def test_pce_jacobian(self):
degree = 2
alpha_stat, beta_stat = 2, 3
univariate_variables = [
stats.beta(alpha_stat, beta_stat, 0, 1),
stats.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)
fd_jac = approx_jacobian(lambda x: poly(x[:, np.newaxis])[0, :],
sample[:, 0])
assert np.allclose(jac, fd_jac)
def test_hermite_basis_for_lognormal_variables(self):
def function(x):
return (x.T)**2
degree = 2
# mu_g, sigma_g = 1e1, 0.1
mu_l, sigma_l = 2.1e11, 2.1e10
mu_g = np.log(mu_l**2 / np.sqrt(mu_l**2 + sigma_l**2))
sigma_g = np.sqrt(np.log(1 + sigma_l**2 / mu_l**2))
lognorm = stats.lognorm(s=sigma_g, scale=np.exp(mu_g))
# assert np.allclose([lognorm.mean(), lognorm.std()], [mu_l, sigma_l])
univariate_variables = [stats.norm(mu_g, sigma_g)]
var_trans = AffineRandomVariableTransformation(univariate_variables)
pce = PolynomialChaosExpansion()
pce_opts = define_poly_options_from_variable_transformation(var_trans)
pce.configure(pce_opts)
pce.set_indices(
compute_hyperbolic_indices(var_trans.num_vars(), degree, 1.))
nsamples = int(1e6)
samples = lognorm.rvs(nsamples)[None, :]
values = function(samples)
ntrain_samples = 20
train_samples = lognorm.rvs(ntrain_samples)[None, :]
train_values = function(train_samples)
from pyapprox.quantile_regression import solve_quantile_regression, \
solve_least_squares_regression
coef = solve_quantile_regression(0.5,
np.log(train_samples),
train_values,
pce.basis_matrix,
normalize_vals=True)
pce.set_coefficients(coef)
print(pce.mean(), values.mean())
assert np.allclose(pce.mean(), values.mean(), rtol=1e-3)
def test_float_rv_discrete_chebyshev(self):
N, degree = 10, 5
xk, pk = np.geomspace(1.0, 512.0, num=N), np.ones(N) / N
rv = float_rv_discrete(name='float_rv_discrete', values=(xk, pk))()
var_trans = AffineRandomVariableTransformation([rv])
poly = PolynomialChaosExpansion()
poly_opts = define_poly_options_from_variable_transformation(var_trans)
poly_opts['numerically_generated_poly_accuracy_tolerance'] = 1e-9
poly.configure(poly_opts)
poly.set_indices(np.arange(degree + 1)[np.newaxis, :])
p = poly.basis_matrix(xk[np.newaxis, :])
w = pk
assert np.allclose(np.dot(p.T * w, p), np.eye(degree + 1))
def test_discrete_chebyshev(self):
N, degree = 10, 5
xk, pk = np.arange(N), np.ones(N) / N
rv = float_rv_discrete(name='discrete_chebyshev', values=(xk, pk))()
var_trans = AffineRandomVariableTransformation([rv])
poly = PolynomialChaosExpansion()
poly_opts = define_poly_options_from_variable_transformation(var_trans)
poly.configure(poly_opts)
poly.set_indices(np.arange(degree + 1)[np.newaxis, :])
p = poly.basis_matrix(xk[np.newaxis, :])
w = pk
# print((np.dot(p.T*w, p), np.eye(degree+1)))
assert np.allclose(np.dot(p.T * w, p), np.eye(degree + 1))
def test_krawtchouk_binomial(self):
degree = 4
n, p = 10, 0.5
rv = stats.binom(n, p)
var_trans = AffineRandomVariableTransformation([rv])
poly = PolynomialChaosExpansion()
poly_opts = define_poly_options_from_variable_transformation(var_trans)
