def test_approximate_gaussian_process(self):
from sklearn.gaussian_process.kernels import Matern
num_vars = 1
univariate_variables = [stats.uniform(-1, 2)] * num_vars
variable = pya.IndependentMultivariateRandomVariable(
univariate_variables)
num_samples = 100
train_samples = pya.generate_independent_random_samples(
variable, num_samples)
# Generate random function
nu = np.inf # 2.5
kernel = Matern(0.5, nu=nu)
X = np.linspace(-1, 1, 1000)[np.newaxis, :]
alpha = np.random.normal(0, 1, X.shape[1])
train_vals = kernel(train_samples.T, X.T).dot(alpha)[:, np.newaxis]
gp = approximate(train_samples, train_vals, "gaussian_process", {
"nu": nu,
"noise_level": 1e-8
}).approx
error = np.linalg.norm(gp(X)[:, 0]-kernel(X.T, X.T).dot(alpha)) /\
np.sqrt(X.shape[1])
assert error < 1e-5
def help_cross_validate_pce_degree(self, solver_type, solver_options):
print(solver_type, solver_options)
num_vars = 2
univariate_variables = [stats.uniform(-1, 2)] * num_vars
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)
degree = 3
poly.set_indices(pya.compute_hyperbolic_indices(num_vars, degree, 1.0))
# factor of 2 does not pass test but 2.2 does
num_samples = int(poly.num_terms() * 2.2)
coef = np.random.normal(0, 1, (poly.indices.shape[1], 2))
coef[pya.nchoosek(num_vars + 2, 2):, 0] = 0
# for first qoi make degree 2 the best degree
poly.set_coefficients(coef)
train_samples = pya.generate_independent_random_samples(
variable, num_samples)
train_vals = poly(train_samples)
true_poly = poly
poly = approximate(
train_samples, train_vals, "polynomial_chaos", {
"basis_type": "hyperbolic_cross",
"variable": variable,
"options": {
"verbose": 3,
"solver_type": solver_type,
"min_degree": 1,
"max_degree": degree + 1,
"linear_solver_options": solver_options
}
}).approx
num_validation_samples = 10
validation_samples = pya.generate_independent_random_samples(
variable, num_validation_samples)
assert np.allclose(poly(validation_samples),
true_poly(validation_samples))
poly = copy.deepcopy(true_poly)
approx_res = cross_validate_pce_degree(
poly,
train_samples,
train_vals,
1,
degree + 1,
solver_type=solver_type,
linear_solver_options=solver_options)
assert np.allclose(approx_res.degrees, [2, 3])
def test_pce_basis_expansion(self):
num_vars = 2
univariate_variables = [stats.uniform(-1, 2)] * num_vars
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)
degree, hcross_strength = 7, 0.4
poly.set_indices(
pya.compute_hyperbolic_indices(num_vars, degree, hcross_strength))
num_samples = poly.num_terms() * 2
degrees = poly.indices.sum(axis=0)
coef = np.random.normal(
0, 1, (poly.indices.shape[1], 2)) / (degrees[:, np.newaxis] + 1)**2
# set some coefficients to zero to make sure that different qoi
# are treated correctly.
II = np.random.permutation(coef.shape[0])[:coef.shape[0] // 2]
coef[II, 0] = 0
II = np.random.permutation(coef.shape[0])[:coef.shape[0] // 2]
coef[II, 1] = 0
poly.set_coefficients(coef)
train_samples = pya.generate_independent_random_samples(
variable, num_samples)
train_vals = poly(train_samples)
true_poly = poly
poly = approximate(
train_samples, train_vals, "polynomial_chaos", {
"basis_type": "expanding_basis",
"variable": variable,
"options": {
"max_num_expansion_steps_iter": 1,
"verbose": 3,
"max_num_terms": 1000,
"max_num_step_increases": 2,
"max_num_init_terms": 33
}
}).approx
num_validation_samples = 100
validation_samples = pya.generate_independent_random_samples(
variable, num_validation_samples)
validation_samples = train_samples
error = np.linalg.norm(
poly(validation_samples) -
true_poly(validation_samples)) / np.sqrt(num_validation_samples)
assert np.allclose(poly(validation_samples),
