def test_pce_sensitivities_of_sobol_g_function(self):
nsamples = 2000
nvars, degree = 3, 8
a = np.array([1, 2, 5])[:nvars]
univariate_variables = [uniform(0, 1)] * nvars
variable = pya.IndependentMultivariateRandomVariable(
univariate_variables)
var_trans = pya.AffineRandomVariableTransformation(variable)
poly = pya.PolynomialChaosExpansion()
poly_opts = pya.define_poly_options_from_variable_transformation(
var_trans)
poly.configure(poly_opts)
indices = pya.tensor_product_indices([degree] * nvars)
poly.set_indices(indices)
#print('No. PCE Terms',indices.shape[1])
samples = pya.generate_independent_random_samples(
var_trans.variable, nsamples)
samples = (np.cos(np.random.uniform(0, np.pi,
(nvars, nsamples))) + 1) / 2
values = sobol_g_function(a, samples)
basis_matrix = poly.basis_matrix(samples)
weights = 1 / np.sum(basis_matrix**2, axis=1)[:, np.newaxis]
coef = np.linalg.lstsq(basis_matrix * weights,
values * weights,
rcond=None)[0]
poly.set_coefficients(coef)
nvalidation_samples = 1000
validation_samples = pya.generate_independent_random_samples(
var_trans.variable, nvalidation_samples)
validation_values = sobol_g_function(a, validation_samples)
poly_validation_vals = poly(validation_samples)
rel_error = np.linalg.norm(poly_validation_vals - validation_values
) / np.linalg.norm(validation_values)
print('Rel. Error', rel_error)
pce_main_effects, pce_total_effects =\
pya.get_main_and_total_effect_indices_from_pce(
poly.get_coefficients(), poly.get_indices())
interaction_terms, pce_sobol_indices = get_sobol_indices(
poly.get_coefficients(), poly.get_indices(), max_order=3)
mean, variance, main_effects, total_effects, sobol_indices = \
get_sobol_g_function_statistics(a, interaction_terms)
assert np.allclose(poly.mean(), mean, atol=1e-2)
# print((poly.variance(),variance))
assert np.allclose(poly.variance(), variance, atol=1e-2)
# print(pce_main_effects,main_effects)
assert np.allclose(pce_main_effects, main_effects, atol=1e-2)
# print(pce_total_effects,total_effects)
assert np.allclose(pce_total_effects, total_effects, atol=1e-2)
assert np.allclose(pce_sobol_indices, sobol_indices, atol=1e-2)
def test_pce_sensitivities_of_ishigami_function(self):
nsamples = 1500
nvars, degree = 3, 18
univariate_variables = [uniform(-np.pi, 2 * np.pi)] * nvars
variable = pya.IndependentMultivariateRandomVariable(
univariate_variables)
var_trans = pya.AffineRandomVariableTransformation(variable)
poly = pya.PolynomialChaosExpansion()
poly_opts = pya.define_poly_options_from_variable_transformation(
var_trans)
poly.configure(poly_opts)
indices = pya.compute_hyperbolic_indices(nvars, degree, 1.0)
poly.set_indices(indices)
#print('No. PCE Terms',indices.shape[1])
samples = pya.generate_independent_random_samples(
var_trans.variable, nsamples)
values = ishigami_function(samples)
basis_matrix = poly.basis_matrix(samples)
coef = np.linalg.lstsq(basis_matrix, values, rcond=None)[0]
poly.set_coefficients(coef)
nvalidation_samples = 1000
validation_samples = pya.generate_independent_random_samples(
var_trans.variable, nvalidation_samples)
validation_values = ishigami_function(validation_samples)
poly_validation_vals = poly(validation_samples)
abs_error = np.linalg.norm(poly_validation_vals - validation_values