poly.configure(poly_opts)
poly.set_indices(np.arange(degree + 1)[np.newaxis, :])
xk = np.arange(0, n + 1)[np.newaxis, :]
p = poly.basis_matrix(xk)
w = rv.pmf(xk[0, :])
assert np.allclose(np.dot(p.T * w, p), np.eye(degree + 1))
def test_hahn_hypergeometric(self):
degree = 4
M, n, N = 20, 7, 12
rv = stats.hypergeom(M, n, N)
var_trans = AffineRandomVariableTransformation([rv])
poly = PolynomialChaosExpansion()
poly_opts = define_poly_options_from_variable_transformation(var_trans)
poly.configure(poly_opts)
poly.set_indices(np.arange(degree + 1)[np.newaxis, :])
xk = np.arange(0, n + 1)[np.newaxis, :]
p = poly.basis_matrix(xk)
w = rv.pmf(xk[0, :])
assert np.allclose(np.dot(p.T * w, p), np.eye(degree + 1))
def test_pce_for_gumbel_variable(self):
degree = 3
mean, std = 1e4, 7.5e3
beta = std * np.sqrt(6) / np.pi
mu = mean - beta * np.euler_gamma
rv1 = stats.gumbel_r(loc=mu, scale=beta)
assert np.allclose(rv1.mean(), mean) and np.allclose(rv1.std(), std)
rv2 = stats.lognorm(1)
for rv in [rv2, rv1]:
var_trans = AffineRandomVariableTransformation([rv])
poly = PolynomialChaosExpansion()
poly_opts = define_poly_options_from_variable_transformation(
var_trans)
poly_opts['numerically_generated_poly_accuracy_tolerance'] = 1e-9
poly.configure(poly_opts)
poly.set_indices(np.arange(degree + 1)[np.newaxis, :])
poly.set_coefficients(np.ones((poly.indices.shape[1], 1)))
def integrand(x):
p = poly.basis_matrix(x[np.newaxis, :])
G = np.empty((x.shape[0], p.shape[1]**2))
kk = 0
for ii in range(p.shape[1]):
for jj in range(p.shape[1]):
G[:, kk] = p[:, ii] * p[:, jj]
kk += 1
return G * rv.pdf(x)[:, None]
lb, ub = rv.interval(1)
interval_size = rv.interval(0.99)[1] - rv.interval(0.99)[0]
interval_size *= 10
res = \
integrate_using_univariate_gauss_legendre_quadrature_unbounded(
integrand, lb, ub, 10, interval_size=interval_size,
verbose=0, max_steps=10000)
res = np.reshape(res,
(poly.indices.shape[1], poly.indices.shape[1]),
order='C')
# print('r', res-np.eye(degree+1))
assert np.allclose(res, np.eye(degree + 1), atol=1e-6)
def test_conditional_moments_of_polynomial_chaos_expansion(self):
num_vars = 3
degree = 2
inactive_idx = [0, 2]
np.random.seed(1)
# keep variables on canonical domain to make constructing
# tensor product quadrature rule, used for testing, easier
var = [stats.uniform(-1, 2), stats.beta(2, 2, -1, 2), stats.norm(0, 1)]
quad_rules = [
partial(gauss_jacobi_pts_wts_1D, alpha_poly=0, beta_poly=0),
partial(gauss_jacobi_pts_wts_1D, alpha_poly=1, beta_poly=1),
partial(gauss_hermite_pts_wts_1D)
]
var_trans = AffineRandomVariableTransformation(var)
poly = PolynomialChaosExpansion()
poly_opts = define_poly_options_from_variable_transformation(var_trans)
poly.configure(poly_opts)
poly.set_indices(compute_hyperbolic_indices(num_vars, degree, 1.0))
poly.set_coefficients(
np.arange(poly.indices.shape[1], dtype=float)[:, np.newaxis])
fixed_samples = np.array(
