def test_sample_based_apc_orthonormality(self):
num_vars = 1
alpha_stat = 2
beta_stat = 5
degree = 2
pce_var_trans = define_iid_random_variable_transformation(
stats.uniform(0, 1), num_vars)
pce_opts = define_poly_options_from_variable_transformation(
pce_var_trans)
random_var_trans = define_iid_random_variable_transformation(
stats.beta(alpha_stat, beta_stat), num_vars)
num_moment_samples = 10000
moment_matrix_samples = generate_independent_random_samples(
random_var_trans.variable, num_moment_samples)
compute_moment_matrix_function = partial(
compute_moment_matrix_from_samples, samples=moment_matrix_samples)
pce = APC(compute_moment_matrix_function)
pce.configure(pce_opts)
num_samples = 10000
samples = generate_independent_random_samples(
random_var_trans.variable, num_samples)
indices = compute_hyperbolic_indices(num_vars, degree, 1.0)
pce.set_indices(indices)
basis_matrix = pce.basis_matrix(samples)
assert np.allclose(np.dot(basis_matrix.T, basis_matrix) / num_samples,
np.eye(basis_matrix.shape[1]),
atol=1e-1)
def get_AB_sample_sets_for_sobol_sensitivity_analysis(variables,
nsamples,
method,
qmc_start_index=0):
if method == 'random':
samplesA = generate_independent_random_samples(variables, nsamples)
samplesB = generate_independent_random_samples(variables, nsamples)
elif method == 'halton' or 'sobol':
nvars = variables.num_vars()
if method == 'halton':
qmc_samples = halton_sequence(2 * nvars, qmc_start_index,
qmc_start_index + nsamples)
else:
qmc_samples = sobol_sequence(2 * nvars, nsamples, qmc_start_index)
samplesA = qmc_samples[:nvars, :]
samplesB = qmc_samples[nvars:, :]
for ii, rv in enumerate(variables.all_variables()):
lb, ub = rv.interval(1)
# transformation is undefined at [0,1] for unbouned random variables
# create bounds for unbounded interval that exclude 1e-8
# of the total probability
t1, t2 = rv.interval(1 - 1e-8)
nlb, nub = rv.cdf([t1, t2])
if not np.isfinite(lb):
samplesA[ii, samplesA[ii, :] == 0] = nlb
samplesB[ii, samplesB[ii, :] == 0] = nlb
if not np.isfinite(ub):
samplesA[ii, samplesA[ii, :] == 1] = nub
samplesB[ii, samplesB[ii, :] == 1] = nub
samplesA[ii, :] = rv.ppf(samplesA[ii, :])
samplesB[ii, :] = rv.ppf(samplesB[ii, :])
else:
raise Exception(f'Sampling method {method} not supported')
return samplesA, samplesB
def generate_probability_samples_tolerance(pce,
nindices,
cond_tol,
samples=None,
verbosity=0):
r"""
Add samples in integer increments of nindices.
E.g. if try nsamples = nindices, 2*nindices, 3*nindices
until condition number is less than tolerance.
Parameters
----------
samples : np.ndarray
samples in the canonical domain of the polynomial
Returns
-------
new_samples : np.ndarray(nvars, nnew_samples)
New samples appended to samples. must be in canonical space
"""
variable = pce.var_trans.variable
if samples is None:
new_samples = generate_independent_random_samples(variable, nindices)
new_samples = pce.var_trans.map_to_canonical_space(new_samples)
else:
new_samples = samples.copy()
cond = compute_preconditioned_canonical_basis_matrix_condition_number(
pce, new_samples)
if verbosity > 0:
print('\tCond No.', cond, 'No. samples', new_samples.shape[1],
'cond tol', cond_tol)
cnt = 1
max_nsamples = 1000 * pce.indices.shape[1]
while cond > cond_tol:
tmp = generate_independent_random_samples(variable, cnt * nindices)
tmp = pce.var_trans.map_to_canonical_space(tmp)
new_samples = np.hstack((new_samples, tmp))
cond = compute_preconditioned_canonical_basis_matrix_condition_number(
pce, new_samples)
if verbosity > 0:
print('\tCond No.', cond, 'No. samples', new_samples.shape[1],
'cond tol', cond_tol)
# double number of samples added so loop does not take to long
cnt *= 2
if new_samples.shape[1] > max_nsamples:
msg = "Basis and sample combination is ill conditioned"
raise RuntimeError(msg)
return new_samples
def find_deterministic_beam_design():
lower_bound = 0
uq_vars, objective, constraint_functions, bounds, init_design_sample = \
setup_beam_design()
uq_nominal_sample = uq_vars.get_statistics('mean')
#constraints_info = [{'type':'deterministic',
# 'lower_bound':lower_bound}]*len(constraint_functions)
#constraints = setup_inequality_constraints(
# constraint_functions,constraints_info,uq_nominal_sample)
constraint_info = {'lower_bound':lower_bound,
'uq_nominal_sample':uq_nominal_sample}
individual_constraints = [
DeterministicConstraint(f,constraint_info)
for f in constraint_functions]
constraints = MultipleConstraints(individual_constraints)
optim_options={'ftol': 1e-9, 'disp': 3, 'maxiter':1000}
res, opt_history = run_design(
objective,init_design_sample,constraints,bounds,optim_options)
nsamples = 10000
uq_samples = generate_independent_random_samples(uq_vars,nsamples)
return objective,constraints,constraint_functions,uq_samples,res,\
opt_history
def test_multiply_pce(self):
np.random.seed(1)
np.set_printoptions(precision=16)
univariate_variables = [norm(), uniform()]
variable = IndependentMultivariateRandomVariable(univariate_variables)
degree1, degree2 = 1, 2
poly1 = get_polynomial_from_variable(variable)