) / np.sqrt(nvalidation_samples)
#print('Abs. Error',abs_error)
pce_main_effects, pce_total_effects =\
pya.get_main_and_total_effect_indices_from_pce(
poly.get_coefficients(), poly.get_indices())
mean, variance, main_effects, total_effects, sobol_indices, \
sobol_interaction_indices = get_ishigami_funciton_statistics()
assert np.allclose(poly.mean(), mean)
assert np.allclose(poly.variance(), variance)
assert np.allclose(pce_main_effects, main_effects)
assert np.allclose(pce_total_effects, total_effects)
interaction_terms, pce_sobol_indices = get_sobol_indices(
poly.get_coefficients(), poly.get_indices(), max_order=3)
assert np.allclose(pce_sobol_indices, sobol_indices)
def test_marginalize_polynomial_chaos_expansions(self):
univariate_variables = [uniform(-1, 2), norm(0, 1), uniform(-1, 2)]
variable = pya.IndependentMultivariateRandomVariable(
univariate_variables)
var_trans = pya.AffineRandomVariableTransformation(variable)
num_vars = len(univariate_variables)
poly = pya.PolynomialChaosExpansion()
poly_opts = pya.define_poly_options_from_variable_transformation(
var_trans)
poly.configure(poly_opts)
degree = 2
indices = pya.compute_hyperbolic_indices(num_vars, degree, 1)
poly.set_indices(indices)
poly.set_coefficients(np.ones((indices.shape[1], 1)))
pce_main_effects, pce_total_effects =\
pya.get_main_and_total_effect_indices_from_pce(
poly.get_coefficients(), poly.get_indices())
print(poly.num_terms())
for ii in range(num_vars):
# Marginalize out 2 variables
xx = np.linspace(-1, 1, 101)
inactive_idx = np.hstack(
(np.arange(ii), np.arange(ii + 1, num_vars)))
marginalized_pce = pya.marginalize_polynomial_chaos_expansion(
poly, inactive_idx, center=True)
mvals = marginalized_pce(xx[None, :])
variable_ii = variable.all_variables()[ii:ii + 1]
var_trans_ii = pya.AffineRandomVariableTransformation(variable_ii)
poly_ii = pya.PolynomialChaosExpansion()
poly_opts_ii = \
pya.define_poly_options_from_variable_transformation(
var_trans_ii)
poly_ii.configure(poly_opts_ii)
indices_ii = compute_hyperbolic_indices(1, degree, 1.)
poly_ii.set_indices(indices_ii)
poly_ii.set_coefficients(np.ones((indices_ii.shape[1], 1)))
pvals = poly_ii(xx[None, :])
# import matplotlib.pyplot as plt
# plt.plot(xx, pvals)
# plt.plot(xx, mvals, '--')
# plt.show()
assert np.allclose(mvals, pvals - poly.mean())
assert np.allclose(poly_ii.variance() / poly.variance(),
pce_main_effects[ii])
poly_ii.coefficients /= np.sqrt(poly.variance())
assert np.allclose(poly_ii.variance(), pce_main_effects[ii])
# Marginalize out 1 variable
xx = pya.cartesian_product([xx] * 2)
inactive_idx = np.array([ii])
marginalized_pce = pya.marginalize_polynomial_chaos_expansion(
poly, inactive_idx, center=True)
mvals = marginalized_pce(xx)
variable_ii = variable.all_variables()[:ii] +\
variable.all_variables()[ii+1:]
var_trans_ii = pya.AffineRandomVariableTransformation(variable_ii)
poly_ii = pya.PolynomialChaosExpansion()
poly_opts_ii = \
pya.define_poly_options_from_variable_transformation(
var_trans_ii)
poly_ii.configure(poly_opts_ii)
indices_ii = pya.compute_hyperbolic_indices(2, degree, 1.)
poly_ii.set_indices(indices_ii)
poly_ii.set_coefficients(np.ones((indices_ii.shape[1], 1)))
pvals = poly_ii(xx)
assert np.allclose(mvals, pvals - poly.mean())