[[vv.rvs() for vv in np.array(var)[inactive_idx]]]).T
mean, variance = conditional_moments_of_polynomial_chaos_expansion(
poly, fixed_samples, inactive_idx, True)
active_idx = np.setdiff1d(np.arange(num_vars), inactive_idx)
random_samples, weights = get_tensor_product_quadrature_rule(
[2 * degree] * len(active_idx), len(active_idx),
[quad_rules[ii] for ii in range(num_vars) if ii in active_idx])
samples = get_all_sample_combinations(fixed_samples, random_samples)
temp = samples[len(inactive_idx):].copy()
samples[inactive_idx] = samples[:len(inactive_idx)]
samples[active_idx] = temp
true_mean = (poly(samples).T.dot(weights).T)
true_variance = ((poly(samples)**2).T.dot(weights).T) - true_mean**2
assert np.allclose(true_mean, mean)
assert np.allclose(true_variance, variance)
def test_pce_product_of_beta_variables(self):
def fun(x):
return np.sqrt(x.prod(axis=0))[:, None]
dist_alpha1, dist_beta1 = 1, 1
dist_alpha2, dist_beta2 = dist_alpha1 + 0.5, dist_beta1
nvars = 2
x_1d, w_1d = [], []
nquad_samples_1d = 100
x, w = gauss_jacobi_pts_wts_1D(nquad_samples_1d, dist_beta1 - 1,
dist_alpha1 - 1)
x = (x + 1) / 2
x_1d.append(x)
w_1d.append(w)
x, w = gauss_jacobi_pts_wts_1D(nquad_samples_1d, dist_beta2 - 1,
dist_alpha2 - 1)
x = (x + 1) / 2
x_1d.append(x)
w_1d.append(w)
quad_samples = cartesian_product(x_1d)
quad_weights = outer_product(w_1d)
mean = fun(quad_samples)[:, 0].dot(quad_weights)
variance = (fun(quad_samples)[:, 0]**2).dot(quad_weights) - mean**2
assert np.allclose(mean,
stats.beta(dist_alpha1 * 2, dist_beta1 * 2).mean())
assert np.allclose(variance,
stats.beta(dist_alpha1 * 2, dist_beta1 * 2).var())
degree = 10
poly = PolynomialChaosExpansion()
# the distribution and ranges of univariate variables is ignored
# when var_trans.set_identity_maps([0]) is used
initial_variables = [stats.uniform(0, 1)]
# TODO get quad rules from initial variables
quad_rules = [(x, w) for x, w in zip(x_1d, w_1d)]
univariate_variables = [
rv_function_indpndt_vars(fun, initial_variables, quad_rules)
]
variable = IndependentMultivariateRandomVariable(univariate_variables)
var_trans = AffineRandomVariableTransformation(variable)
poly_opts = define_poly_options_from_variable_transformation(var_trans)
poly.configure(poly_opts)
poly.set_indices(tensor_product_indices([degree]))
train_samples = (np.linspace(0, np.pi, 101)[None, :] + 1) / 2
train_vals = train_samples.T
coef = np.linalg.lstsq(poly.basis_matrix(train_samples),
train_vals,
rcond=None)[0]
poly.set_coefficients(coef)
assert np.allclose(poly.mean(),
stats.beta(dist_alpha1 * 2, dist_beta1 * 2).mean())
assert np.allclose(poly.variance(),
stats.beta(dist_alpha1 * 2, dist_beta1 * 2).var())
poly = PolynomialChaosExpansion()
initial_variables = [stats.uniform(0, 1)]
funs = [lambda x: np.sqrt(x)] * nvars
quad_rules = [(x, w) for x, w in zip(x_1d, w_1d)]
# TODO get quad rules from initial variables
univariate_variables = [