poly1.set_indices(
compute_hyperbolic_indices(variable.num_vars(), degree1))
poly2 = get_polynomial_from_variable(variable)
poly2.set_indices(
compute_hyperbolic_indices(variable.num_vars(), degree2))
#coef1 = np.random.normal(0,1,(poly1.indices.shape[1],1))
#coef2 = np.random.normal(0,1,(poly2.indices.shape[1],1))
coef1 = np.arange(poly1.indices.shape[1])[:, np.newaxis]
coef2 = np.arange(poly2.indices.shape[1])[:, np.newaxis]
poly1.set_coefficients(coef1)
poly2.set_coefficients(coef2)
poly3 = poly1 * poly2
samples = generate_independent_random_samples(variable, 10)
assert np.allclose(poly3(samples), poly1(samples) * poly2(samples))
for order in range(4):
poly = poly1**order
assert np.allclose(poly(samples), poly1(samples)**order)
def test_multiply_multivariate_orthonormal_polynomial_expansions(self):
univariate_variables = [norm(), uniform()]
variable = IndependentMultivariateRandomVariable(univariate_variables)
degree1, degree2 = 3, 2
poly1 = get_polynomial_from_variable(variable)
poly1.set_indices(
compute_hyperbolic_indices(variable.num_vars(), degree1))
poly1.set_coefficients(
np.random.normal(0, 1, (poly1.indices.shape[1], 1)))
poly2 = get_polynomial_from_variable(variable)
poly2.set_indices(
compute_hyperbolic_indices(variable.num_vars(), degree2))
poly2.set_coefficients(
np.random.normal(0, 1, (poly2.indices.shape[1], 1)))
max_degrees1 = poly1.indices.max(axis=1)
max_degrees2 = poly2.indices.max(axis=1)
product_coefs_1d = compute_product_coeffs_1d_for_each_variable(
poly1, max_degrees1, max_degrees2)
indices, coefs = multiply_multivariate_orthonormal_polynomial_expansions(
product_coefs_1d, poly1.get_indices(), poly1.get_coefficients(),
poly2.get_indices(), poly2.get_coefficients())
poly3 = get_polynomial_from_variable(variable)
poly3.set_indices(indices)
poly3.set_coefficients(coefs)
samples = generate_independent_random_samples(variable, 10)
# print(poly3(samples),poly1(samples)*poly2(samples))
assert np.allclose(poly3(samples), poly1(samples) * poly2(samples))
def test_compute_multivariate_orthonormal_basis_product(self):
univariate_variables = [norm(), uniform()]
variable = IndependentMultivariateRandomVariable(univariate_variables)
poly1 = get_polynomial_from_variable(variable)
poly2 = get_polynomial_from_variable(variable)
max_degrees1, max_degrees2 = [3, 3], [2, 2]
product_coefs_1d = compute_product_coeffs_1d_for_each_variable(
poly1, max_degrees1, max_degrees2)
for ii in range(max_degrees1[0]):
for jj in range(max_degrees1[1]):
poly_index_ii, poly_index_jj = np.array([ii, jj
]), np.array([ii, jj])
poly1.set_indices(poly_index_ii[:, np.newaxis])
poly1.set_coefficients(np.ones([1, 1]))
poly2.set_indices(poly_index_jj[:, np.newaxis])
poly2.set_coefficients(np.ones([1, 1]))
product_indices, product_coefs = \
compute_multivariate_orthonormal_basis_product(
product_coefs_1d, poly_index_ii, poly_index_jj,
max_degrees1, max_degrees2)
poly_prod = get_polynomial_from_variable(variable)
poly_prod.set_indices(product_indices)
poly_prod.set_coefficients(product_coefs)
samples = generate_independent_random_samples(variable, 5)
# print(poly_prod(samples),poly1(samples)*poly2(samples))
assert np.allclose(poly_prod(samples),
poly1(samples) * poly2(samples))
def test_pce_jacobian(self):
degree = 2
alpha_stat, beta_stat = 2, 3
univariate_variables = [beta(alpha_stat, beta_stat, 0, 1), norm(-1, 2)]
variable = IndependentMultivariateRandomVariable(univariate_variables)
var_trans = AffineRandomVariableTransformation(variable)
num_vars = len(univariate_variables)
poly = PolynomialChaosExpansion()
poly_opts = define_poly_options_from_variable_transformation(var_trans)
poly.configure(poly_opts)
indices = compute_hyperbolic_indices(num_vars, degree, 1.0)
poly.set_indices(indices)
sample = generate_independent_random_samples(variable, 1)
coef = np.ones((indices.shape[1], 2))
coef[:, 1] *= 2
poly.set_coefficients(coef)
jac = poly.jacobian(sample)
from pyapprox.optimization import approx_jacobian
fd_jac = approx_jacobian(lambda x: poly(x[:, np.newaxis])[0, :],
sample[:, 0])
assert np.allclose(jac, fd_jac)
def test_identity_map_subset(self):
num_vars = 3
var_trans = define_iid_random_variable_transformation(
uniform(0, 1), num_vars)
var_trans.set_identity_maps([1])
samples = np.random.uniform(0, 1, (num_vars, 4))
canonical_samples = var_trans.map_to_canonical_space(samples)
assert np.allclose(canonical_samples[1, :], samples[1, :])
assert np.allclose(
var_trans.map_from_canonical_space(canonical_samples), samples)
univariate_variables = [
uniform(-1, 2), beta(1, 1, -1, 2), norm(-1, np.sqrt(4)), uniform(),
uniform(-1, 2), beta(2, 1, -2, 3)]
var_trans = AffineRandomVariableTransformation(univariate_variables)
var_trans.set_identity_maps([4, 2])
from pyapprox.probability_measure_sampling import \
generate_independent_random_samples
samples = generate_independent_random_samples(var_trans.variable, 10)
canonical_samples = var_trans.map_to_canonical_space(samples)
assert np.allclose(canonical_samples[[2, 4], :], samples[[2, 4], :])
