def test_design_function(self):
obs = TestSpaceTimeKroneckerElement.SimulatedObservationStructure()
spde_space = SphereMeshSPDE(level=1)
spde_time = LatticeSPDE.construct(dimension_specification=[(23, 27, 5)
],
basis_function=WendlandC4Basis(),
overlap_factor=2.5)
design = SpaceTimeKroneckerDesign(observationstructure=obs,
spatial_model=spde_space,
alpha=2,
temporal_model=spde_time,
H=1.01)
# Get and check indices of nonzero design elements
A_space = spde_space.build_A(obs.location_polar_coordinates())
observation_indices, vertex_indices = A_space.sorted_indices().nonzero(
)
numpy.testing.assert_equal(observation_indices, [0, 0, 0, 1, 1, 1])
self.assertEqual((6, ), vertex_indices.shape)
# Vertices of observations 0 and 1
# (one row per vertex, one column per coordinate)
vertices0 = spde_space.triangulation.points[vertex_indices[0:3], :]
vertices1 = spde_space.triangulation.points[vertex_indices[3:6], :]
# Multiply vertices by the weights from A and sum to get cartesian locations
testpoint0 = A_space[0, vertex_indices[0:3]] * vertices0
testpoint1 = A_space[1, vertex_indices[3:6]] * vertices1
# Check results correspond to original polar coordinates
numpy.testing.assert_almost_equal(cartesian_to_polar2d(testpoint0),
[[15.0, -7.0]])
numpy.testing.assert_almost_equal(cartesian_to_polar2d(testpoint1),
[[5.0, 100.0]])
# So the function with those indices set should give required values, provided it's evaluated at time index == 24
state_vector = numpy.zeros((210, ))
state_vector[42 + vertex_indices[0:3]] = 40.0
state_vector[42 + vertex_indices[3:6]] = 28.0
numpy.testing.assert_almost_equal(design.design_function(state_vector),
[40.0, 28.0])
# But if we change to a time far away then we get nothing
state_vector = numpy.zeros((210, ))
state_vector[(42 * 4) + vertex_indices[0:3]] = 40.0
state_vector[(42 * 4) + vertex_indices[3:6]] = 28.0
numpy.testing.assert_almost_equal(design.design_function(state_vector),
[0.0, 0.0])
def test_wrap(self):
# Model time resolution
node_spacing = 1./12.0
# Model start/end bound a decimal year
model_start = 1850.0
model_end = 1851.0
# number of model nodes in as 12 nodes within year
n_nodes = 12
element = AnnualKroneckerElement(n_triangulation_divisions=1, alpha=2, starttime=model_start, endtime=model_end, n_nodes=n_nodes, overlap_factor=2.5, H=1.0, wrap_dimensions = [True,])
# Check unit diagonal basis function values when point at beginning of years and a each node in the spatial triangulation
node_locations_space = cartesian_to_polar2d( element.spatial_model.triangulation.points )
node_locations_time = element.temporal_model.lattice.nodes
obs = TestSpaceTimeKroneckerElement.SimulatedObservationStructure(node_locations_space, datetime(1849, 1, 1) )
numpy.testing.assert_almost_equal( numpy.ones(obs.number_of_observations()), element.element_design(obs).design_matrix().diagonal() )
obs = TestSpaceTimeKroneckerElement.SimulatedObservationStructure(node_locations_space, datetime(1850, 1, 1) )
numpy.testing.assert_almost_equal( numpy.ones(obs.number_of_observations()), element.element_design(obs).design_matrix().diagonal() )
obs = TestSpaceTimeKroneckerElement.SimulatedObservationStructure(node_locations_space, datetime(1851, 1, 1) )
numpy.testing.assert_almost_equal( numpy.ones(obs.number_of_observations()), element.element_design(obs).design_matrix().diagonal() )
def test_build_A_space(self):
spde = SphereMeshSPDE(level=1, sparse_format=SPARSEFORMAT)
design = SeasonalElementDesign(
TestSeasonalElement.SimulatedObservationStructure(),
spde,
n_harmonics=3,
include_local_mean=True)
A = design.build_A_space()