rv_product_indpndt_vars(funs, initial_variables, quad_rules)
]
variable = IndependentMultivariateRandomVariable(univariate_variables)
var_trans = AffineRandomVariableTransformation(variable)
poly_opts = define_poly_options_from_variable_transformation(var_trans)
poly.configure(poly_opts)
poly.set_indices(tensor_product_indices([degree]))
train_samples = (np.linspace(0, np.pi, 101)[None, :] + 1) / 2
train_vals = train_samples.T
coef = np.linalg.lstsq(poly.basis_matrix(train_samples),
train_vals,
rcond=None)[0]
poly.set_coefficients(coef)
assert np.allclose(poly.mean(),
stats.beta(dist_alpha1 * 2, dist_beta1 * 2).mean())
assert np.allclose(poly.variance(),
stats.beta(dist_alpha1 * 2, dist_beta1 * 2).var())
def test_composition_of_orthonormal_polynomials(self):
def fn1(z):
# return W_1
return (z[0, :] + 3 * z[0, :]**2)[:, None]
def fn2(z):
# return W_2
return (1 + z[0, :] * z[1, :])[:, None]
def fn3(z):
# z is just random variables
return z[0:1, :].T + 35 * (3 * fn1(z)**2 -
1) + 3 * z[0:1, :].T * fn2(z)
def fn3_trans(x):
"""
x is z_1, W_1, W_2
"""
return (x[0:1, :] + 35 * (3 * x[1:2, :]**2 - 1) +
3 * x[0:1, :] * x[2:3, :])
nvars = 2
samples = np.random.uniform(-1, 1, (nvars, 100))
values = fn3(samples)
indices = compute_hyperbolic_indices(nvars, 4, 1)
poly = PolynomialChaosExpansion()
var_trans = define_iid_random_variable_transformation(
stats.uniform(-1, 2), nvars)
poly_opts = define_poly_options_from_variable_transformation(var_trans)
poly.configure(poly_opts)
poly.set_indices(indices)
basis_mat = poly.basis_matrix(samples)
coef = np.linalg.lstsq(basis_mat, values, rcond=None)[0]
mean = coef[0]
variance = np.sum(coef[1:]**2)
# print(mean, variance, 2595584/15-mean**2, 2059769/15)
assert np.allclose(mean, 189)
assert np.allclose(variance, 2595584 / 15 - mean**2)
samples = np.random.uniform(-1, 1, (nvars, 100000))
basis_mat = poly.basis_matrix(samples)
x = samples[0:1, :].T
y = samples[1:2, :].T
assert np.allclose(
basis_mat.dot(coef),
-35 + 4 * x + 3 * x**2 * y + 105 * x**2 + 630 * x**3 + 945 * x**4)
assert np.allclose(2 / np.sqrt(5) * basis_mat[:, 3:4], (3 * x**2 - 1))
assert np.allclose(
basis_mat.dot(coef), 4 * x + 3 * x**2 * y +
2 / np.sqrt(5) * 35 * basis_mat[:, 3:4] + 630 * x**3 + 945 * x**4)
assert np.allclose(
basis_mat.dot(coef),
382 * x + 3 * x**2 * y + 2 / np.sqrt(5) * 35 * basis_mat[:, 3:4] +
2 / np.sqrt(7) * 126 * basis_mat[:, 6:7] + 945 * x**4)
assert np.allclose(
basis_mat.dot(coef),
382 * x + 3 * x**2 * y + 2 / np.sqrt(5) * 35 * basis_mat[:, 3:4] +
2 / np.sqrt(7) * 126 * basis_mat[:, 6:7] +
8 / np.sqrt(9) * 27 * basis_mat[:, 10:11] + 810 * x**2 - 81)
assert np.allclose(
basis_mat.dot(coef), -81 + 270 + 382 * x + 3 * x**2 * y +
2 / np.sqrt(5) * 305 * basis_mat[:, 3:4] +
2 / np.sqrt(7) * 126 * basis_mat[:, 6:7] +
8 / np.sqrt(9) * 27 * basis_mat[:, 10:11])
assert np.allclose(
basis_mat.dot(coef),
189 + 382 * x + 2 / np.sqrt(5) * 305 * basis_mat[:, 3:4] +