assert np.allclose(
var_trans.map_from_canonical_space(canonical_samples), samples)
def test_feed_forward_system_of_polynomials_estimate_coupling_bounds(self):
graph, variables, graph_data = get_3_recursive_polynomial_components()
# overwrite functions, they have the same network structure so this
# is fine
graph_data['functions'] = get_polynomial_models()
graph = build_chain_graph(3, graph_data)
network = SystemNetwork(graph)
approx = DecoupledSystemSurrogate(network,
variables,
estimate_coupling_ranges=True,
verbose=2,
nrefinement_samples=1e4)
univariate_quad_rule_info = [
clenshaw_curtis_in_polynomial_order, clenshaw_curtis_rule_growth
]
refinement_indicator = None
options = [{
'univariate_quad_rule_info': univariate_quad_rule_info,
'max_nsamples': 30,
'tol': 0,
'verbose': 0,
'refinement_indicator': refinement_indicator
}] * 3
surr_graph = approx.surrogate_network.graph
# under estimate coupling ranges
output_bounds = [[0.5, 1.5], [2, 5]]
coupling_variables = {
0: [],
1: [stats.uniform(output_bounds[0][0], np.diff(output_bounds[0]))],
2: [stats.uniform(output_bounds[1][0], np.diff(output_bounds[1]))]
}
approx.set_coupling_variables(coupling_variables)
approx.initialize_component_surrogates(options)
approx.build(terminate_test=TerminateTest(max_work=30))
nvalidation_samples = 10
validation_samples = generate_independent_random_samples(
variables, nvalidation_samples)
validation_values = network(validation_samples,
component_ids=[0, 1, 2])
validation_values_approx = []
for node_id in surr_graph.nodes:
component_surr = surr_graph.nodes[node_id]['functions']
idx = graph_data["global_random_var_indices"][node_id]
if node_id == 0:
component_vals_approx = component_surr(
validation_samples[idx, :])
else:
jdx = graph_data["local_random_var_indices"][node_id][0]
component_samples = np.empty((2, validation_samples.shape[1]))
component_samples[jdx] = validation_samples[idx, :]
component_samples[1 - jdx] = validation_values_approx[-1].T
component_vals_approx = component_surr(component_samples)
validation_values_approx.append(component_vals_approx)
assert np.allclose(validation_values[node_id],
component_vals_approx)
assert np.allclose(approx(validation_samples), validation_values[-1])
def test_peer_feed_forward_system_of_polynomials_multiple_qoi(self):
graph, variables, graph_data = \
get_3_peer_polynomial_components_multiple_qoi()
network = SystemNetwork(graph)
nsamples = 10
samples = generate_independent_random_samples(variables, nsamples)
values = network(samples)
component_nvars = network.component_nvars()
assert component_nvars == [1, 2, 4]
funs = graph_data['functions']
global_random_var_indices = graph_data['global_random_var_indices']
values0 = funs[0](samples[global_random_var_indices[0], :])
values1 = funs[1](samples[global_random_var_indices[1], :])
values2 = funs[2](np.vstack([
samples[global_random_var_indices[2], :], values1.T, values0[:,
1:2].T
]))
assert np.allclose(values, values2)
network_values = network(samples, [0, 1, 2])
assert np.allclose(network_values[0], values0)
assert np.allclose(network_values[1], values1)
assert np.allclose(network_values[2], values2)
# test when component_ids are specified
ncomponent_outputs = [2, 2, 1]
ncomponent_coupling_vars = [0, 0, 3]
noutputs = np.sum(ncomponent_outputs)
ncoupling_vars = np.sum(ncomponent_coupling_vars)
adjacency_matrix = np.zeros((ncoupling_vars, noutputs))
adjacency_matrix[0, 2] = 1
adjacency_matrix[1, 3] = 1
adjacency_matrix[2, 1] = 1
adjacency_matrix = scipy_sparse.csr_matrix(adjacency_matrix)
exog_extraction_indices = [[0], [0, 1], [2]]
coup_extraction_indices = [[], [], [0, 1, 2]]
qoi_extraction_indices = [0, 2, 3, 4]
network = GaussJacobiSystemNetwork(graph)
network.set_adjacency_matrix(adjacency_matrix)
network.set_extraction_indices(exog_extraction_indices,
coup_extraction_indices,
qoi_extraction_indices,
ncomponent_outputs)
network.set_initial_coupling_sample(np.ones((ncoupling_vars, 1)))
component_ids = [0, 1, 2]
outputs = network(samples, component_ids, init_coup_samples=None)
true_outputs = [values0, values1, values2]
# print(outputs[0], true_outputs[0])
# print(outputs[1], true_outputs[1])
# print(outputs[2], true_outputs[2])
assert np.allclose(np.hstack(outputs), np.hstack(true_outputs))
outputs = network(samples, component_ids=None, init_coup_samples=None)
assert np.allclose(outputs, np.hstack(true_outputs)[:, [0, 2, 3, 4]])
def increment_probability_samples(pce,
cond_tol,
samples,
indices,
new_indices,
verbosity=0):
r"""
Parameters
----------
samples : np.ndarray
samples in the canonical domain of the polynomial
Returns
-------
new_samples : np.ndarray(nvars, nnew_samples)
New samples appended to samples. must be in canonical space
"""
# allocate at one sample for every new basis
tmp = generate_independent_random_samples(pce.var_trans.variable,
new_indices.shape[1])
tmp = pce.var_trans.map_to_canonical_space(tmp)
new_samples = np.hstack((samples, tmp))
# keep sampling until condition number is below cond_tol
new_samples = generate_probability_samples_tolerance(
