cholesky_decomposition_in_place(
u,
epsilon=1.e-8)
assert gwm.epsilon == 1.e-8
u = c.matrix_upper_triangle_as_packed_u()
gwm = scitbx.linalg.gill_murray_wright_cholesky_decomposition_in_place(u)
assert gwm.epsilon == scitbx.math.floating_point_epsilon_double_get()
assert gwm.packed_u.id() == u.id()
p, e = gwm.pivots, gwm.e
r = matrix.sqr(u.matrix_packed_u_as_upper_triangle())
if a.n != (0,0):
rtr = r.transpose() * r
pm = p_as_mx(p)
papt = pm * a * pm.transpose()
paept = papt + matrix.diag(e)
delta_decomposition = scitbx.linalg.matrix_equality_ratio(paept, rtr)
assert delta_decomposition < 10, delta_decomposition
b = flex.random_double(size=a.n[0], factor=2)-1
x = gwm.solve(b=b)
px = pm * matrix.col(x)
pb = pm * matrix.col(b)
if 0:
eigen = scitbx.linalg.eigensystem.real_symmetric(
paept.as_flex_double_matrix())
lambda_ = eigen.values()
print "condition number: ", lambda_[0]/lambda_[-1]
delta_solve = scitbx.linalg.matrix_cholesky_test_ratio(
a=paept.as_flex_double_matrix(),
x=flex.double(px),
b=flex.double(pb),
epsilon=gwm.epsilon)
assert delta_solve < 10, delta_solve
return p, e, r
# empty matrix
a = matrix.sqr([])
p, e, r = core(a)
assert p.size() == 0
assert e.size() == 0
assert len(r) == 0
n_max = 15
n_trials_per_n = 10
# identity matrices
for n in xrange(1,n_max+1):
a = matrix.diag([1]*n)
p, e, r = core(a)
assert list(p) == range(n)
assert approx_equal(e, [0]*n)
assert approx_equal(r, a)
# null matrices
for n in xrange(1,n_max+1):
a = matrix.sqr([0]*n*n)
p, e, r = core(a)
assert list(p) == range(n)
assert list(e) == [scitbx.math.floating_point_epsilon_double_get()]*n
for i in xrange(n):
for j in xrange(n):
if (i != j): r(i,j) == 0
else: r(i,j) == r(0,0)
# random semi-positive diagonal matrices
for n in xrange(1,n_max+1):
for i_trial in xrange(n_trials_per_n):
a = matrix.diag(flex.random_double(size=n))
p, e, r = core(a)
assert approx_equal(e, [0]*n)
for i in xrange(n):
for j in xrange(n):
if (i != j): approx_equal(r(i,j), 0)
# random diagonal matrices
for n in xrange(1,n_max+1):
for i_trial in xrange(n_trials_per_n):
a = matrix.diag(flex.random_double(size=n, factor=2)-1)
p, e, r = core(a)
for i in xrange(n):
for j in xrange(n):
if (i != j): approx_equal(r(i,j), 0)
# random semi-positive definite matrices
for n in xrange(1,n_max+1):
for i_trial in xrange(n_trials_per_n):
m = matrix.sqr(flex.random_double(size=n*n, factor=2)-1)
a = m.transpose_multiply()
p, e, r = core(a)
assert approx_equal(e, [0]*n)
# random matrices
for n in xrange(1,n_max+1):
size = n*(n+1)//2
for i_trial in xrange(n_trials_per_n):
a = (flex.random_double(size=size, factor=2)-1) \
.matrix_packed_u_as_symmetric()
core(matrix.sqr(a))
a.matrix_diagonal_set_in_place(0)
core(matrix.sqr(a))