true_poly(validation_samples),
atol=1e-8), error
def test_approximate_neural_network(self):
np.random.seed(2)
benchmark = setup_benchmark("ishigami", a=7, b=0.1)
nvars = benchmark.variable.num_vars()
nqoi = 1
maxiter = 30000
print(benchmark.variable)
# var_trans = pya.AffineRandomVariableTransformation(
# [stats.uniform(-2, 4)]*nvars)
var_trans = pya.AffineRandomVariableTransformation(benchmark.variable)
network_opts = {
"activation_func": "sigmoid",
"layers": [nvars, 75, nqoi],
"loss_func": "squared_loss",
"var_trans": var_trans,
"lag_mult": 0
}
optimizer_opts = {
"method": "L-BFGS-B",
"options": {
"maxiter": maxiter,
"iprint": -1,
"gtol": 1e-6
}
}
opts = {
"network_opts": network_opts,
"verbosity": 3,
"optimizer_opts": optimizer_opts
}
ntrain_samples = 500
train_samples = pya.generate_independent_random_samples(
var_trans.variable, ntrain_samples)
train_samples = var_trans.map_from_canonical_space(
np.cos(np.random.uniform(0, np.pi, (nvars, ntrain_samples))))
train_vals = benchmark.fun(train_samples)
opts = {
"network_opts": network_opts,
"verbosity": 3,
"optimizer_opts": optimizer_opts,
"x0": 10
}
approx = approximate(train_samples, train_vals, "neural_network",
opts).approx
nsamples = 100
error = compute_l2_error(approx, benchmark.fun, var_trans.variable,
nsamples)
print(error)
assert error < 6e-2
def test_analyze_sensitivity_polynomial_chaos(self):
from pyapprox.benchmarks.benchmarks import setup_benchmark
from pyapprox.approximate import approximate
benchmark = setup_benchmark("ishigami", a=7, b=0.1)
num_samples = 1000
train_samples = pya.generate_independent_random_samples(
benchmark.variable, num_samples)
train_vals = benchmark.fun(train_samples)
pce = approximate(
train_samples, train_vals, 'polynomial_chaos', {
'basis_type': 'hyperbolic_cross',
'variable': benchmark.variable,
'options': {
'max_degree': 8
}
}).approx
res = analyze_sensitivity_polynomial_chaos(pce)
assert np.allclose(res.main_effects, benchmark.main_effects, atol=2e-3)
def test_approximate_polynomial_chaos_custom_poly_type(self):
benchmark = setup_benchmark("ishigami", a=7, b=0.1)
nvars = benchmark.variable.num_vars()
# this test purposefully select wrong variable to make sure
# poly_type overide is activated
univariate_variables = [stats.beta(5, 5, -np.pi, 2 * np.pi)] * nvars
variable = pya.IndependentMultivariateRandomVariable(
univariate_variables)
var_trans = pya.AffineRandomVariableTransformation(variable)
# specify correct basis so it is not chosen from var_trans.variable
poly_opts = {"var_trans": var_trans}
# but rather from another variable which will invoke Legendre polys
basis_opts = pya.define_poly_options_from_variable(
pya.IndependentMultivariateRandomVariable([stats.uniform()] *
nvars))
poly_opts["poly_types"] = basis_opts
options = {
"poly_opts": poly_opts,
"variable": variable,
"options": {
"max_num_step_increases": 1
}
}
ntrain_samples = 400
train_samples = np.random.uniform(-np.pi, np.pi,
(nvars, ntrain_samples))
train_vals = benchmark.fun(train_samples)
approx = approximate(train_samples,
train_vals,
method="polynomial_chaos",
options=options).approx
nsamples = 100
error = compute_l2_error(approx,
benchmark.fun,
approx.var_trans.variable,
nsamples,
rel=True)
# print(error)
assert error < 1e-4
assert np.allclose(approx.mean(), benchmark.mean, atol=error)
def test_cross_validate_approximation_after_regularization_selection(self):
"""
This test is useful as it shows how to use cross_validate_approximation
to produce a list of approximations on each cross validation fold
once regularization parameters have been chosen.
These can be used to show variance in predictions of values,
sensitivity indices, etc.