# Check sparse format
self.assertEqual(SPARSEFORMAT, A.getformat())
# Shape should be number of obs x number of vertices (basis functions)
self.assertEqual((2, 42), A.shape)
# Two triangles means 6 points are non-zero
self.assertEqual(6, len(A.nonzero()[0]))
# Get and check indices of nonzero design elements
observation_indices, vertex_indices = A.sorted_indices().nonzero()
numpy.testing.assert_equal(observation_indices, [0, 0, 0, 1, 1, 1])
self.assertEqual((6, ), vertex_indices.shape)
# Vertices of observations 0 and 1
# (one row per vertex, one column per coordinate)
vertices0 = spde.triangulation.points[vertex_indices[0:3], :]
vertices1 = spde.triangulation.points[vertex_indices[3:6], :]
# Multiply vertices by the weights from A and sum to get cartesian locations
testpoint0 = A[0, vertex_indices[0:3]] * vertices0
testpoint1 = A[1, vertex_indices[3:6]] * vertices1
# Check results correspond to original polar coordinates
numpy.testing.assert_almost_equal(cartesian_to_polar2d(testpoint0),
[[15.0, -7.0]])
numpy.testing.assert_almost_equal(cartesian_to_polar2d(testpoint1),
[[5.0, 100.0]])
def test_mini_world_large_and_local(self):
# Use a number of time steps
number_of_simulated_time_steps = 30
# Large-scale spatial variability
simulated_large_variation = 10.0
# Local variability
simulated_local_variation = 1.0
# Iterations to use
number_of_solution_iterations = 5
# Build system
# Large-scale factor
element_large = SpaceTimeKroneckerElement(
n_triangulation_divisions=1,
alpha=2,
starttime=0,
endtime=number_of_simulated_time_steps + 1,
n_nodes=number_of_simulated_time_steps + 2,
overlap_factor=2.5,
H=1)
initial_hyperparameters_large = SpaceTimeSPDEHyperparameters(
space_log_sigma=0.0,
space_log_rho=numpy.log(numpy.radians(5.0)),
time_log_rho=numpy.log(1.0 / 365.0))
component_large = SpaceTimeComponent(
ComponentStorage_InMemory(element_large,
initial_hyperparameters_large),
SpaceTimeComponentSolutionStorage_InMemory())
# And a local process
component_local = SpatialComponent(
ComponentStorage_InMemory(
LocalElement(n_triangulation_divisions=3),
LocalHyperparameters(log_sigma=0.0,
log_rho=numpy.log(numpy.radians(5.0)))),
SpatialComponentSolutionStorage_InMemory())
analysis_system = AnalysisSystem([component_large, component_local],
ObservationSource.TMEAN,
log=StringIO())
# analysis_system = AnalysisSystem([ component_large ], ObservationSource.TMEAN)
# analysis_system = AnalysisSystem([ component_local ], ObservationSource.TMEAN)
# use fixed locations from icosahedron
fixed_locations = cartesian_to_polar2d(
MeshIcosahedronSubdivision.build(3).points)
# random measurement at each location
numpy.random.seed(8976)
field_basis = simulated_large_variation * numpy.random.randn(
fixed_locations.shape[0])
# some time function that varies over a year
time_basis = numpy.cos(
numpy.linspace(0.1, 1.75 * numpy.pi,
number_of_simulated_time_steps))
# kronecker product of the two
large_scale_process = numpy.kron(field_basis,
numpy.expand_dims(time_basis, 1))
# Random local changes where mean change at each time is zero
# local_process = simulated_local_variation * numpy.random.randn(large_scale_process.shape[0], large_scale_process.shape[1])
# local_process -= numpy.tile(local_process.mean(axis=1), (local_process.shape[1], 1)).T
local_process = numpy.zeros(large_scale_process.shape)
somefield = simulated_local_variation * numpy.random.randn(
1, large_scale_process.shape[1])
somefield -= somefield.ravel().mean()
local_process[10, :] = somefield
local_process[11, :] = -somefield
# Add the two processes
measurement = large_scale_process + local_process