2 / np.sqrt(7) * 126 * basis_mat[:, 6:7] +
8 / np.sqrt(9) * 27 * basis_mat[:, 10:11] +
2 / np.sqrt(15) * basis_mat[:, 8:9] + y)
assert np.allclose(
basis_mat.dot(coef),
189 + 1 / np.sqrt(3) * 382 * basis_mat[:, 1:2] +
2 / np.sqrt(5) * 305 * basis_mat[:, 3:4] +
2 / np.sqrt(7) * 126 * basis_mat[:, 6:7] +
8 / np.sqrt(9) * 27. * basis_mat[:, 10:11] +
2 / np.sqrt(15) * 1.0 * basis_mat[:, 8:9] +
1 / np.sqrt(3) * 1.0 * basis_mat[:, 2:3])
assert np.allclose(
variance, 382**2 / 3 + (2 * 305)**2 / 5 + (2 * 126)**2 / 7 +
(8 * 27)**2 / 9 + 4 / 15 + 1 / 3)
def test_evaluate_multivariate_mixed_basis_pce_moments(self):
degree = 2
alpha_stat, beta_stat = 2, 3
univariate_variables = [
stats.beta(alpha_stat, beta_stat, 0, 1),
stats.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)
univariate_quadrature_rules = [
partial(gauss_jacobi_pts_wts_1D,
alpha_poly=beta_stat - 1,
beta_poly=alpha_stat - 1), gauss_hermite_pts_wts_1D
]
samples, weights = get_tensor_product_quadrature_rule(
degree + 1, num_vars, univariate_quadrature_rules,
var_trans.map_from_canonical_space)
coef = np.ones((indices.shape[1], 2))
coef[:, 1] *= 2
poly.set_coefficients(coef)
basis_matrix = poly.basis_matrix(samples)
values = basis_matrix.dot(coef)
true_mean = values.T.dot(weights)
true_variance = (values.T**2).dot(weights) - true_mean**2
assert np.allclose(poly.mean(), true_mean)
assert np.allclose(poly.variance(), true_variance)
assert np.allclose(np.diag(poly.covariance()), poly.variance())
assert np.allclose(poly.covariance()[0, 1], coef[1:, 0].dot(coef[1:,
1]))
def test_evaluate_multivariate_monomial_pce(self):
num_vars = 2
degree = 2
poly = PolynomialChaosExpansion()
var_trans = define_iid_random_variable_transformation(
rv_continuous(name="continuous_monomial")(), num_vars)
poly_opts = define_poly_options_from_variable_transformation(var_trans)
poly.configure(poly_opts)
def univariate_quadrature_rule(nn):
x, w = gauss_jacobi_pts_wts_1D(nn, 0, 0)
x = (x + 1) / 2.
return x, w
samples, weights = get_tensor_product_quadrature_rule(
degree, num_vars, univariate_quadrature_rule)
indices = compute_hyperbolic_indices(num_vars, degree, 1.0)
# sort lexographically to make testing easier
II = np.lexsort((indices[0, :], indices[1, :], indices.sum(axis=0)))
indices = indices[:, II]
poly.set_indices(indices)
basis_matrix = poly.basis_matrix(samples, {'deriv_order': 1})
exact_basis_vals_1d = []
exact_basis_derivs_1d = []
for dd in range(num_vars):
x = samples[dd, :]
exact_basis_vals_1d.append(np.asarray([1 + 0. * x, x, x**2]).T)
exact_basis_derivs_1d.append(
np.asarray([0. * x, 1.0 + 0. * x, 2. * x]).T)
exact_basis_matrix = np.asarray([
exact_basis_vals_1d[0][:, 0], exact_basis_vals_1d[0][:, 1],
exact_basis_vals_1d[1][:, 1], exact_basis_vals_1d[0][:, 2],
exact_basis_vals_1d[0][:, 1] * exact_basis_vals_1d[1][:, 1],
exact_basis_vals_1d[1][:, 2]
]).T
# x1 derivative
exact_basis_matrix = np.vstack(
(exact_basis_matrix,
np.asarray([