Ideally this could be avoided if sklearn stored the coefficients
and alphas for each fold and then we can just find the coefficients
that correspond to the first time the path drops below the best_alpha
"""
num_vars = 2
univariate_variables = [stats.uniform(-1, 2)] * num_vars
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)
degree, hcross_strength = 7, 0.4
poly.set_indices(
pya.compute_hyperbolic_indices(num_vars, degree, hcross_strength))
num_samples = poly.num_terms() * 2
degrees = poly.indices.sum(axis=0)
coef = np.random.normal(
0, 1, (poly.indices.shape[1], 2)) / (degrees[:, np.newaxis] + 1)**2
# set some coefficients to zero to make sure that different qoi
# are treated correctly.
II = np.random.permutation(coef.shape[0])[:coef.shape[0] // 2]
coef[II, 0] = 0
II = np.random.permutation(coef.shape[0])[:coef.shape[0] // 2]
coef[II, 1] = 0
poly.set_coefficients(coef)
train_samples = pya.generate_independent_random_samples(
variable, num_samples)
train_vals = poly(train_samples)
# true_poly = poly
result = approximate(train_samples, train_vals, "polynomial_chaos", {
"basis_type": "expanding_basis",
"variable": variable
})
# Even with the same folds, iterative methods such as Lars, LarsLasso
# and OMP will not have cv_score from approximate and cross validate
# approximation exactly the same because iterative methods interpolate
# residuals to compute cross validation scores
nfolds = 10
linear_solver_options = [{
"alpha": result.reg_params[0]
}, {
"alpha": result.reg_params[1]
}]
indices = [
result.approx.indices[:, np.where(np.absolute(c) > 0)[0]]
for c in result.approx.coefficients.T
]
options = {
"basis_type": "fixed",
"variable": variable,
"options": {
"linear_solver_options": linear_solver_options,
"indices": indices
}
}
approx_list, residues_list, cv_score = \
cross_validate_approximation(
train_samples, train_vals, options, nfolds, "polynomial_chaos",
random_folds="sklearn")
assert (np.all(cv_score < 6e-14) and np.all(result.scores < 4e-13))
def test_analytic_sobol_indices_from_gaussian_process(self):
from pyapprox.benchmarks.benchmarks import setup_benchmark
from pyapprox.approximate import approximate
benchmark = setup_benchmark("ishigami", a=7, b=0.1)
nvars = benchmark.variable.num_vars()
ntrain_samples = 500
# train_samples = pya.generate_independent_random_samples(
# benchmark.variable, ntrain_samples)
train_samples = pya.sobol_sequence(nvars,
ntrain_samples,
variable=benchmark.variable)
train_vals = benchmark.fun(train_samples)
approx = approximate(train_samples, train_vals, 'gaussian_process', {
'nu': np.inf,
'normalize_y': True,
'alpha': 1e-10
}).approx
nsobol_samples = int(1e4)
from pyapprox.approximate import compute_l2_error
error = compute_l2_error(approx,
benchmark.fun,
benchmark.variable,
nsobol_samples,
rel=True)
print(error)
order = 2
interaction_terms = compute_hyperbolic_indices(nvars, order)
interaction_terms = interaction_terms[:,
np.where(
interaction_terms.max(
axis=0) == 1)[0]]
result = analytic_sobol_indices_from_gaussian_process(
approx,
benchmark.variable,
interaction_terms,
ngp_realizations=1000,
stat_functions=(np.mean, np.std),
ninterpolation_samples=2000,
ncandidate_samples=3000,
use_cholesky=False,
alpha=1e-8)
mean_mean = result['mean']['mean']
mean_sobol_indices = result['sobol_indices']['mean']
mean_total_effects = result['total_effects']['mean']
mean_main_effects = mean_sobol_indices[:nvars]
print(result['mean']['values'][-1])
print(result['variance']['values'][-1])
print(benchmark.main_effects[:, 0] - mean_main_effects)
print(benchmark.total_effects[:, 0] - mean_total_effects)
print(benchmark.sobol_indices[:-1, 0] - mean_sobol_indices)
assert np.allclose(mean_mean, benchmark.mean, rtol=1e-3, atol=3e-3)
assert np.allclose(mean_main_effects,
benchmark.main_effects[:, 0],
rtol=1e-3,
atol=3e-3)
assert np.allclose(mean_total_effects,
benchmark.total_effects[:, 0],
rtol=1e-3,
atol=3e-3)
assert np.allclose(mean_sobol_indices,
benchmark.sobol_indices[:-1, 0],
rtol=1e-3,
atol=3e-3)
def test_sampling_based_sobol_indices_from_gaussian_process(self):
from pyapprox.benchmarks.benchmarks import setup_benchmark