# J. Nocedal and S. Wright:
# Numerical Optimization.
# Springer, New York, 1999, pp. 145-150.
for i in xrange(3):
for j in xrange(3):
a = flex.double([[4,2,1],[2,6,3],[1,3,-0.004]])
a.matrix_swap_rows_in_place(i=i, j=j)
a.matrix_swap_columns_in_place(i=i, j=j)
p, e, r = core(matrix.sqr(a))
if (i == 0 and j == 0):
assert list(p) == [1,1,2] # swap row 0 and 1 and nothing else
assert approx_equal(e, [0.0, 0.0, 3.008])
assert approx_equal(r,
[2.4494897427831779, 0.81649658092772592, 1.2247448713915889,
0.0, 1.8257418583505538, 0.0,
0.0, 0.0, 1.2263767773404712])
def exercise_cif_from_cctbx():
quartz = xray.structure(
crystal_symmetry=crystal.symmetry(
(5.01,5.01,5.47,90,90,120), "P6222"),
scatterers=flex.xray_scatterer([
xray.scatterer("Si", (1/2.,1/2.,1/3.)),
xray.scatterer("O", (0.197,-0.197,0.83333))]))
for sc in quartz.scatterers():
sc.flags.set_grad_site(True)
s = StringIO()
loop = geometry.distances_as_cif_loop(
quartz.pair_asu_table(distance_cutoff=2),
site_labels=quartz.scatterers().extract_labels(),
sites_frac=quartz.sites_frac()).loop
print >> s, loop
assert not show_diff(s.getvalue(), """\
loop_
_geom_bond_atom_site_label_1
_geom_bond_atom_site_label_2
_geom_bond_distance
_geom_bond_site_symmetry_2
Si O 1.6160 4_554
Si O 1.6160 2_554
Si O 1.6160 3_664
Si O 1.6160 5_664
""")
s = StringIO()
loop = geometry.angles_as_cif_loop(
quartz.pair_asu_table(distance_cutoff=2),
site_labels=quartz.scatterers().extract_labels(),
sites_frac=quartz.sites_frac()).loop
print >> s, loop
assert not show_diff(s.getvalue(), """\
loop_
_geom_angle_atom_site_label_1
_geom_angle_atom_site_label_2
_geom_angle_atom_site_label_3
_geom_angle
_geom_angle_site_symmetry_1
_geom_angle_site_symmetry_3
O Si O 101.3 2_554 4_554
O Si O 111.3 3_664 4_554
O Si O 116.1 3_664 2_554
O Si O 116.1 5_664 4_554
O Si O 111.3 5_664 2_554
O Si O 101.3 5_664 3_664
Si O Si 146.9 3 5
""")
# with a covariance matrix
flex.set_random_seed(1)
vcv_matrix = matrix.diag(
flex.random_double(size=quartz.n_parameters(), factor=1e-5))\
.as_flex_double_matrix().matrix_symmetric_as_packed_u()
s = StringIO()
loop = geometry.distances_as_cif_loop(
quartz.pair_asu_table(distance_cutoff=2),
site_labels=quartz.scatterers().extract_labels(),
sites_frac=quartz.sites_frac(),
covariance_matrix=vcv_matrix,
parameter_map=quartz.parameter_map()).loop
print >> s, loop
assert not show_diff(s.getvalue(), """\
loop_
_geom_bond_atom_site_label_1
_geom_bond_atom_site_label_2
_geom_bond_distance
_geom_bond_site_symmetry_2
Si O 1.616(14) 4_554
Si O 1.616(12) 2_554
Si O 1.616(14) 3_664
Si O 1.616(12) 5_664
""")
s = StringIO()
loop = geometry.angles_as_cif_loop(
quartz.pair_asu_table(distance_cutoff=2),
site_labels=quartz.scatterers().extract_labels(),
sites_frac=quartz.sites_frac(),
covariance_matrix=vcv_matrix,
parameter_map=quartz.parameter_map()).loop
print >> s, loop
assert not show_diff(s.getvalue(), """\
loop_
_geom_angle_atom_site_label_1
_geom_angle_atom_site_label_2
_geom_angle_atom_site_label_3
_geom_angle
_geom_angle_site_symmetry_1
_geom_angle_site_symmetry_3
O Si O 101.3(8) 2_554 4_554
O Si O 111.3(10) 3_664 4_554