pce, new_indices.shape[1], cond_tol, new_samples, verbosity)
if verbosity > 0:
print('No. samples', new_samples.shape[1])
print('No. initial samples', samples.shape[1])
print('No. indices', indices.shape[1], pce.indices.shape[1])
print('No. new indices', new_indices.shape[1])
print('No. new samples', new_samples.shape[1] - samples.shape[1])
return new_samples
def allocate_initial_samples(self):
if self.induced_sampling:
return generate_induced_samples_migliorati_tolerance(
self.pce, self.cond_tol)
else:
return generate_independent_random_samples(
self.pce.var_trans.variable,
self.sample_ratio * self.pce.num_terms())
def analytic_sobol_indices_from_gaussian_process(
gp, variable, interaction_terms, ngp_realizations=1,
stat_functions=(np.mean, np.median, np.min, np.max),
ninterpolation_samples=500, nvalidation_samples=100,
ncandidate_samples=1000, nquad_samples=50, use_cholesky=True, alpha=0):
x_train, y_train, K_inv, lscale, kernel_var, transform_quad_rules = \
extract_gaussian_process_attributes_for_integration(gp)
if ngp_realizations > 0:
gp_realizations = generate_gp_realizations(
gp, ngp_realizations, ninterpolation_samples, nvalidation_samples,
ncandidate_samples, variable, use_cholesky, alpha)
# Check how accurate realizations
validation_samples = generate_independent_random_samples(
variable, 1000)
mean_vals, std = gp(validation_samples, return_std=True)
realization_vals = gp_realizations(validation_samples)
print(mean_vals[:, 0].mean())
# print(std,realization_vals.std(axis=1))
print('std of realizations error',
np.linalg.norm(std-realization_vals.std(axis=1))/np.linalg.norm(
std))
print('var of realizations error',
np.linalg.norm(std**2-realization_vals.var(axis=1)) /
np.linalg.norm(std**2))
print('mean interpolation error',
np.linalg.norm((mean_vals[:, 0]-realization_vals[:, -1])) /
np.linalg.norm(mean_vals[:, 0]))
x_train = gp_realizations.selected_canonical_samples
# gp_realizations.train_vals is normalized so unnormalize
y_train = gp._y_train_std*gp_realizations.train_vals
# kernel_var has already been adjusted by call to
# extract_gaussian_process_attributes_for_integration
K_inv = np.linalg.inv(gp_realizations.L.dot(gp_realizations.L.T))
K_inv /= gp._y_train_std**2
sobol_values, total_values, means, variances = \
_compute_expected_sobol_indices(
gp, variable, interaction_terms, nquad_samples,
x_train, y_train, K_inv, lscale, kernel_var,
transform_quad_rules, gp._y_train_mean)
sobol_values = sobol_values.T
total_values = total_values.T
result = dict()
data = [sobol_values, total_values, variances, means]
data_names = ['sobol_indices', 'total_effects', 'variance', 'mean']
for item, name in zip(data, data_names):
subdict = dict()
for ii, sfun in enumerate(stat_functions):
subdict[sfun.__name__] = sfun(item, axis=(0))
subdict['values'] = item
result[name] = subdict
return result
def test_beta_2d_preconditioning(self):
"""
Interpolate a set of points using preconditioing. First select
all initial points then adding a subset of the remaining points.
x in Beta(2,5)[0,1]^2
"""
num_vars = 2
alpha_stat = 2
beta_stat = 5
var_trans = define_iid_random_variable_transformation(
beta(alpha_stat, beta_stat, -1, 2), num_vars)
pce_opts = define_poly_options_from_variable_transformation(var_trans)
# Set oli options
oli_opts = {'verbosity': 0, 'assume_non_degeneracy': False}
basis_generator = \
lambda num_vars, degree: (degree+1, compute_hyperbolic_level_indices(
num_vars, degree, 1.0))
# from scipy.special import beta as beta_fn
# def beta_pdf(x,alpha_poly,beta_poly):
# values = (1.-x)**(alpha_poly) * (1.+x)**(beta_poly)
# values /= 2.**(beta_poly+alpha_poly+1)*beta_fn(
# beta_poly+1,alpha_poly+1)
# return values
# univariate_pdf = partial(beta_pdf,alpha_poly=beta_stat-1,beta_poly=alpha_stat-1)
univariate_beta_pdf = partial(beta.pdf, a=alpha_stat, b=beta_stat)
def univariate_pdf(x):
return univariate_beta_pdf((x + 1.) / 2.) / 2.
preconditioning_function = partial(tensor_product_pdf,
univariate_pdfs=univariate_pdf)
# define target function
def model(x):
return np.asarray([(x[0]**2 - 1) + (x[1]**2 - 1) + x[0] * x[1]]).T
# define points to interpolate
pts = generate_independent_random_samples(var_trans.variable, 12)
initial_pts = np.array([pts[:, 0]]).T
helper_least_factorization(
pts,
model,
var_trans,
pce_opts,
oli_opts,
basis_generator,
initial_pts=initial_pts,
max_num_pts=12,
preconditioning_function=preconditioning_function)
def increment_samples(self,current_poly_indices,unique_poly_indices):
if self.induced_sampling:
samples = increment_induced_samples_migliorati(
self.pce,self.cond_tol,self.samples,
current_poly_indices, unique_poly_indices)
else:
samples = generate_independent_random_samples(self.pce.var_trans.variable,self.sample_ratio*unique_poly_indices.shape[1])
samples = self.pce.var_trans.map_to_canonical_space(samples)
samples = np.hstack([self.samples,samples])
return samples
def compute_l2_error(f, g, variable, nsamples, rel=False):
r"""
Compute the :math:`\ell^2` error of the output of two functions f and g, i.e.
