def setup_check_variance_reduction_model_ensemble_short_column(
nmodels=5,npilot_samples=None):
example = ShortColumnModelEnsemble()
model_ensemble = pya.ModelEnsemble(
[example.models[ii] for ii in range(nmodels)])
univariate_variables = [
uniform(5,10),uniform(15,10),norm(500,100),norm(2000,400),
lognorm(s=0.5,scale=np.exp(5))]
variable=pya.IndependentMultivariateRandomVariable(univariate_variables)
generate_samples=partial(
pya.generate_independent_random_samples,variable)
if npilot_samples is not None:
# The number of pilot samples effects ability of numerical estimate
# of variance reduction to match theoretical value
cov, samples, weights = pya.estimate_model_ensemble_covariance(
npilot_samples,generate_samples,model_ensemble)
else:
# it is difficult to create a quadrature rule for the lognormal
# distribution so instead define the variable as normal and then
# apply log transform
univariate_variables = [
uniform(5,10),uniform(15,10),norm(500,100),norm(2000,400),
norm(loc=5,scale=0.5)]
variable=pya.IndependentMultivariateRandomVariable(
univariate_variables)
example.apply_lognormal=True
cov = example.get_covariance_matrix(variable)[:nmodels,:nmodels]
example.apply_lognormal=False
return model_ensemble, cov, generate_samples
def test_generate_samples_and_values_mfmc(self):
functions = ShortColumnModelEnsemble()
model_ensemble = pya.ModelEnsemble(
[functions.m0, functions.m1, functions.m2])
univariate_variables = [
uniform(5, 10),
uniform(15, 10),
norm(500, 100),
norm(2000, 400),
lognorm(s=0.5, scale=np.exp(5))
]
variable = pya.IndependentMultivariateRandomVariable(
univariate_variables)
generate_samples = partial(pya.generate_independent_random_samples,
variable)
nhf_samples = 10
nsample_ratios = [2, 4]
samples,values =\
pya.generate_samples_and_values_mfmc(
nhf_samples,nsample_ratios,model_ensemble,generate_samples)
for jj in range(1, len(samples)):
assert samples[jj][1].shape[1] == nsample_ratios[jj -
1] * nhf_samples
idx = 1
if jj == 1:
idx = 0
assert np.allclose(samples[jj][0], samples[jj - 1][idx])
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 setup_oakley_function():
r"""
Setup the Oakely function benchmark
.. math:: f(z) = a_1^Tz + a_2^T\sin(z) + a_3^T\cos(z) + z^TMz
where :math:`z` consists of 15 I.I.D. standard Normal variables and the data :math:`a_1,a_2,a_3` and :math:`M` are defined in the function :func:`pyapprox.benchmarks.sensitivity_benchmarks.get_oakley_function_data`.
>>> from pyapprox.benchmarks.benchmarks import setup_benchmark
>>> benchmark=setup_benchmark('oakley')
>>> print(benchmark.keys())
dict_keys(['fun', 'variable', 'mean', 'variance', 'main_effects'])
Returns
-------
benchmark : pya.Benchmark
Object containing the benchmark attributes
References
----------
.. [OakelyOJRSB2004] `Oakley, J.E. and O'Hagan, A. (2004), Probabilistic sensitivity analysis of complex models: a Bayesian approach. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 66: 751-769. <https://doi.org/10.1111/j.1467-9868.2004.05304.x>`_
"""
univariate_variables = [stats.norm()] * 15
variable = pya.IndependentMultivariateRandomVariable(univariate_variables)
mean, variance, main_effects = oakley_function_statistics()
return Benchmark({
'fun': oakley_function,
'variable': variable,
'mean': mean,
'variance': variance,
'main_effects': main_effects
})
def __init__(self, theta1, shifts=None):
"""
Parameters
----------
theta0 : float
Angle controling
Notes
-----
The choice of A0, A1, A2 here results in unit variance for each model
"""
self.A0 = np.sqrt(11)
self.A1 = np.sqrt(7)
self.A2 = np.sqrt(3)
self.nmodels = 3
self.theta0 = np.pi / 2
self.theta1 = theta1
self.theta2 = np.pi / 6
assert self.theta0 > self.theta1 and self.theta1 > self.theta2
self.shifts = shifts
if self.shifts is None:
self.shifts = [0, 0]
assert len(self.shifts) == 2
self.models = [self.m0, self.m1, self.m2]
univariate_variables = [uniform(-1, 2), uniform(-1, 2)]
self.variable = pya.IndependentMultivariateRandomVariable(
univariate_variables)
self.generate_samples = partial(
pya.generate_independent_random_samples, self.variable)
def test_adaptive_approximate_increment_degree(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 = 3