0. * x, exact_basis_derivs_1d[0][:, 1], 0. * x,
exact_basis_derivs_1d[0][:, 2],
exact_basis_derivs_1d[0][:, 1] * exact_basis_vals_1d[1][:, 1],
0. * x
]).T))
# x2 derivative
exact_basis_matrix = np.vstack(
(exact_basis_matrix,
np.asarray([
0. * x, 0. * x, exact_basis_derivs_1d[1][:, 1], 0. * x,
exact_basis_vals_1d[0][:, 1] * exact_basis_derivs_1d[1][:, 1],
exact_basis_derivs_1d[1][:, 2]
]).T))
assert np.allclose(exact_basis_matrix, basis_matrix)
def test_evaluate_multivariate_mixed_basis_pce(self):
degree = 2
gauss_mean, gauss_var = -1, 4
univariate_variables = [
stats.uniform(-1, 2),
stats.norm(gauss_mean, np.sqrt(gauss_var)),
stats.uniform(0, 3)
]
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)
univariate_quadrature_rules = [
partial(gauss_jacobi_pts_wts_1D, alpha_poly=0, beta_poly=0),
gauss_hermite_pts_wts_1D,
partial(gauss_jacobi_pts_wts_1D, alpha_poly=0, beta_poly=0)
]
samples, weights = get_tensor_product_quadrature_rule(
degree + 1, num_vars, univariate_quadrature_rules,
var_trans.map_from_canonical_space)
indices = compute_hyperbolic_indices(num_vars, degree, 1.0)
# sort lexographically to make testing easier
indices = sort_indices_lexiographically(indices)
poly.set_indices(indices)
basis_matrix = poly.basis_matrix(samples, {'deriv_order': 1})
vals_basis_matrix = basis_matrix[:samples.shape[1], :]
inner_products = (vals_basis_matrix.T * weights).dot(vals_basis_matrix)
assert np.allclose(inner_products, np.eye(basis_matrix.shape[1]))
exact_basis_vals_1d = []
exact_basis_derivs_1d = []
for dd in range(num_vars):
x = samples[dd, :].copy()
if dd == 0 or dd == 2:
if dd == 2:
# y = x/3
# z = 2*y-1=2*x/3-1=2/3*x-3/2*2/3=2/3*(x-3/2)=(x-3/2)/(3/2)
loc, scale = 3 / 2, 3 / 2
x = (x - loc) / scale
exact_basis_vals_1d.append(
np.asarray([1 + 0. * x, x, 0.5 * (3. * x**2 - 1)]).T)
exact_basis_derivs_1d.append(
np.asarray([0. * x, 1.0 + 0. * x, 3. * x]).T)
exact_basis_vals_1d[-1] /= np.sqrt(
1. / (2 * np.arange(degree + 1) + 1))
exact_basis_derivs_1d[-1] /= np.sqrt(
1. / (2 * np.arange(degree + 1) + 1))
# account for affine transformation in derivs
if dd == 2:
exact_basis_derivs_1d[-1] /= scale
if dd == 1:
loc, scale = gauss_mean, np.sqrt(gauss_var)
x = (x - loc) / scale
exact_basis_vals_1d.append(
np.asarray([1 + 0. * x, x, x**2 - 1]).T)
exact_basis_derivs_1d.append(
np.asarray([0. * x, 1.0 + 0. * x, 2. * x]).T)
exact_basis_vals_1d[-1] /= np.sqrt(
sp.factorial(np.arange(degree + 1)))
exact_basis_derivs_1d[-1] /= np.sqrt(
sp.factorial(np.arange(degree + 1)))
# account for affine transformation in derivs
exact_basis_derivs_1d[-1] /= scale
exact_basis_matrix = np.asarray([
exact_basis_vals_1d[0][:, 0], exact_basis_vals_1d[0][:, 1],
exact_basis_vals_1d[1][:, 1], exact_basis_vals_1d[2][:, 1],
exact_basis_vals_1d[0][:, 2],
exact_basis_vals_1d[0][:, 1] * exact_basis_vals_1d[1][:, 1],
exact_basis_vals_1d[1][:, 2],
exact_basis_vals_1d[0][:, 1] * exact_basis_vals_1d[2][:, 1],