.. math:: \lVertf(z)-g(z)\rVert\approx \sum_{m=1}^M f(z^{(m)})
from a set of random draws :math:`\mathcal{Z}=\{z^{(m)}\}_{m=1}^M`
from the PDF of :math:`z`.
Parameters
----------
f : callable
Function with signature
``g(z) -> np.ndarray``
where ``z`` is a 2D np.ndarray with shape (nvars,nsamples) and the
output is a 2D np.ndarray with shaoe (nsamples,nqoi)
g : callable
Function with signature
``f(z) -> np.ndarray``
where ``z`` is a 2D np.ndarray with shape (nvars,nsamples) and the
output is a 2D np.ndarray with shaoe (nsamples,nqoi)
variable : pya.IndependentMultivariateRandomVariable
Object containing information of the joint density of the inputs z.
This is used to generate random samples from this join density
nsamples : integer
The number of samples used to compute the :math:`\ell^2` error
rel : boolean
True - compute relative error
False - compute absolute error
Returns
-------
error : np.ndarray (nqoi)
"""
validation_samples = generate_independent_random_samples(
variable, nsamples)
validation_vals = f(validation_samples)
approx_vals = g(validation_samples)
assert validation_vals.shape == approx_vals.shape
error = np.linalg.norm(approx_vals - validation_vals, axis=0)
if not rel:
error /= np.sqrt(validation_samples.shape[1])
else:
error /= np.linalg.norm(validation_vals, axis=0)
return error
def test_bayesian_importance_sampling_avar(self):
np.random.seed(1)
nrandom_vars = 2
Amat = np.array([[-0.5, 1]])
noise_std = 0.1
prior_variable = IndependentMultivariateRandomVariable(
[stats.norm(0, 1)] * nrandom_vars)
prior_mean = prior_variable.get_statistics('mean')
prior_cov = np.diag(prior_variable.get_statistics('var')[:, 0])
prior_cov_inv = np.linalg.inv(prior_cov)
noise_cov_inv = np.eye(Amat.shape[0]) / noise_std**2
true_sample = np.array([.4] * nrandom_vars)[:, None]
collected_obs = Amat.dot(true_sample)
collected_obs += np.random.normal(0, noise_std, (collected_obs.shape))
exact_post_mean, exact_post_cov = \
laplace_posterior_approximation_for_linear_models(
Amat, prior_mean, prior_cov_inv, noise_cov_inv,
collected_obs)
chol_factor = np.linalg.cholesky(exact_post_cov)
chol_factor_inv = np.linalg.inv(chol_factor)
def g_model(samples):
return np.exp(
np.sum(chol_factor_inv.dot(samples - exact_post_mean),
axis=0))[:, None]
nsamples = int(1e6)
prior_samples = generate_independent_random_samples(
prior_variable, nsamples)
posterior_samples = chol_factor.dot(
np.random.normal(0, 1, (nrandom_vars, nsamples))) + exact_post_mean
g_mu, g_sigma = 0, np.sqrt(nrandom_vars)
f, f_cdf, f_pdf, VaR, CVaR, ssd, ssd_disutil = \
get_lognormal_example_exact_quantities(g_mu, g_sigma)
beta = .1
cvar_exact = CVaR(beta)
cvar_mc = conditional_value_at_risk(g_model(posterior_samples), beta)
prior_pdf = prior_variable.pdf
post_pdf = stats.multivariate_normal(mean=exact_post_mean[:, 0],
cov=exact_post_cov).pdf
weights = post_pdf(prior_samples.T) / prior_pdf(prior_samples)[:, 0]
weights /= weights.sum()
cvar_im = conditional_value_at_risk(g_model(prior_samples), beta,
weights)
# print(cvar_exact, cvar_mc, cvar_im)
assert np.allclose(cvar_exact, cvar_mc, rtol=1e-3)
assert np.allclose(cvar_exact, cvar_im, rtol=2e-3)
def allocate_initial_samples(self):
if self.induced_sampling:
return generate_induced_samples_migliorati_tolerance(
self.pce, self.cond_tol)
if self.cond_tol < 0:
sample_ratio = -self.cond_tol
return generate_independent_random_samples(
self.pce.var_trans.variable,
sample_ratio * self.pce.num_terms())
return generate_probability_samples_tolerance(self.pce,
self.pce.num_terms(),
self.cond_tol)
def test_peer_feed_forward_system_of_polynomials(self):
graph, variables, graph_data, system_fun = \
get_3_peer_polynomial_components()
network = SystemNetwork(graph)
nsamples = 10
samples = generate_independent_random_samples(variables, nsamples)
values = network(samples, [0, 1, 2])
component_nvars = network.component_nvars()
assert component_nvars == [1, 2, 3]
true_values = system_fun(samples)
assert np.allclose(values, true_values)
def test_add_pce(self):
univariate_variables = [norm(), uniform()]
variable = IndependentMultivariateRandomVariable(univariate_variables)
degree1, degree2 = 2, 3
poly1 = get_polynomial_from_variable(variable)
poly1.set_indices(
compute_hyperbolic_indices(variable.num_vars(), degree1))
poly1.set_coefficients(
np.random.normal(0, 1, (poly1.indices.shape[1], 1)))
poly2 = get_polynomial_from_variable(variable)
poly2.set_indices(
compute_hyperbolic_indices(variable.num_vars(), degree2))
poly2.set_coefficients(
np.random.normal(0, 1, (poly2.indices.shape[1], 1)))
poly3 = poly1 + poly2 + poly2
samples = generate_independent_random_samples(variable, 10)
# print(poly3(samples),poly1(samples)*poly2(samples))
assert np.allclose(poly3(samples), poly1(samples) + 2 * poly2(samples))
poly4 = poly1 - poly2
samples = generate_independent_random_samples(variable, 10)