O Si O 116.1(9) 3_664 2_554
O Si O 116.1(9) 5_664 4_554
O Si O 111.3(10) 5_664 2_554
O Si O 101.3(8) 5_664 3_664
Si O Si 146.9(9) 3 5
""")
cell_vcv = flex.pow2(matrix.diag(flex.random_double(size=6,factor=1e-1))\
.as_flex_double_matrix().matrix_symmetric_as_packed_u())
s = StringIO()
loop = geometry.distances_as_cif_loop(
quartz.pair_asu_table(distance_cutoff=2),
site_labels=quartz.scatterers().extract_labels(),
sites_frac=quartz.sites_frac(),
covariance_matrix=vcv_matrix,
cell_covariance_matrix=cell_vcv,
parameter_map=quartz.parameter_map()).loop
print >> s, loop
assert not show_diff(s.getvalue(), """\
loop_
_geom_bond_atom_site_label_1
_geom_bond_atom_site_label_2
_geom_bond_distance
_geom_bond_site_symmetry_2
Si O 1.616(15) 4_554
Si O 1.616(19) 2_554
Si O 1.616(15) 3_664
Si O 1.616(19) 5_664
""")
def exercise_hbond_as_cif_loop():
xs = sucrose()
for sc in xs.scatterers():
sc.flags.set_grad_site(True)
radii = [
covalent_radii.table(elt).radius() for elt in
xs.scattering_type_registry().type_index_pairs_as_dict() ]
asu_mappings = xs.asu_mappings(
buffer_thickness=2*max(radii) + 0.5)
pair_asu_table = crystal.pair_asu_table(asu_mappings)
pair_asu_table.add_covalent_pairs(
xs.scattering_types(),
tolerance=0.5)
hbonds = [
geometry.hbond(1,5, sgtbx.rt_mx('-X,0.5+Y,2-Z')),
geometry.hbond(5,14, sgtbx.rt_mx('-X,-0.5+Y,1-Z')),
geometry.hbond(7,10, sgtbx.rt_mx('1+X,+Y,+Z')),
geometry.hbond(10,0),
geometry.hbond(12,14, sgtbx.rt_mx('-1-X,0.5+Y,1-Z')),
geometry.hbond(14,12, sgtbx.rt_mx('-1-X,-0.5+Y,1-Z')),
geometry.hbond(16,7)
]
loop = geometry.hbonds_as_cif_loop(
hbonds, pair_asu_table, xs.scatterers().extract_labels(),
sites_frac=xs.sites_frac()).loop
s = StringIO()
print >> s, loop
assert not show_diff(s.getvalue(), """\
loop_
_geom_hbond_atom_site_label_D
_geom_hbond_atom_site_label_H
_geom_hbond_atom_site_label_A
_geom_hbond_distance_DH
_geom_hbond_distance_HA
_geom_hbond_distance_DA
_geom_hbond_angle_DHA
_geom_hbond_site_symmetry_A
O2 H2 O4 0.8200 2.0636 2.8635 165.0 2_557
O4 H4 O9 0.8200 2.0559 2.8736 174.9 2_546
O5 H5 O7 0.8200 2.0496 2.8589 169.0 1_655
O7 H7 O1 0.8200 2.0573 2.8617 166.8 .
O8 H8 O9 0.8200 2.1407 2.8943 152.8 2_456
O9 H9 O8 0.8200 2.1031 2.8943 162.1 2_446
O10 H10 O5 0.8200 2.0167 2.7979 159.1 .
""")
# with a covariance matrix
flex.set_random_seed(1)
vcv_matrix = matrix.diag(
flex.random_double(size=xs.n_parameters(), factor=1e-5))\
.as_flex_double_matrix().matrix_symmetric_as_packed_u()
loop = geometry.hbonds_as_cif_loop(
hbonds, pair_asu_table, xs.scatterers().extract_labels(),
sites_frac=xs.sites_frac(),
covariance_matrix=vcv_matrix,
parameter_map=xs.parameter_map()).loop
s = StringIO()
print >> s, loop
assert not show_diff(s.getvalue(), """\
loop_
_geom_hbond_atom_site_label_D
_geom_hbond_atom_site_label_H
_geom_hbond_atom_site_label_A
_geom_hbond_distance_DH
_geom_hbond_distance_HA
_geom_hbond_distance_DA
_geom_hbond_angle_DHA
_geom_hbond_site_symmetry_A
O2 H2 O4 0.82(3) 2.06(3) 2.86(3) 165.0(18) 2_557
O4 H4 O9 0.82(4) 2.06(4) 2.87(4) 175(2) 2_546
O5 H5 O7 0.82(2) 2.05(2) 2.859(19) 169.0(18) 1_655
O7 H7 O1 0.82(2) 2.06(2) 2.86(2) 167(2) .