exact_basis_vals_1d[1][:, 1] * exact_basis_vals_1d[2][:, 1],
exact_basis_vals_1d[2][:, 2]
]).T
# x1 derivative
exact_basis_matrix = np.vstack(
(exact_basis_matrix,
np.asarray([
0. * x, exact_basis_derivs_1d[0][:, 1], 0. * x, 0 * x,
exact_basis_derivs_1d[0][:, 2],
exact_basis_derivs_1d[0][:, 1] * exact_basis_vals_1d[1][:, 1],
0. * x,
exact_basis_derivs_1d[0][:, 1] * exact_basis_vals_1d[2][:, 1],
0. * x, 0. * x
]).T))
# x2 derivative
exact_basis_matrix = np.vstack(
(exact_basis_matrix,
np.asarray([
0. * x, 0. * x, exact_basis_derivs_1d[1][:, 1], 0. * x, 0 * x,
exact_basis_derivs_1d[1][:, 1] * exact_basis_vals_1d[0][:, 1],
exact_basis_derivs_1d[1][:, 2], 0. * x,
exact_basis_derivs_1d[1][:, 1] * exact_basis_vals_1d[2][:, 1],
0. * x
]).T))
# x3 derivative
exact_basis_matrix = np.vstack(
(exact_basis_matrix,
np.asarray([
0. * x, 0. * x, 0. * x, exact_basis_derivs_1d[2][:, 1], 0 * x,
0 * x, 0 * x,
exact_basis_derivs_1d[2][:, 1] * exact_basis_vals_1d[0][:, 1],
exact_basis_derivs_1d[2][:, 1] * exact_basis_vals_1d[1][:, 1],
exact_basis_derivs_1d[2][:, 2]
]).T))
func = poly.basis_matrix
exact_basis_matrix_derivs = exact_basis_matrix[samples.shape[1]:]
basis_matrix_derivs_fd = np.empty_like(exact_basis_matrix_derivs)
for ii in range(samples.shape[1]):
basis_matrix_derivs_fd[ii::samples.shape[1], :] = approx_fprime(
samples[:, ii:ii + 1], func)
# print(np.linalg.stats.norm(
# exact_basis_matrix_derivs-basis_matrix_derivs_fd,
# ord=np.inf))
assert np.allclose(exact_basis_matrix_derivs,
basis_matrix_derivs_fd,
atol=1e-7,
rtol=1e-7)
assert np.allclose(exact_basis_matrix, basis_matrix)
def test_evaluate_multivariate_hermite_pce(self):
num_vars = 2
degree = 2
poly = PolynomialChaosExpansion()
var_trans = define_iid_random_variable_transformation(
stats.norm(0, 1), num_vars)
poly_opts = define_poly_options_from_variable_transformation(var_trans)
poly.configure(poly_opts)
samples, weights = get_tensor_product_quadrature_rule(
degree + 1, num_vars, gauss_hermite_pts_wts_1D)
indices = compute_hyperbolic_indices(num_vars, degree, 1.0)
# sort lexographically to make testing easier
II = np.lexsort((indices[0, :], indices[1, :], indices.sum(axis=0)))
indices = indices[:, II]
poly.set_indices(indices)
basis_matrix = poly.basis_matrix(samples, {'deriv_order': 1})
vals_basis_matrix = basis_matrix[:samples.shape[1], :]
inner_products = (vals_basis_matrix.T * weights).dot(vals_basis_matrix)
assert np.allclose(inner_products, np.eye(basis_matrix.shape[1]))
exact_basis_vals_1d = []
exact_basis_derivs_1d = []
for dd in range(num_vars):
x = samples[dd, :]
exact_basis_vals_1d.append(np.asarray([1 + 0. * x, x, x**2 - 1]).T)
exact_basis_derivs_1d.append(
np.asarray([0. * x, 1.0 + 0. * x, 2. * x]).T)
exact_basis_vals_1d[-1] /= np.sqrt(
sp.factorial(np.arange(degree + 1)))
exact_basis_derivs_1d[-1] /= np.sqrt(
sp.factorial(np.arange(degree + 1)))
exact_basis_matrix = np.asarray([