# print(poly3(samples),poly1(samples)*poly2(samples))
assert np.allclose(poly4(samples), poly1(samples) - poly2(samples))
def test_gauss_jacobi_fpi_feedback(self):
ncomponent_outputs = [2, 2, 1]
ncomponent_coupling_vars = [2, 2, 2]
noutputs = np.sum(ncomponent_outputs)
ncoupling_vars = np.sum(ncomponent_coupling_vars)
adjacency_matrix = np.zeros((ncoupling_vars, noutputs))
adjacency_matrix[0, 0] = 1 # xi_00 = C1
adjacency_matrix[1, 2] = 1 # xi_01 = C2
adjacency_matrix[2, 0] = 1 # xi_10 = C1
adjacency_matrix[3, 2] = 1 # xi_11 = C2
adjacency_matrix[4, 1] = 1 # xi_20 = y1
adjacency_matrix[5, 3] = 1 # xi_21 = y2
adjacency_matrix = scipy_sparse.csr_matrix(adjacency_matrix)
# plot_adjacency_matrix(
# adjacency_matrix, (ncomponent_coupling_vars, ncomponent_outputs))
# from matplotlib import pyplot as plt
# plt.show()
component_funs, variables = get_chaudhuri_3_component_system()
# output_extraction_indices = [[0, 1], [1, 2], [3]]
exog_extraction_indices = [[0, 1, 2], [0, 3, 4], []]
coup_extraction_indices = [[0, 1], [2, 3], [4, 5]]
nsamples = 10
exog_samples = generate_independent_random_samples(
variables, nsamples - 1)
exog_samples = np.hstack(
(exog_samples, variables.get_statistics("mean")))
init_coup_samples = 5 * np.ones((adjacency_matrix.shape[0], nsamples))
outputs = gauss_jacobi_fixed_point_iteration(adjacency_matrix,
exog_extraction_indices,
coup_extraction_indices,
component_funs,
init_coup_samples,
exog_samples,
tol=1e-12,
max_iters=20,
verbose=0,
anderson_memory=1)[0]
# Mathematica Solution
# Solve[(0.02 + (9.7236*x + 0.2486*y)/Sqrt[x^2 + y^2] - x == 0) &&
# 0.03 + (9.7764*y + 0.2486*x)/Sqrt[x^2 + y^2] - y == 0, {x, y}]
# print(outputs[-1, [0, 2]], [6.63852, 7.52628])
assert np.allclose(outputs[-1, [0, 2]], [6.63852, 7.52628])
def find_uncertainty_aware_beam_design(constraint_type='quantile'):
quantile=0.1
print(quantile)
quantile_lower_bound = 0
uq_vars, objective, constraint_functions, bounds, init_design_sample = \
setup_beam_design()
if constraint_type=='quantile':
constraint_info = [{'type':'quantile',
'lower_bound':quantile_lower_bound,
'quantile':quantile}]*len(constraint_functions)
elif constraint_type=='cvar':
constraint_info = [{'type':'cvar',
'lower_bound':0.05,
'quantile':quantile,
'smoothing_eps':1e-1}]*len(
constraint_functions)
else:
raise Exception()
nsamples=10000
uq_samples = generate_independent_random_samples(uq_vars,nsamples)
#constraints = setup_inequality_constraints(
# constraint_functions,constraint_info,uq_samples)
def generate_samples():
# always use the same samples avoid noise in constraint vals
np.random.seed(1)
return uq_samples
individual_constraints = [
MCStatisticConstraint(f,generate_samples,constraint_info[0])
for f in constraint_functions]
constraints = MultipleConstraints(individual_constraints)
optim_options={'ftol': 1e-9, 'disp': 3, 'maxiter':1000}
res, opt_history = run_design(
objective,init_design_sample,constraints,bounds,optim_options)
return objective,constraints,constraint_functions,uq_samples,res,\
opt_history
def test_recursive_feed_forward_system_of_polynomials(self):
graph, variables, graph_data = get_3_recursive_polynomial_components()
network = SystemNetwork(graph)
nsamples = 10
samples = generate_independent_random_samples(variables, nsamples)
values = network(samples)
component_nvars = network.component_nvars()
assert component_nvars == [1, 2, 2]
funs = graph_data['functions']
global_random_var_indices = graph_data['global_random_var_indices']
values0 = funs[0](samples[global_random_var_indices[0], :])
values1 = funs[1](np.vstack(
[values0.T, samples[global_random_var_indices[1], :]]))
true_values = funs[2](np.vstack(
[samples[global_random_var_indices[2], :], values1.T]))
assert np.allclose(values, true_values)
def helper(self, function, var_trans, pce, max_level, error_tol):
max_level_1d = [max_level]*(pce.num_vars)
max_num_samples = 100
admissibility_function = partial(
max_level_admissibility_function, max_level, max_level_1d,
max_num_samples, error_tol)
refinement_indicator = variance_pce_refinement_indicator
pce.set_function(function, var_trans)
pce.set_refinement_functions(
refinement_indicator, admissibility_function,
clenshaw_curtis_rule_growth)
pce.build()
validation_samples = generate_independent_random_samples(
var_trans.variable, int(1e3))
validation_vals = function(validation_samples)
pce_vals = pce(validation_samples)
error = np.linalg.norm(pce_vals-validation_vals)/np.sqrt(
validation_samples.shape[1])
return error, pce
def get_coupling_variables_via_sampling(network,
random_variables,
nsamples,
expansion_factor=0.1,
filename=None):
"""
Compute the bounds on the coupling variables (output of upstream models)
using Monte Carlo sampling. Return uniform variables over a slgithly larger
range
"""
if filename is not None and os.path.exists(filename):
print(f'loading file {filename}')
values = np.load(filename)['values']
else:
samples = generate_independent_random_samples(random_variables,
nsamples)
component_ids = np.arange(len(network.graph.nodes))
values = network(samples, component_ids)
if filename is not None:
np.savez(filename, values=values, samples=samples)
coupling_bounds = np.array([[v.min(axis=0), v.max(axis=0)]
for v in values])
coupling_variables = {}
graph = network.graph
for jj in graph.nodes:
coupling_variables[jj] = []
indices = graph.nodes[jj]['global_coupling_component_indices']
for ii in range(len(indices) // 2):
lb = coupling_bounds[indices[2 * ii]][0][indices[2 * ii + 1]]
ub = coupling_bounds[indices[2 * ii]][1][indices[2 * ii + 1]]
diff = ub - lb
lb = lb - diff * expansion_factor / 2
ub = ub + diff * expansion_factor / 2
coupling_variables[jj].append(stats.uniform(lb, ub - lb))
return coupling_variables
def genz_example(max_num_samples, precond_type):
error_tol = 1e-12
univariate_variables = [uniform(), beta(3, 3)]
variable = IndependentMultivariateRandomVariable(univariate_variables)
var_trans = AffineRandomVariableTransformation(variable)
c = np.array([10, 0.00])
model = GenzFunction("oscillatory",
variable.num_vars(),
c=c,
w=np.zeros_like(c))
# model.set_coefficients(4,'exponential-decay')
validation_samples = generate_independent_random_samples(
var_trans.variable, int(1e3))
validation_values = model(validation_samples)
errors = []
num_samples = []
def callback(pce):
error = compute_l2_error(validation_samples, validation_values, pce)
errors.append(error)
num_samples.append(pce.samples.shape[1])
candidate_samples = -np.cos(
np.random.uniform(0, np.pi, (var_trans.num_vars(), int(1e4))))
pce = AdaptiveLejaPCE(var_trans.num_vars(),
candidate_samples,
factorization_type='fast')
if precond_type == 'density':
def precond_function(basis_matrix, samples):
trans_samples = var_trans.map_from_canonical_space(samples)
vals = np.ones(samples.shape[1])
for ii in range(len(univariate_variables)):
rv = univariate_variables[ii]
vals *= np.sqrt(rv.pdf(trans_samples[ii, :]))
return vals
elif precond_type == 'christoffel':
precond_function = chistoffel_preconditioning_function
else:
raise Exception(f'Preconditioner: {precond_type} not supported')
pce.set_preconditioning_function(precond_function)
max_level = np.inf
max_level_1d = [max_level] * (pce.num_vars)
admissibility_function = partial(max_level_admissibility_function,
max_level, max_level_1d, max_num_samples,
error_tol)
growth_rule = partial(constant_increment_growth_rule, 2)
#growth_rule = clenshaw_curtis_rule_growth
pce.set_function(model, var_trans)
pce.set_refinement_functions(variance_pce_refinement_indicator,
admissibility_function, growth_rule)
while (not pce.active_subspace_queue.empty()
or pce.subspace_indices.shape[1] == 0):
pce.refine()
pce.recompute_active_subspace_priorities()
if callback is not None:
callback(pce)
from pyapprox.sparse_grid import plot_sparse_grid_2d
plot_sparse_grid_2d(pce.samples, np.ones(pce.samples.shape[1]),
pce.pce.indices, pce.subspace_indices)
plt.figure()
plt.loglog(num_samples, errors, 'o-')
plt.show()
input_path = file_settings()[1]
filename = f'{input_path}parameter.csv'
variable = variables_prep(filename, product_uniform=False)
index_product = np.load(f'{input_path}index_product.npy', allow_pickle=True)
# Check whether the Beta distribution is a proper option
filename = f'{input_path}parameter-adjust.csv'
param_adjust = pd.read_csv(filename)
beta_index = param_adjust[param_adjust['distribution']== 'beta'].\
index.to_list()
# prepare the loc and scale argument for Beta fit
locs = np.array(param_adjust.loc[beta_index, ['min', 'max']])
locs[:, 1] = locs[:, 1] - locs[:, 0]
param_names = param_adjust.loc[beta_index, 'Veneer_name'].values
num_samples = 20000
samples_uniform = generate_independent_random_samples(variable, num_samples)
beta_fit = np.zeros(shape=(len(param_names), 4))
for ii in range(list(index_product.shape)[0]):
index_temp = index_product[ii]
samples_uniform[index_temp[0], :] = np.prod(samples_uniform[index_temp, :],
axis=0)
# fit the Beta distribution
rv_product = samples_uniform[index_temp[0]]
beta_aguments = beta.fit(rv_product, floc=locs[ii][0], fscale=locs[ii][1])
beta_fit[ii, :] = np.round(beta_aguments, 4)
# calculate the KS-statistic
num_boot = 1000
ks_stat = []
for i in range(num_boot):
def test_gauss_jacobi_fpi_peer(self):
ncomponent_outputs = [1, 1, 1]
ncomponent_coupling_vars = [0, 0, 2]
noutputs = np.sum(ncomponent_outputs)
ncoupling_vars = np.sum(ncomponent_coupling_vars)
adjacency_matrix = np.zeros((ncoupling_vars, noutputs))
adjacency_matrix[0, 0] = 1
adjacency_matrix[1, 1] = 1
adjacency_matrix = scipy_sparse.csr_matrix(adjacency_matrix)
# plot_adjacency_matrix(adjacency_matrix, component_shapes)
# from matplotlib import pyplot as plt
# plt.show()
# output_extraction_indices = [[0], [1], [2]]
exog_extraction_indices = [[0], [0, 1], [2]]
coup_extraction_indices = [[], [], [1, 0]]
qoi_ext_indices = [2]