# Simulated inputs
simulated_input_loader = SimulatedInputLoader(fixed_locations,
measurement, 0.001)
# Simulate evaluation of this time index
simulated_time_indices = range(number_of_simulated_time_steps)
# All systems linear so single update should be ok
analysis_system.update([simulated_input_loader],
simulated_time_indices)
# Get all results
result = numpy.zeros(measurement.shape)
for t in range(number_of_simulated_time_steps):
result[t, :] = analysis_system.evaluate_expected_value(
'MAP',
SimulatedObservationStructure(t, fixed_locations, None, None),
flag='POINTWISE')
disparity_large_scale = (numpy.abs(result -
large_scale_process)).ravel().max()
# print 'large scale disparity: ', disparity_large_scale
disparity_overall = (numpy.abs(result - measurement)).ravel().max()
# print 'overall disparity: ', disparity_overall
numpy.testing.assert_almost_equal(result, measurement, decimal=4)
self.assertTrue(disparity_overall < 1E-4)
def test_mini_world_noiseless(self):
number_of_simulated_time_steps = 1
# Build system
element = SeasonalElement(n_triangulation_divisions=3,
n_harmonics=5,
include_local_mean=True)
hyperparameters = SeasonalHyperparameters(n_spatial_components=6,
common_log_sigma=0.0,
common_log_rho=0.0)
component = SpaceTimeComponent(
ComponentStorage_InMemory(element, hyperparameters),
SpaceTimeComponentSolutionStorage_InMemory())
analysis_system = AnalysisSystem([component],
ObservationSource.TMEAN,
log=StringIO())
# use fixed locations from icosahedron
fixed_locations = cartesian_to_polar2d(
MeshIcosahedronSubdivision.build(3).points)
# random measurement at each location
numpy.random.seed(8976)
field_basis = numpy.random.randn(fixed_locations.shape[0])
#print(field_basis.shape)
#time_basis = numpy.array(harmonics_list)
# some time function that varies over a year
#decimal_years = numpy.array([datetime_to_decimal_year(epoch_plus_days(step)) for step in range(number_of_simulated_time_steps)])
time_basis = numpy.cos(
numpy.linspace(0.1, 1.75 * numpy.pi,
number_of_simulated_time_steps))
# kronecker product of the two
#print(numpy.expand_dims(time_basis, 1))
measurement = numpy.kron(field_basis, numpy.expand_dims(
time_basis, 1)) #numpy.expand_dims(time_basis, 1))
#print(measurement.shape)
# Simulated inputs
simulated_input_loader = SimulatedInputLoader(fixed_locations,
measurement, 0.0001)
# Simulate evaluation of this time index
simulated_time_indices = range(number_of_simulated_time_steps)
# Iterate
for iteration in range(5):
analysis_system.update([simulated_input_loader],
simulated_time_indices)
# Get all results
result = numpy.zeros(measurement.shape)
for t in range(number_of_simulated_time_steps):
result[t, :] = analysis_system.evaluate_expected_value(
'MAP',
SimulatedObservationStructure(t, fixed_locations, None, None),
flag='POINTWISE')
# Should be very close to original because specified noise is low
numpy.testing.assert_almost_equal(result, measurement)
max_disparity = (numpy.abs(result - measurement)).ravel().max()
self.assertTrue(max_disparity < 1E-5)
# test output gridding, pointwise limit
outputstructure = OutputRectilinearGridStructure(
2,
epoch_plus_days(2),
latitudes=numpy.linspace(-60., 60., num=5),
longitudes=numpy.linspace(-90., 90, num=10))
pointwise_result = analysis_system.evaluate_expected_value(
'MAP', outputstructure, 'POINTWISE')
pointwise_limit_result = analysis_system.evaluate_expected_value(
'MAP', outputstructure, 'GRID_CELL_AREA_AVERAGE', [1, 1], 10)
numpy.testing.assert_array_almost_equal(pointwise_result,
pointwise_limit_result)
def test_mini_world_noiseless(self):
# Use a number of time steps
number_of_simulated_time_steps = 1