O8 H8 O9 0.82(3) 2.14(3) 2.89(3) 153(3) 2_456
O9 H9 O8 0.82(3) 2.10(3) 2.89(3) 162(2) 2_446
O10 H10 O5 0.82(3) 2.02(3) 2.80(3) 159(3) .
""")
cell_vcv = flex.pow2(matrix.diag(flex.random_double(size=6,factor=1e-1))\
.as_flex_double_matrix().matrix_symmetric_as_packed_u())
loop = geometry.hbonds_as_cif_loop(
hbonds, pair_asu_table, xs.scatterers().extract_labels(),
sites_frac=xs.sites_frac(),
covariance_matrix=vcv_matrix,
cell_covariance_matrix=cell_vcv,
parameter_map=xs.parameter_map()).loop
s = StringIO()
print >> s, loop
assert not show_diff(s.getvalue(), """\
loop_
_geom_hbond_atom_site_label_D
_geom_hbond_atom_site_label_H
_geom_hbond_atom_site_label_A
_geom_hbond_distance_DH
_geom_hbond_distance_HA
_geom_hbond_distance_DA
_geom_hbond_angle_DHA
_geom_hbond_site_symmetry_A
O2 H2 O4 0.82(3) 2.06(4) 2.86(4) 165.0(18) 2_557
O4 H4 O9 0.82(4) 2.06(4) 2.87(4) 175(2) 2_546
O5 H5 O7 0.82(2) 2.05(2) 2.86(2) 169.0(18) 1_655
O7 H7 O1 0.82(2) 2.06(3) 2.86(3) 167(2) .
O8 H8 O9 0.82(3) 2.14(4) 2.89(4) 153(3) 2_456
O9 H9 O8 0.82(3) 2.10(3) 2.89(4) 162(2) 2_446
O10 H10 O5 0.82(3) 2.02(3) 2.80(3) 159(3) .
""")
def exercise_gill_murray_wright_cholesky_decomposition():
def p_as_mx(p):
n = len(p)
m = flex.double(flex.grid(n, n))
m.matrix_diagonal_set_in_place(1)
for i in xrange(n):
if p[i] != i:
m.matrix_swap_rows_in_place(i, p[i])
return matrix.sqr(m)
def core(a):
c = flex.double(a)
c.resize(flex.grid(a.n))
u = c.matrix_upper_triangle_as_packed_u()
gwm = scitbx.linalg.gill_murray_wright_cholesky_decomposition_in_place(
u, epsilon=1.e-8)
assert gwm.epsilon == 1.e-8
u = c.matrix_upper_triangle_as_packed_u()
gwm = scitbx.linalg.gill_murray_wright_cholesky_decomposition_in_place(
u)
assert gwm.epsilon == scitbx.math.floating_point_epsilon_double_get()
assert gwm.packed_u.id() == u.id()
p, e = gwm.pivots, gwm.e
r = matrix.sqr(u.matrix_packed_u_as_upper_triangle())
if a.n != (0, 0):
rtr = r.transpose() * r
pm = p_as_mx(p)
papt = pm * a * pm.transpose()
paept = papt + matrix.diag(e)
delta_decomposition = scitbx.linalg.matrix_equality_ratio(
paept, rtr)
assert delta_decomposition < 10, delta_decomposition
b = flex.random_double(size=a.n[0], factor=2) - 1
x = gwm.solve(b=b)
px = pm * matrix.col(x)
pb = pm * matrix.col(b)
if 0:
eigen = scitbx.linalg.eigensystem.real_symmetric(
paept.as_flex_double_matrix())
lambda_ = eigen.values()
print "condition number: ", lambda_[0] / lambda_[-1]
delta_solve = scitbx.linalg.matrix_cholesky_test_ratio(
a=paept.as_flex_double_matrix(),
x=flex.double(px),
b=flex.double(pb),
epsilon=gwm.epsilon)
assert delta_solve < 10, delta_solve
return p, e, r
# empty matrix
a = matrix.sqr([])
p, e, r = core(a)
assert p.size() == 0
assert e.size() == 0
assert len(r) == 0
n_max = 15
n_trials_per_n = 10
# identity matrices
for n in xrange(1, n_max + 1):
a = matrix.diag([1] * n)
p, e, r = core(a)
assert list(p) == range(n)
assert approx_equal(e, [0] * n)
assert approx_equal(r, a)
# null matrices
for n in xrange(1, n_max + 1):
a = matrix.sqr([0] * n * n)
p, e, r = core(a)