exact_basis_vals_1d[0][:, 0], exact_basis_vals_1d[0][:, 1],
exact_basis_vals_1d[1][:, 1], exact_basis_vals_1d[0][:, 2],
exact_basis_vals_1d[0][:, 1] * exact_basis_vals_1d[1][:, 1],
exact_basis_vals_1d[1][:, 2]
]).T
# x1 derivative
exact_basis_matrix = np.vstack(
(exact_basis_matrix,
np.asarray([
0. * x, exact_basis_derivs_1d[0][:, 1], 0. * x,
exact_basis_derivs_1d[0][:, 2],
exact_basis_derivs_1d[0][:, 1] * exact_basis_vals_1d[1][:, 1],
0. * x
]).T))
# x2 derivative
exact_basis_matrix = np.vstack(
(exact_basis_matrix,
np.asarray([
0. * x, 0. * x, exact_basis_derivs_1d[1][:, 1], 0. * x,
exact_basis_vals_1d[0][:, 1] * exact_basis_derivs_1d[1][:, 1],
exact_basis_derivs_1d[1][:, 2]
]).T))
assert np.allclose(exact_basis_matrix, basis_matrix)
def test_evaluate_multivariate_jacobi_pce(self):
num_vars = 2
degree = 2
poly = PolynomialChaosExpansion()
var_trans = define_iid_random_variable_transformation(
stats.uniform(-1, 2), num_vars)
poly_opts = define_poly_options_from_variable_transformation(var_trans)
poly.configure(poly_opts)
samples, weights = get_tensor_product_quadrature_rule(
degree - 1, num_vars, np.polynomial.legendre.leggauss)
indices = compute_hyperbolic_indices(num_vars, degree, 1.0)
# sort lexographically to make testing easier
II = np.lexsort((indices[0, :], indices[1, :], indices.sum(axis=0)))
indices = indices[:, II]
# remove [0,2] index so max_level is not the same for every dimension
# also remove [1,0] and [1,1] to make sure can handle index sets that
# have missing univariate degrees not at the ends
J = [1, 5, 4]
reduced_indices = np.delete(indices, J, axis=1)
poly.set_indices(reduced_indices)
basis_matrix = poly.basis_matrix(samples, {'deriv_order': 1})
exact_basis_vals_1d = []
exact_basis_derivs_1d = []
for dd in range(num_vars):
x = samples[dd, :]
exact_basis_vals_1d.append(
np.asarray([1 + 0. * x, x, 0.5 * (3. * x**2 - 1)]).T)
exact_basis_derivs_1d.append(
np.asarray([0. * x, 1.0 + 0. * x, 3. * x]).T)
exact_basis_vals_1d[-1] /= np.sqrt(1. /
(2 * np.arange(degree + 1) + 1))
exact_basis_derivs_1d[-1] /= np.sqrt(
1. / (2 * np.arange(degree + 1) + 1))
exact_basis_matrix = np.asarray([
exact_basis_vals_1d[0][:, 0], exact_basis_vals_1d[0][:, 1],
exact_basis_vals_1d[1][:, 1], exact_basis_vals_1d[0][:, 2],
exact_basis_vals_1d[0][:, 1] * exact_basis_vals_1d[1][:, 1],
exact_basis_vals_1d[1][:, 2]
]).T
# x1 derivative
exact_basis_matrix = np.vstack(
(exact_basis_matrix,
np.asarray([
0. * x, exact_basis_derivs_1d[0][:, 1], 0. * x,
exact_basis_derivs_1d[0][:, 2],
exact_basis_derivs_1d[0][:, 1] * exact_basis_vals_1d[1][:, 1],
0. * x
]).T))
# x2 derivative
exact_basis_matrix = np.vstack(
(exact_basis_matrix,
np.asarray([
0. * x, 0. * x, exact_basis_derivs_1d[1][:, 1], 0. * x,
exact_basis_vals_1d[0][:, 1] * exact_basis_derivs_1d[1][:, 1],
exact_basis_derivs_1d[1][:, 2]
]).T))
exact_basis_matrix = np.delete(exact_basis_matrix, J, axis=1)
assert np.allclose(exact_basis_matrix, basis_matrix)