# Build system
element = SpaceTimeFactorElement(
n_triangulation_divisions=3,
alpha=2,
starttime=0,
endtime=number_of_simulated_time_steps + 1,
n_nodes=number_of_simulated_time_steps + 2,
overlap_factor=2.5,
H=1)
initial_hyperparameters = SpaceTimeSPDEHyperparameters(
space_log_sigma=0.0,
space_log_rho=numpy.log(numpy.radians(45.0)),
time_log_rho=numpy.log(3.0 / 365.0))
component = SpaceTimeComponent(
ComponentStorage_InMemory(element, initial_hyperparameters),
SpaceTimeComponentSolutionStorage_InMemory())
analysis_system = AnalysisSystem([component],
ObservationSource.TMEAN,
log=StringIO())
# use fixed locations from icosahedron
fixed_locations = cartesian_to_polar2d(
MeshIcosahedronSubdivision.build(3).points)
# random measurement at each location
numpy.random.seed(8976)
field_basis = 10.0 * numpy.random.randn(fixed_locations.shape[0])
# some time function that varies over a year
time_basis = numpy.cos(
numpy.linspace(0.1, 1.75 * numpy.pi,
number_of_simulated_time_steps))
# kronecker product of the two
measurement = numpy.kron(field_basis, numpy.expand_dims(time_basis, 1))
# Simulated inputs
simulated_input_loader = SimulatedInputLoader(fixed_locations,
measurement, 0.01)
# Simulate evaluation of this time index
simulated_time_indices = range(number_of_simulated_time_steps)
# Iterate
for iteration in range(5):
analysis_system.update([simulated_input_loader],
simulated_time_indices)
# Get all results
result = numpy.zeros(measurement.shape)
for t in range(number_of_simulated_time_steps):
result[t, :] = analysis_system.evaluate_expected_value(
'MAP',
SimulatedObservationStructure(t, fixed_locations, None, None),
flag='POINTWISE')
# Should be very close to original because specified noise is low
numpy.testing.assert_almost_equal(result, measurement)
max_disparity = (numpy.abs(result - measurement)).ravel().max()
self.assertTrue(max_disparity < 1E-5)
def demo_non_stationary():
full_resolution_level = 5
neighbourhood_level = 2
full_spde = SphereMeshViewGlobal(level=full_resolution_level)
active_triangles = full_spde.neighbours_at_level(neighbourhood_level, 0)
n_regions = full_spde.n_triangles_at_level(neighbourhood_level)
merge_method = 'new'
if merge_method == 'old':
local_spdes = []
local_hyperparameters = []
for region_index in range(n_regions):
local_spdes.append(
SphereMeshViewSuperTriangle(full_resolution_level,
neighbourhood_level, region_index))
hyperparameters = numpy.array(
[numpy.float64(region_index),
numpy.float64(region_index)])
hyperparameters = numpy.log(
numpy.concatenate([
numpy.random.uniform(1.0, 3.0, 1),
numpy.random.uniform(5.0, 30.0, 1) * numpy.pi / 180.
]))
#hyperparameters = numpy.log( numpy.concatenate( [numpy.ones(1), numpy.random.uniform(15.0,45.0, 1) *numpy.pi/180.] ) )
#hyperparameters = numpy.array([2.0, 3.0])
#hyperparameters = numpy.log([2.0, numpy.pi/4])
local_hyperparameters.append(hyperparameters)
global_hyperparameters, global_sigma_design, global_rho_design = full_spde.merge_local_parameterisations(
local_spdes, local_hyperparameters, merge_method='exp_average')
log_sigmas = global_sigma_design.dot(global_hyperparameters)
log_rhos = global_rho_design.dot(global_hyperparameters)
elif merge_method == 'new':
sigma_accumulator = None
rho_accumulator = None
contribution_counter = None
for region_index in range(n_regions):
local_spde = SphereMeshViewSuperTriangle(full_resolution_level,
neighbourhood_level,
region_index)
local_hyperparameters = hyperparameters = numpy.log(
numpy.concatenate([
numpy.random.uniform(1.0, 5.0, 1),
numpy.random.uniform(10.0, 45.0, 1) * numpy.pi / 180.