poly.set_indices(pya.compute_hyperbolic_indices(num_vars, degree))
poly.set_coefficients(
np.random.normal(0, 1, (poly.indices.shape[1], 1)))
fun = poly
max_degree = degree + 2
result = adaptive_approximate_polynomial_chaos_increment_degree(
fun,
variable,
max_degree,
max_nsamples=31,
cond_tol=1e4,
sample_growth_factor=2,
verbose=0,
oversampling_ratio=None,
solver_type='lstsq',
callback=None)
print('Ntrain samples', result.train_samples.shape[1])
assert np.allclose(
result.approx.coefficients[:poly.coefficients.shape[0]],
poly.coefficients)
def setup_rosenbrock_function(nvars):
r"""
Setup the Rosenbrock function benchmark
.. math:: f(z) = \sum_{i=1}^{d/2}\left[100(z_{2i-1}^{2}-z_{2i})^{2}+(z_{2i-1}-1)^{2}\right]
using
>>> from pyapprox.benchmarks.benchmarks import setup_benchmark
>>> benchmark=setup_benchmark('rosenbrock',nvars=2)
>>> print(benchmark.keys())
dict_keys(['fun', 'jac', 'hessp', 'variable'])
Parameters
----------
nvars : integer
The number of variables of the Rosenbrock function
Returns
-------
benchmark : pya.Benchmark
Object containing the benchmark attributes
References
----------
.. [DixonSzego1990] `Dixon, L. C. W.; Mills, D. J. "Effect of Rounding Errors on the Variable Metric Method". Journal of Optimization Theory and Applications. 80: 175–179. 1994 <https://doi.org/10.1007%2FBF02196600>`_
"""
univariate_variables = [stats.uniform(-2,4)]*nvars
variable=pya.IndependentMultivariateRandomVariable(univariate_variables)
return Benchmark(
{'fun':rosenbrock_function,'jac':rosenbrock_function_jacobian,
'hessp':rosenbrock_function_hessian_prod,'variable':variable})
def test_bootstrap_control_variate_estimator(self):
example = TunableModelEnsemble(np.pi / 2 * 0.95)
model_ensemble = pya.ModelEnsemble(example.models)
univariate_variables = [uniform(-1, 2), uniform(-1, 2)]
variable = pya.IndependentMultivariateRandomVariable(
univariate_variables)
cov_matrix = example.get_covariance_matrix()
model_costs = [1, 0.5, 0.4]
est = ACVMF(cov_matrix, model_costs)
target_cost = 1000
nhf_samples, nsample_ratios = est.allocate_samples(target_cost)[:2]
generate_samples = partial(pya.generate_independent_random_samples,
variable)
samples, values = est.generate_data(nhf_samples, nsample_ratios,
generate_samples, model_ensemble)
mc_cov_matrix = compute_covariance_from_control_variate_samples(values)
#assert np.allclose(cov_matrix,mc_cov_matrix,atol=1e-2)
est = ACVMF(mc_cov_matrix, model_costs)
weights = get_mfmc_control_variate_weights(
example.get_covariance_matrix())
bootstrap_mean,bootstrap_variance = \
pya.bootstrap_mfmc_estimator(values,weights,10000)
est_mean = est(values)
est_variance = est.get_variance(nhf_samples, nsample_ratios)
print(abs((est_variance - bootstrap_variance) / est_variance))
assert abs((est_variance - bootstrap_variance) / est_variance) < 6e-2
def test_adaptive_approximate_gaussian_process_normalize_inputs(self):
from sklearn.gaussian_process.kernels import Matern
num_vars = 1
univariate_variables = [stats.beta(5, 10, 0, 2)] * num_vars
# Generate random function
nu = np.inf # 2.5
kernel = Matern(0.1, nu=nu)
X = np.linspace(-1, 1, 1000)[np.newaxis, :]
alpha = np.random.normal(0, 1, X.shape[1])
def fun(x):
return kernel(x.T, X.T).dot(alpha)[:, np.newaxis]
# return np.cos(2*np.pi*x.sum(axis=0)/num_vars)[:, np.newaxis]
errors = []
validation_samples = pya.generate_independent_random_samples(
pya.IndependentMultivariateRandomVariable(univariate_variables),
100)
validation_values = fun(validation_samples)
def callback(gp):
gp_vals = gp(validation_samples)
assert gp_vals.shape == validation_values.shape
error = np.linalg.norm(gp_vals - validation_values
) / np.linalg.norm(validation_values)
print(error, gp.y_train_.shape[0])
errors.append(error)
weight_function = partial(
pya.tensor_product_pdf,
univariate_pdfs=[v.pdf for v in univariate_variables])
gp = adaptive_approximate(
fun, univariate_variables, "gaussian_process", {
"nu": nu,
"noise_level": None,
"normalize_y": True,
"alpha": 1e-10,
"normalize_inputs": True,
"weight_function": weight_function,
"ncandidate_samples": 1e3,
"callback": callback
}).approx
