class SpatialBiasElement(ElementLinear):
"""
Class for evaluating design and precision matrices for SPDE models on a triangulated sphere.
"""
def __init__(self, groupname, n_triangulation_divisions):
"""
Setup triangulation of sphere for LocalBias model.
Args:
* groupname:
The name of the bias group that this bias corresponds to. Not currently used.
* n_triangulation_divisions (int):
Number of subdivisions of icosohedron for triangulation of sphere.
"""
super(SpatialBiasElement, self).__init__()
self.groupname = groupname
self.spde = SphereMeshSPDE(level=n_triangulation_divisions,
sparse_format=SPARSEFORMAT)
self.number_of_biases = self.spde.n_latent_variables()
def element_design(self, observationstructure):
return SpatialBiasDesign(observationstructure, self.spde,
self.groupname)
def element_prior(self, hyperparameters):
"""
Return LocalPrior instance for local model, using given hyperparameters.
"""
return LocalPrior(hyperparameters, self.spde)
def optimisation_demo():
"""
Demonstrate construction of an SPDE model on a tringular mesh on a sphere.
The mesh is constructed as a subdivided icosahedron. Observations are simulated
and a latent variable model is learnt.
"""
import matplotlib.pyplot as plt
from advanced_standard.linalg import sparse
from advanced_standard.utilities import plot_tools
print "############"
print "# Setup #"
print "############"
my_spde = SphereMeshSPDE(level=4, project_onto_sphere=True)
print "number of latent variables:"
print my_spde.n_latent_variables()
print "############"
print "# Q matrix #"
print "############"
Q_parameters = np.log(np.array([2., 1.0]))
print "Q_parameters:", Q_parameters
Q = my_spde.build_Q(Q_parameters=Q_parameters, alpha=2)
#dQdp0 = my_spde.build_dQdp(Q_parameters = np.log(np.array([1., 0.5])), alpha = 2, parameter_index = 0)
#dQdp1 = my_spde.build_dQdp(Q_parameters = np.log(np.array([1., 0.5])), alpha = 2, parameter_index = 1)
print "############"
print "# Q sample #"
print "############"
c = sparse.sample_mvn(Q, 1)
print "############"
print "# Obs gen #"
print "############"
number_of_observations = 25000 #2592
test_points = np.random.uniform(-1, 1, (number_of_observations, 3))
standard_error = 1.0
Q_e = scipy.sparse.eye(number_of_observations) * (1.0 / standard_error**2)
observation_error = sparse.sample_mvn(Q_e, 1)
print "############"
print "# A matrix #"
print "############"
A = my_spde.build_A(test_points)
truth = A * c
#y = np.atleast_2d(truth + observation_error).T
y = truth + observation_error
print "############"
print "# solving #"
print "############"
Q_increment = A.T * Q_e * A
i_increment = A.T * Q_e * y
import advanced_standard.linalg.optimisation as optimisation
Qkwargs = {'alpha': 2, 'sparse_format': 'csc'}
# interval = 1e-5
#
# cost_function = optimisation.cost_function(Q_parameters=Q_parameters,
# i_c = np.zeros( (Q.shape[0],1) ),
# i_increment= i_increment,
# Q_increment=Q_increment,
# Q_function = my_spde.build_Q,
# dQ_function = my_spde.build_dQdp, set_point_mode = 'fixed',
# fixed_set_point = np.zeros( (Q.shape[0],1) ),
# observation_loglik=None,
# **Qkwargs)
#
# vost_function = optimisation.cost_function(Q_parameters=Q_parameters +np.array([0.0, interval]),
# i_c = np.zeros( (Q.shape[0],1) ),
# i_increment= i_increment,
# Q_increment=Q_increment,
# Q_function = my_spde.build_Q,
# dQ_function = my_spde.build_dQdp, set_point_mode = 'fixed',
# fixed_set_point = np.zeros( (Q.shape[0],1) ),
# observation_loglik=None,
# **Qkwargs)
#
# dcost_function = optimisation.cost_function_derivative(Q_parameters=Q_parameters,
# i_c = np.zeros( (Q.shape[0],1) ),
# i_increment= i_increment,
# Q_increment=Q_increment,
# Q_function = my_spde.build_Q,
# dQ_function = my_spde.build_dQdp, set_point_mode = 'fixed',
# fixed_set_point = np.zeros( (Q.shape[0],1) ),
# **Qkwargs)
#
#
# print (vost_function - cost_function) / interval
#
# print dcost_function
# print cost_function
Q_init = np.log(np.array([2.0, np.pi / 4]))
mu_c = np.zeros((Q.shape[0], 1))
i_increment = i_increment
Q_increment = Q_increment
Q_function = my_spde.build_Q
dQ_function = my_spde.build_dQdp
set_point_mode = 'fixed'
fixed_set_point = np.zeros((Q.shape[0], 1))
observation_loglik = None
opt_args = (mu_c, i_increment, Q_increment, Q_function, dQ_function,
set_point_mode, fixed_set_point, observation_loglik, Qkwargs)
print scipy.optimize.check_grad(optimisation.cost_function,
optimisation.cost_function_derivative,
Q_init, *opt_args)
def opt_printer(x):
print x
fit = scipy.optimize.minimize(optimisation.cost_function,
Q_init,
args=opt_args,
method='BFGS',
