def beta_leja_quadrature_rule(alpha_stat,
beta_stat,
level,
growth_rule=leja_growth_rule,
samples_filename=None,
return_weights_for_all_levels=True,
initial_points=None):
"""
Return the samples and weights of the Leja quadrature rule for the beta
probability measure.
By construction these rules have polynomial ordering.
Parameters
----------
level : integer
The level of the isotropic sparse grid.
alpha_stat : integer
The alpha shape parameter of the Beta distribution
beta_stat : integer
The beta shape parameter of the Beta distribution
samples_filename : string
Name of file to save leja samples and weights to
Return
------
ordered_samples_1d : np.ndarray (num_samples_1d)
The reordered samples.
ordered_weights_1d : np.ndarray (num_samples_1d)
The reordered weights.
"""
from pyapprox.multivariate_polynomials import PolynomialChaosExpansion
from pyapprox.leja_sequences import get_leja_sequence_1d,\
get_quadrature_weights_from_samples
num_vars = 1
num_leja_samples = growth_rule(level)
#print(('num_leja_samples',num_leja_samples))
# freezing beta rv like below has huge overhead
# it creates a docstring each time which adds up to many seconds
# for repeated calls to pdf
#univariate_weight_function=lambda x: beta_rv(
# alpha_stat,beta_stat).pdf((x+1)/2)/2
#univariate_weight_function = lambda x: beta_rv.pdf(
# (x+1)/2,alpha_stat,beta_stat)/2
univariate_weight_function = lambda x: beta_pdf(alpha_stat, beta_stat,
(x + 1) / 2) / 2
univariate_weight_function_deriv = lambda x: beta_pdf_derivative(
alpha_stat, beta_stat, (x + 1) / 2) / 4
weight_function = partial(evaluate_tensor_product_function,
[univariate_weight_function] * num_vars)
weight_function_deriv = partial(gradient_of_tensor_product_function,
[univariate_weight_function] * num_vars,
[univariate_weight_function_deriv] *
num_vars)
# assert np.allclose(
# (univariate_weight_function(0.5+1e-8)-
# univariate_weight_function(0.5))/1e-8,
# univariate_weight_function_deriv(0.5),atol=1e-6)
poly = PolynomialChaosExpansion()
# must be imported locally otherwise I have a circular dependency
from pyapprox.variable_transformations import \
define_iid_random_variable_transformation
from scipy.stats import uniform
var_trans = define_iid_random_variable_transformation(
uniform(-1, 2), num_vars)
poly_opts = {
'poly_type': 'jacobi',
'alpha_poly': beta_stat - 1,
'beta_poly': alpha_stat - 1,
'var_trans': var_trans
}
poly.configure(poly_opts)
if samples_filename is None or not os.path.exists(samples_filename):
ranges = [-1, 1]
from scipy.stats import beta as beta_rv
if initial_points is None:
initial_points = np.asarray(
[[2 * beta_rv(alpha_stat, beta_stat).ppf(0.5) - 1]]).T
leja_sequence = get_leja_sequence_1d(num_leja_samples, initial_points,
poly, weight_function,
weight_function_deriv, ranges)
if samples_filename is not None:
np.savez(samples_filename, samples=leja_sequence)
else:
leja_sequence = np.load(samples_filename)['samples']
#print (leja_sequence.shape[1],growth_rule(level),level)
assert leja_sequence.shape[1] >= growth_rule(level)
leja_sequence = leja_sequence[:, :growth_rule(level)]
indices = np.arange(growth_rule(level))[np.newaxis, :]
poly.set_indices(indices)
ordered_weights_1d = get_leja_sequence_quadrature_weights(
leja_sequence, growth_rule, poly.basis_matrix, weight_function, level,
return_weights_for_all_levels)
return leja_sequence[0, :], ordered_weights_1d
def pdf(x):
return beta_pdf(1, 1, (x + 1) / 2) / 2
def pdf(x):
return beta_pdf(alpha_stat, beta_stat, (x + 1) / 2) / 2
def univariate_weight_function(x): return beta_pdf(
alpha_stat, beta_stat, (x+1)/2)/2
def univariate_weight_function_deriv(x): return beta_pdf_derivative(