def test_beta_leja_quadrature(self):
level = 12
alpha_stat, beta_stat = 2, 10
x_quad, w_quad = beta_leja_quadrature_rule(
alpha_stat, beta_stat, level, return_weights_for_all_levels=False)
import sympy as sp
x = sp.Symbol('x')
weight_function = beta_pdf_on_ab(alpha_stat, beta_stat, -1, 1, x)
ranges = [-1, 1]
exact_integral = float(
sp.integrate(weight_function * x**3, (x, ranges[0], ranges[1])))
assert np.allclose(exact_integral, np.dot(x_quad**3, w_quad))
level = 12
alpha_stat, beta_stat = 2, 10
x_quad, w_quad = beta_leja_quadrature_rule(
alpha_stat, beta_stat, level, return_weights_for_all_levels=False)
x_quad = (x_quad + 1) / 2
import sympy as sp
x = sp.Symbol('x')
weight_function = beta_pdf_on_ab(alpha_stat, beta_stat, 0, 1, x)
ranges = [0, 1]
exact_integral = float(
sp.integrate(weight_function * x**3, (x, ranges[0], ranges[1])))
assert np.allclose(exact_integral, np.dot(x_quad**3, w_quad))
def test_beta_leja_quadrature(self):
level = 12
alpha_stat, beta_stat = 2, 10
quad_rule = get_univariate_leja_quadrature_rule(
stats.beta(alpha_stat, beta_stat),
leja_growth_rule,
return_weights_for_all_levels=False)
x_quad, w_quad = quad_rule(level)
x = sp.Symbol('x')
weight_function = beta_pdf_on_ab(alpha_stat, beta_stat, -1, 1, x)
ranges = [-1, 1]
exact_integral = float(
sp.integrate(weight_function * x**3, (x, ranges[0], ranges[1])))
assert np.allclose(exact_integral, np.dot(x_quad**3, w_quad))
level = 12
alpha_stat, beta_stat = 2, 10
x_quad, w_quad = quad_rule(level)
x_quad = (x_quad + 1) / 2
x = sp.Symbol('x')
weight_function = beta_pdf_on_ab(alpha_stat, beta_stat, 0, 1, x)
ranges = [0, 1]
exact_integral = float(
sp.integrate(weight_function * x**3, (x, ranges[0], ranges[1])))
assert np.allclose(exact_integral, np.dot(x_quad**3, w_quad))
def setup(self,num_vars,alpha_stat,beta_stat):
#univariate_weight_function=lambda x: beta(alpha_stat,beta_stat).pdf(
# (x+1)/2)/2
univariate_weight_function=lambda x: beta_pdf_on_ab(
alpha_stat,beta_stat,-1,1,x)
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-6)-
univariate_weight_function(0.5))/1e-6,
univariate_weight_function_deriv(0.5),atol=1e-6)
poly = PolynomialChaosExpansion()
var_trans = define_iid_random_variable_transformation(
uniform(-2,1),num_vars)
poly_opts = {'alpha_poly':beta_stat-1,'beta_poly':alpha_stat-1,
'var_trans':var_trans,'poly_type':'jacobi'}
poly.configure(poly_opts)
return weight_function, weight_function_deriv, poly
def univariate_weight_function(x):
return beta_pdf_on_ab(alpha_stat, beta_stat, -1, 1, x)