# import matplotlib.pyplot as plt
# plt.plot(gp.X_train_.T[0, :], 0*gp.X_train_.T[0, :], "s")
# plt.plot(gp.get_training_samples()[0, :], 0*gp.get_training_samples()[0, :], "x")
# plt.plot(gp.sampler.candidate_samples[0, :], 0*gp.sampler.candidate_samples[0, :], "^")
# plt.plot(validation_samples[0, :], validation_values[:, 0], "o")
# var = univariate_variables[0]
# lb, ub = var.interval(1)
# xx = np.linspace(lb, ub, 101)
# plt.plot(xx, var.pdf(xx), "r-")
# plt.show()
print(errors[-1])
assert errors[-1] < 1e-7
def test_approximate_fixed_pce(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.
I = np.random.permutation(coef.shape[0])[:coef.shape[0] // 2]
coef[I, 0] = 0
I = np.random.permutation(coef.shape[0])[:coef.shape[0] // 2]
coef[I, 1] = 0
poly.set_coefficients(coef)
train_samples = pya.generate_independent_random_samples(
variable, num_samples)
train_vals = poly(train_samples)
indices = compute_hyperbolic_indices(num_vars, 1, 1)
nfolds = 10
method = 'polynomial_chaos'
options = {
'basis_type': 'fixed',
'variable': variable,
'options': {
'linear_solver_options': {},
'indices': indices,
'solver_type': 'lstsq'
}
}
approx_list, residues_list, cv_score = cross_validate_approximation(
train_samples,
train_vals,
options,
nfolds,
method,
random_folds=False)
solver = LinearLeastSquaresCV(cv=nfolds, random_folds=False)
poly.set_indices(indices)
basis_matrix = poly.basis_matrix(train_samples)
solver.fit(basis_matrix, train_vals[:, 0:1])
assert np.allclose(solver.cv_score_, cv_score[0])
solver.fit(basis_matrix, train_vals[:, 1:2])
assert np.allclose(solver.cv_score_, cv_score[1])
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 __init__(self):
self.nmodels = 5
self.nvars = 1
self.models = [self.m0, self.m1, self.m2, self.m3, self.m4]
univariate_variables = [uniform(0, 1)]
self.variable = pya.IndependentMultivariateRandomVariable(
univariate_variables)
self.generate_samples = partial(
pya.generate_independent_random_samples, self.variable)
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_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 __init__(self):
self.nmodels=5
self.nvars=5
self.models = [self.m0,self.m1,self.m2,self.m3,self.m4]
self.apply_lognormal=False
univariate_variables = [
uniform(5,10),uniform(15,10),norm(500,100),norm(2000,400),
lognorm(s=0.5,scale=np.exp(5))]
self.variable = pya.IndependentMultivariateRandomVariable(
univariate_variables)
self.generate_samples=partial(
pya.generate_independent_random_samples,self.variable)
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 setup_ishigami_function(a, b):
r"""
Setup the Ishigami function benchmark
.. math:: f(z) = \sin(z_1)+a\sin^2(z_2) + bz_3^4\sin(z_0)
using
>>> from pyapprox.benchmarks.benchmarks import setup_benchmark
>>> benchmark=setup_benchmark('ishigami',a=7,b=0.1)
>>> print(benchmark.keys())
dict_keys(['fun', 'jac', 'hess', 'variable', 'mean', 'variance', 'main_effects', 'total_effects', 'sobol_indices'])
Parameters
----------
a : float
The hyper-parameter a
b : float
The hyper-parameter b
Returns
-------
benchmark : pya.Benchmark
Object containing the benchmark attributes
References
----------
.. [Ishigami1990] `T. Ishigami and T. Homma, "An importance quantification technique in uncertainty analysis for computer models," [1990] Proceedings. First International Symposium on Uncertainty Modeling and Analysis, College Park, MD, USA, 1990, pp. 398-403 <https://doi.org/10.1109/ISUMA.1990.151285>`_
"""
univariate_variables = [stats.uniform(-np.pi, 2 * np.pi)] * 3
variable = pya.IndependentMultivariateRandomVariable(univariate_variables)
mean, variance, main_effects, total_effects, sobol_indices, \
sobol_interaction_indices = get_ishigami_funciton_statistics()
return Benchmark({
'fun': partial(ishigami_function, a=a, b=b),
'jac': partial(ishigami_function_jacobian, a=a, b=b),
'hess': partial(ishigami_function_hessian, a=a, b=b),
'variable': variable,
'mean': mean,
'variance': variance,
'main_effects': main_effects,
'total_effects': total_effects,
'sobol_indices': sobol_indices,
'sobol_interaction_indices': sobol_interaction_indices
})
def variables_prep(filename, product_uniform=False, dummy=False):
"""
Help function for preparing the data training data to fit PCE.