jac=optimisation.cost_function_derivative,
tol=None,
callback=opt_printer,
options={
'disp': False,
'gtol': 1e-04,
'eps': 1e-08,
'return_all': False,
'maxiter': 20
})
print fit
fitted_hyperparamters = fit['x']
new_Q = Q_function(Q_parameters=fitted_hyperparamters, **Qkwargs)
Q_estimate = sparse.compute_Q_u(Q_e, A, new_Q)
i_estimate = sparse.compute_i_u(
y, Q_e, A, np.zeros((my_spde.n_latent_variables(), 1)))
c_estimate = sparse.compute_mu_u(i_estimate, Q_estimate)
reconstruction = A * c_estimate
print "############"
print "# plotting #"
print "############"
vmin = -6
vmax = 6
plt.figure()
vertices_coords_2d = cartesian_to_polar(my_spde.triangulation.vertices)
plot_tools.scatter2d(vertices_coords_2d[:, 1] * 180 / np.pi,
vertices_coords_2d[:, 0] * 180 / np.pi,
cs=c.ravel(),
vmax=vmax,
vmin=vmin)
plt.figure()
test_point_coords_2d = cartesian_to_polar(test_points)
plot_tools.scatter2d(test_point_coords_2d[:, 1] * 180 / np.pi,
test_point_coords_2d[:, 0] * 180 / np.pi,
cs=truth.ravel(),
vmax=vmax,
vmin=vmin)
plt.figure()
test_point_coords_2d = cartesian_to_polar(test_points)
plot_tools.scatter2d(test_point_coords_2d[:, 1] * 180 / np.pi,
test_point_coords_2d[:, 0] * 180 / np.pi,
cs=np.asarray(y).ravel(),
vmax=vmax,
vmin=vmin)
plt.figure()
vertices_coords_2d = cartesian_to_polar(my_spde.triangulation.vertices)
plot_tools.scatter2d(vertices_coords_2d[:, 1] * 180 / np.pi,
vertices_coords_2d[:, 0] * 180 / np.pi,
cs=c_estimate.ravel(),
vmax=vmax,
vmin=vmin)
plt.figure()
test_point_coords_2d = cartesian_to_polar(test_points)
plot_tools.scatter2d(test_point_coords_2d[:, 1] * 180 / np.pi,
test_point_coords_2d[:, 0] * 180 / np.pi,
cs=reconstruction.ravel(),
vmax=vmax,
vmin=vmin)
plt.show()
def spde_demo():
"""
Demonstrate construction of an SPDE model on a tringular mesh on a sphere.
The mesh is constructed as a subdivided icosahedron. Observations are simulated
and a latent variable model is learnt.
"""
import matplotlib.pyplot as plt
from advanced_standard.linalg import sparse
from advanced_standard.utilities import plot_tools
print "############"
print "# Setup #"
print "############"
my_spde = SphereMeshSPDE(level=4, project_onto_sphere=True)
print "number of latent variables:"
print my_spde.n_latent_variables()
print "############"
print "# Q matrix #"
print "############"
Q = my_spde.build_Q(Q_parameters=np.log(np.array([2., 0.5])), alpha=2)
#dQdp0 = my_spde.build_dQdp(Q_parameters = np.log(np.array([1., 0.5])), alpha = 2, parameter_index = 0)
#dQdp1 = my_spde.build_dQdp(Q_parameters = np.log(np.array([1., 0.5])), alpha = 2, parameter_index = 1)
print "############"
print "# Q sample #"
print "############"
c = sparse.sample_mvn(Q, 1)
print "############"
print "# Obs gen #"
print "############"
number_of_observations = 2592
test_points = np.random.uniform(-1, 1, (number_of_observations, 3))
standard_error = 0.5
Q_e = scipy.sparse.eye(number_of_observations) * (1.0 / standard_error**2)
observation_error = sparse.sample_mvn(Q_e, 1)
print "############"
print "# A matrix #"
print "############"
A = my_spde.build_A(test_points)
truth = A * c
y = truth + observation_error
print "############"
print "# solving #"
print "############"
Q_estimate = sparse.compute_Q_u(Q_e, A, Q)
i_estimate = sparse.compute_i_u(y, Q_e, A,
np.zeros((my_spde.n_latent_variables())))
c_estimate = sparse.compute_mu_u(i_estimate, Q_estimate)
reconstruction = A * c_estimate
print "############"
print "# plotting #"
print "############"
vmin = -3
vmax = 3
plt.figure()
vertices_coords_2d = cartesian_to_polar(my_spde.triangulation.vertices)
plot_tools.scatter2d(vertices_coords_2d[:, 1] * 180 / np.pi,
vertices_coords_2d[:, 0] * 180 / np.pi,
cs=c,
vmax=vmax,
vmin=vmin)
plt.figure()
test_point_coords_2d = cartesian_to_polar(test_points)
plot_tools.scatter2d(test_point_coords_2d[:, 1] * 180 / np.pi,
test_point_coords_2d[:, 0] * 180 / np.pi,
cs=truth,
vmax=vmax,
vmin=vmin)
plt.figure()
test_point_coords_2d = cartesian_to_polar(test_points)
plot_tools.scatter2d(test_point_coords_2d[:, 1] * 180 / np.pi,
test_point_coords_2d[:, 0] * 180 / np.pi,
cs=y,
vmax=vmax,
vmin=vmin)
plt.figure()
vertices_coords_2d = cartesian_to_polar(my_spde.triangulation.vertices)
plot_tools.scatter2d(vertices_coords_2d[:, 1] * 180 / np.pi,
vertices_coords_2d[:, 0] * 180 / np.pi,
cs=c_estimate.ravel(),
vmax=vmax,
vmin=vmin)
plt.figure()
test_point_coords_2d = cartesian_to_polar(test_points)
plot_tools.scatter2d(test_point_coords_2d[:, 1] * 180 / np.pi,
test_point_coords_2d[:, 0] * 180 / np.pi,
cs=reconstruction.ravel(),
vmax=vmax,
vmin=vmin)
plt.show()