from pyapprox.approximate import approximate
benchmark = setup_benchmark("ishigami", a=7, b=0.1)
nvars = benchmark.variable.num_vars()
# nsobol_samples and ntrain_samples effect assert tolerances
ntrain_samples = 500
nsobol_samples = int(1e4)
train_samples = pya.generate_independent_random_samples(
benchmark.variable, ntrain_samples)
# from pyapprox import CholeskySampler
# sampler = CholeskySampler(nvars, 10000, benchmark.variable)
# kernel = pya.Matern(
# np.array([1]*nvars), length_scale_bounds='fixed', nu=np.inf)
# sampler.set_kernel(kernel)
# train_samples = sampler(ntrain_samples)[0]
train_vals = benchmark.fun(train_samples)
approx = approximate(train_samples, train_vals, 'gaussian_process', {
'nu': np.inf,
'normalize_y': True
}).approx
from pyapprox.approximate import compute_l2_error
error = compute_l2_error(approx,
benchmark.fun,
benchmark.variable,
nsobol_samples,
rel=True)
print('error', error)
# assert error < 4e-2
order = 2
interaction_terms = compute_hyperbolic_indices(nvars, order)
interaction_terms = interaction_terms[:,
np.where(
interaction_terms.max(
axis=0) == 1)[0]]
result = sampling_based_sobol_indices_from_gaussian_process(
approx,
benchmark.variable,
interaction_terms,
nsobol_samples,
sampling_method='sobol',
ngp_realizations=1000,
normalize=True,
nsobol_realizations=3,
stat_functions=(np.mean, np.std),
ninterpolation_samples=1000,
ncandidate_samples=2000)
mean_mean = result['mean']['mean']
mean_sobol_indices = result['sobol_indices']['mean']
mean_total_effects = result['total_effects']['mean']
mean_main_effects = mean_sobol_indices[:nvars]
print(benchmark.mean - mean_mean)
print(benchmark.main_effects[:, 0] - mean_main_effects)
print(benchmark.total_effects[:, 0] - mean_total_effects)
print(benchmark.sobol_indices[:-1, 0] - mean_sobol_indices)
assert np.allclose(mean_mean, benchmark.mean, atol=3e-2)
assert np.allclose(mean_main_effects,
benchmark.main_effects[:, 0],
atol=1e-2)
assert np.allclose(mean_total_effects,
benchmark.total_effects[:, 0],
atol=1e-2)
assert np.allclose(mean_sobol_indices,
benchmark.sobol_indices[:-1, 0],
atol=1e-2)
for jj in inds:
a, b = variable.all_variables()[jj].interval(1)
x, w = gauss_jacobi_pts_wts_1D(nquad_samples_1d, 0, 0)
x = (x+1)/2 # map to [0, 1]
x = (b-a)*x+a # map to [a,b]
quad_rules.append((x, w))
funs = [identity_fun]*len(inds)
basis_opts['basis%d' % ii] = {'poly_type': 'product_indpnt_vars',
'var_nums': [ii], 'funs': funs,
'quad_rules': quad_rules}
cnt += 1
poly_opts = {'var_trans': re_var_trans}
poly_opts['poly_types'] = basis_opts
#var_trans.set_identity_maps(identity_map_indices) #wrong
re_var_trans.set_identity_maps(identity_map_indices) #right
indices = compute_hyperbolic_indices(re_variable.num_vars(), degree)
nterms = total_degree_space_dimension(samples_adjust.shape[0], degree)
options = {'basis_type': 'fixed', 'variable': re_variable,
'poly_opts': poly_opts,
'options': {'linear_solver_options': dict(),
'indices': indices, 'solver_type': 'lstsq'}}
approx_res = approximate(samples_adjust[:, 0:(2 * nterms)], values[0:(2 * nterms)], 'polynomial_chaos', options).approx
y_hat = approx_res(samples_adjust[:, 2 * nterms:])
print((y_hat - values[2 * nterms:]).mean())
print(f'Mean of samples: {values.mean()}')
print(f'Mean of pce: {approx_res.mean()}')
import numpy as np
import pyapprox as pya
from pyapprox.benchmarks.benchmarks import setup_benchmark
from pyapprox.approximate import approximate
benchmark = setup_benchmark("ishigami", a=7, b=0.1)
num_samples = 1000
train_samples = pya.generate_independent_random_samples(
benchmark.variable, num_samples)
train_vals = benchmark.fun(train_samples)
approx_res = approximate(
train_samples, train_vals, 'polynomial_chaos', {
'basis_type': 'hyperbolic_cross',
'variable': benchmark.variable,
'options': {
'max_degree': 8
}
})
pce = approx_res.approx
res = pya.analyze_sensitivity_polynomial_chaos(pce)
#%%
#Now lets compare the estimated values with the exact value
print(res.main_effects[:, 0])
print(benchmark.main_effects[:, 0])
#%%
#We can visualize the sensitivity indices using the following