graph, variables, graph_data, system_fun = \
get_3_peer_polynomial_components()
component_funs = graph_data["functions"]
nsamples = 10
exog_samples = generate_independent_random_samples(variables, nsamples)
init_coup_samples = np.ones((adjacency_matrix.shape[0], nsamples))
outputs = gauss_jacobi_fixed_point_iteration(adjacency_matrix,
exog_extraction_indices,
coup_extraction_indices,
component_funs,
init_coup_samples,
exog_samples,
tol=1e-15,
max_iters=100,
verbose=0)[0]
true_outputs = system_fun(exog_samples)
assert np.allclose(outputs, np.hstack(true_outputs))
# test when component_ids are specified
network = GaussJacobiSystemNetwork(graph)
network.set_adjacency_matrix(adjacency_matrix)
network.set_extraction_indices(exog_extraction_indices,
coup_extraction_indices,
qoi_ext_indices, ncomponent_outputs)
component_ids = [0, 1, 2]
outputs = network(exog_samples,
component_ids,
init_coup_samples=init_coup_samples)
assert np.allclose(np.hstack(outputs), np.hstack(true_outputs))
# test when component_ids are not specified and so qoi indices are
# needed
qoi_ext_indices = [2] # return qoi of last model
network = GaussJacobiSystemNetwork(graph)
network.set_adjacency_matrix(adjacency_matrix)
network.set_extraction_indices(exog_extraction_indices,
coup_extraction_indices,
qoi_ext_indices, ncomponent_outputs)
component_ids = None
outputs = network(exog_samples,
component_ids,
init_coup_samples=init_coup_samples)
# print(outputs, true_outputs[-1])
assert np.allclose(outputs, true_outputs[-1])
def helper_least_factorization(pts, model, var_trans, pce_opts, oli_opts,
basis_generator,
max_num_pts=None, initial_pts=None,
pce_degree=None,
preconditioning_function=None,
verbose=False,
points_non_degenerate=False,
exact_mean=None):
num_vars = pts.shape[0]
pce = PolynomialChaosExpansion()
pce.configure(pce_opts)
oli_solver = LeastInterpolationSolver()
oli_solver.configure(oli_opts)
oli_solver.set_pce(pce)
if preconditioning_function is not None:
oli_solver.set_preconditioning_function(preconditioning_function)
oli_solver.set_basis_generator(basis_generator)
if max_num_pts is None:
max_num_pts = pts.shape[1]
if initial_pts is not None:
# find unique set of points and separate initial pts from pts
# this allows for cases when
# (1) pts intersect initial_pts = empty
# (2) pts intersect initial_pts = initial pts
# (3) 0 < #(pts intersect initial_pts) < #initial_pts
pts = remove_common_rows([pts.T,initial_pts.T]).T
oli_solver.factorize(
pts, initial_pts,
num_selected_pts = max_num_pts)
permuted_pts = oli_solver.get_current_points()
permuted_vals = model( permuted_pts )
pce = oli_solver.get_current_interpolant(
permuted_pts, permuted_vals)
assert permuted_pts.shape[1] == max_num_pts
# Ensure pce interpolates the training data
pce_vals = pce.value( permuted_pts )
assert np.allclose( permuted_vals, pce_vals )
# Ensure pce exactly approximates the polynomial test function (model)
test_pts = generate_independent_random_samples(var_trans.variable,num_samples=10)
test_vals = model(test_pts)
#print 'p',test_pts.T
pce_vals = pce.value(test_pts)
L,U,H=oli_solver.get_current_LUH_factors()
#print L
#print U
#print test_vals
#print pce_vals
#print 'coeff',pce.get_coefficients()
#print oli_solver.selected_basis_indices
assert np.allclose( test_vals, pce_vals )
if initial_pts is not None:
temp = remove_common_rows([permuted_pts.T,initial_pts.T]).T
assert temp.shape[1]==max_num_pts-initial_pts.shape[1]
if oli_solver.enforce_ordering_of_initial_points:
assert np.allclose(
initial_pts,permuted_pts[:,:initial_pts.shape[0]])
elif not oli_solver.get_initial_points_degenerate():
assert allclose_unsorted_matrix_rows(
initial_pts.T, permuted_pts[:,:initial_pts.shape[1]].T)
else:
# make sure that oli tried again to add missing initial
# points after they were found to be degenerate
# often adding one new point will remove degeneracy
assert oli_solver.get_num_initial_points_selected()==\
initial_pts.shape[1]
P = oli_solver.get_current_permutation()
I = np.where(P<initial_pts.shape[1])[0]
assert_allclose_unsorted_matrix_cols(
initial_pts,permuted_pts[:,I])
basis_generator = oli_solver.get_basis_generator()
max_degree = oli_solver.get_current_degree()
basis_cardinality = oli_solver.get_basis_cardinality()
num_terms = 0
for degree in range(max_degree):
__,indices = basis_generator(num_vars,degree)
num_terms += indices.shape[1]
assert num_terms == basis_cardinality[degree]
if points_non_degenerate:
degree_list = oli_solver.get_points_to_degree_map()
num_terms = 1
degree = 0
num_pts = permuted_pts.shape[1]
for i in range(num_pts):
# test assumes non-degeneracy
if i>=num_terms:
degree+=1
indices = PolyIndexVector()
basis_generator.get_degree_basis_indices(
num_vars,degree,indices)
num_terms += indices.size()
assert degree_list[i] == degree
if exact_mean is not None:
mean = pce.get_coefficients()[0,0]
assert np.allclose(mean,exact_mean)