]))
accumulators = SphereMeshViewGlobal.accumulate_local_parameterisations(
sigma_accumulator, rho_accumulator, contribution_counter,
local_spde, local_hyperparameters)
sigma_accumulator, rho_accumulator, contribution_counter = accumulators
log_sigmas, log_rhos = SphereMeshViewGlobal.finalise_local_parameterisation_sigma_rho(
sigma_accumulator, rho_accumulator, contribution_counter)
#print global_hyperparameters, global_sigma_design, global_rho_design
import matplotlib.pyplot as plt
from eustace.analysis.mesh.geometry import cartesian_to_polar2d
polar_coords = cartesian_to_polar2d(full_spde.triangulation.points)
plt.figure()
plt.scatter(polar_coords[:, 1],
polar_coords[:, 0],
c=255. * log_sigmas / numpy.max(numpy.abs(log_sigmas)),
linewidth=0.0,
s=8.0)
plt.figure()
plt.scatter(polar_coords[:, 1],
polar_coords[:, 0],
c=255. * log_rhos / numpy.max(numpy.abs(log_rhos)),
linewidth=0.0,
s=8.0)
#plt.show()
#numpy.testing.assert_almost_equal( log_sigmas, 2.0 * numpy.ones(full_spde.triangulation.points.shape[0]) )
#numpy.testing.assert_almost_equal( log_rhos, 3.0 * numpy.ones(full_spde.triangulation.points.shape[0]) )
from eustace.analysis.advanced_standard.components.storage_inmemory import ComponentStorage_InMemory
from eustace.analysis.advanced_standard.components.storage_inmemory import SpatialComponentSolutionStorage_InMemory
from eustace.analysis.advanced_standard.components.spatialdelayed import DelayedSpatialComponent
from eustace.analysis.advanced_standard.elements.local_view import NonStationaryLocal, ExpandedLocalHyperparameters
from eustace.analysis.advanced_standard.elements.local import LocalElement, LocalHyperparameters
nonstationary_component = DelayedSpatialComponent(
ComponentStorage_InMemory(
NonStationaryLocal(full_resolution_level),
ExpandedLocalHyperparameters(log_sigma=log_sigmas,
log_rho=log_rhos)),
SpatialComponentSolutionStorage_InMemory())
#nonstationary_component = DelayedSpatialComponent(
#ComponentStorage_InMemory(LocalElement(full_resolution_level), LocalHyperparameters(log_sigma = hyperparameters[0], log_rho = hyperparameters[1])),
#SpatialComponentSolutionStorage_InMemory())
#print log_sigmas, log_rhos
#plt.figure()
#plt.scatter(polar_coords[:,1], polar_coords[:,0], c = 255.* process_sample / numpy.max(numpy.abs(process_sample)), linewidth = 0.0, s = 8.0 )
#plt.figure()
#plt.imshow( numpy.asarray( Q.todense() ) )
# setup an output grid
out_lats = numpy.linspace(-89.5, 89.5, 180)
out_lons = numpy.linspace(-179.5, 179.5, 360)
out_lons, out_lats = numpy.meshgrid(out_lons, out_lats)
out_coords = numpy.vstack([out_lats.ravel(), out_lons.ravel()]).T
design_matrix = nonstationary_component.storage.element.spde.build_A(
out_coords)
# setup solver for sampling
from eustace.analysis.advanced_standard.linalg.extendedcholmodwrapper import ExtendedCholmodWrapper
Q = nonstationary_component.storage.element.element_prior(
nonstationary_component.storage.hyperparameters).prior_precision()
factor = ExtendedCholmodWrapper.cholesky(Q)
# draw samples, project onto output grid and plot
random_values = numpy.random.normal(0.0, 1.0, (Q.shape[0], 1))
process_sample = factor.solve_backward_substitution(random_values)
out_values = design_matrix.dot(process_sample)
plt.figure()
plt.scatter(out_coords[:, 1],
out_coords[:, 0],
c=255. * out_values / numpy.max(numpy.abs(out_values)),
linewidth=0.0,
s=8.0)
random_values = numpy.random.normal(0.0, 1.0, (Q.shape[0], 1))
process_sample = factor.solve_backward_substitution(random_values)
out_values = design_matrix.dot(process_sample)
plt.figure()
plt.scatter(out_coords[:, 1],
out_coords[:, 0],
c=255. * out_values / numpy.max(numpy.abs(out_values)),
linewidth=0.0,
s=8.0)
random_values = numpy.random.normal(0.0, 1.0, (Q.shape[0], 1))
process_sample = factor.solve_backward_substitution(random_values)
out_values = design_matrix.dot(process_sample)
plt.figure()
plt.scatter(out_coords[:, 1],
out_coords[:, 0],
c=255. * out_values / numpy.max(numpy.abs(out_values)),
linewidth=0.0,
s=8.0)
plt.show()