assert list(p) == range(n)
assert list(e) == [scitbx.math.floating_point_epsilon_double_get()] * n
for i in xrange(n):
for j in xrange(n):
if (i != j): r(i, j) == 0
else: r(i, j) == r(0, 0)
# random semi-positive diagonal matrices
for n in xrange(1, n_max + 1):
for i_trial in xrange(n_trials_per_n):
a = matrix.diag(flex.random_double(size=n))
p, e, r = core(a)
assert approx_equal(e, [0] * n)
for i in xrange(n):
for j in xrange(n):
if (i != j): approx_equal(r(i, j), 0)
# random diagonal matrices
for n in xrange(1, n_max + 1):
for i_trial in xrange(n_trials_per_n):
a = matrix.diag(flex.random_double(size=n, factor=2) - 1)
p, e, r = core(a)
for i in xrange(n):
for j in xrange(n):
if (i != j): approx_equal(r(i, j), 0)
# random semi-positive definite matrices
for n in xrange(1, n_max + 1):
for i_trial in xrange(n_trials_per_n):
m = matrix.sqr(flex.random_double(size=n * n, factor=2) - 1)
a = m.transpose_multiply()
p, e, r = core(a)
assert approx_equal(e, [0] * n)
# random matrices
for n in xrange(1, n_max + 1):
size = n * (n + 1) // 2
for i_trial in xrange(n_trials_per_n):
a = (flex.random_double(size=size, factor=2)-1) \
.matrix_packed_u_as_symmetric()
core(matrix.sqr(a))
a.matrix_diagonal_set_in_place(0)
core(matrix.sqr(a))
# J. Nocedal and S. Wright:
# Numerical Optimization.
# Springer, New York, 1999, pp. 145-150.
for i in xrange(3):
for j in xrange(3):
a = flex.double([[4, 2, 1], [2, 6, 3], [1, 3, -0.004]])
a.matrix_swap_rows_in_place(i=i, j=j)
a.matrix_swap_columns_in_place(i=i, j=j)
p, e, r = core(matrix.sqr(a))
if (i == 0 and j == 0):
assert list(p) == [1, 1,
2] # swap row 0 and 1 and nothing else
assert approx_equal(e, [0.0, 0.0, 3.008])
assert approx_equal(r, [
2.4494897427831779, 0.81649658092772592, 1.2247448713915889,
0.0, 1.8257418583505538, 0.0, 0.0, 0.0, 1.2263767773404712
])
def glm(x, p = None):
from math import exp , sqrt, log, factorial, lgamma
from scitbx import matrix
from scipy.stats import poisson
beta0 = 0
if p is None:
p = [1.0] * len(x)
while True:
n = beta0
mu = exp(n)
z = [n + (xx - mu) / mu for xx in x]
w = [pp * mu for pp in p]
W = matrix.diag(w)
Z = matrix.col(z)
X = matrix.rec([1]*len(z), (len(z),1))
beta = (X.transpose() * W * X).inverse() * X.transpose()*W*z
if abs(beta[0] - beta0) < 1e-3:
break
beta0 = beta[0]
n = beta[0]
mu = exp(n)
z = [n + (xx - mu) / mu for xx in x]
w = [pp * mu for pp in p]
W = matrix.diag(w)
Z = matrix.col(z)
X = matrix.rec([1]*len(z), (len(z),1))
W12 = matrix.diag([sqrt(ww) for ww in w])
H = X * (X.transpose()*W*X).inverse() * X.transpose()
r = [(xx - mu) / mu for xx in x]
sum1 = 0
sum2 = 0
sum3 = 0
D = []
for xx in x:
d = 0
if xx != 0:
d += xx * log(xx)
d -= xx * log(mu)
d -= (xx - mu)
D.append(2*d)
# r = []
# for xx in x:
# if xx == 0:
# r.append(0)
# else:
# r.append(2*(log(xx) - log(mu)))
# D = []
# for i in range(len(r)):
# h = H[i+i*H.n[0]]
# d = (r[i]/(1 - h))**2 * h/1.0
# D.append(d)
# print min(x), max(x)
print min(D), max(D), F(1, len(x))
return exp(beta[0]), max(D) > F(1, len(x))