Parameters:
===========
filename : str
product_uniform : False do not colapse product into one variable
'uniform' uniform distributions are used for product;
'beta', beta distributions are used for variables which
are adapted considering the correlations
'exact' the true PDF of the product is used
"""
# import parameter inputs and generate the dataframe of analytical ratios between sensitivity indices
if (product_uniform is False) or (product_uniform == 'uniform'):
ranges = np.loadtxt(
filename,delimiter=",",usecols=[2,3],skiprows=1).flatten()
univariate_variables = [uniform(ranges[2*ii],ranges[2*ii+1]-ranges[2*ii]) for ii in range(0, ranges.shape[0]//2)]
else:
param_adjust = pd.read_csv(filename)
beta_index = param_adjust[param_adjust['distribution']== 'beta'].index.to_list()
ranges = np.array(param_adjust.loc[:, ['min','max']])
ranges[:, 1] = ranges[:, 1] - ranges[:, 0]
# param_names = param_adjust.loc[[0, 2, 8], 'Veneer_name'].values
univariate_variables = []
for ii in range(param_adjust.shape[0]):
if ii in beta_index:
shape_ab = param_adjust.loc[ii, ['a','b']].values.astype('float')
univariate_variables.append(beta(shape_ab[0], shape_ab[1],
loc=ranges[ii][0], scale=ranges[ii][1]))
else:
# uniform_args = ranges[ii]
univariate_variables.append(uniform(ranges[ii][0], ranges[ii][1]))
# End if
# End for()
if dummy == True: univariate_variables.append(uniform(0, 1))
# import pdb
# pdb.set_trace()
variable = pya.IndependentMultivariateRandomVariable(univariate_variables)
return variable
#END variables_prep()
def test_rsquared_mfmc(self):
functions = ShortColumnModelEnsemble()
model_ensemble = pya.ModelEnsemble(
[functions.m0, functions.m3, functions.m4])
univariate_variables = [
uniform(5, 10),
uniform(15, 10),
norm(500, 100),
norm(2000, 400),
lognorm(s=0.5, scale=np.exp(5))
]
variable = pya.IndependentMultivariateRandomVariable(
univariate_variables)
generate_samples = partial(pya.generate_independent_random_samples,
variable)
npilot_samples = int(1e4)
pilot_samples = generate_samples(npilot_samples)
config_vars = np.arange(model_ensemble.nmodels)[np.newaxis, :]
pilot_samples = pya.get_all_sample_combinations(
pilot_samples, config_vars)
pilot_values = model_ensemble(pilot_samples)
pilot_values = np.reshape(pilot_values,
(npilot_samples, model_ensemble.nmodels))
cov = np.cov(pilot_values, rowvar=False)
nhf_samples = 10
nsample_ratios = np.asarray([2, 4])
nsamples_per_model = np.concatenate([[nhf_samples],
nsample_ratios * nhf_samples])
eta = pya.get_mfmc_control_variate_weights(cov)
cor = pya.get_correlation_from_covariance(cov)
var_mfmc = cov[0, 0] / nsamples_per_model[0]
for k in range(1, model_ensemble.nmodels):
var_mfmc += (1 / nsamples_per_model[k - 1] -
1 / nsamples_per_model[k]) * (
eta[k - 1]**2 * cov[k, k] + 2 * eta[k - 1] *
cor[0, k] * np.sqrt(cov[0, 0] * cov[k, k]))
assert np.allclose(var_mfmc / cov[0, 0] * nhf_samples,
1 - pya.get_rsquared_mfmc(cov, nsample_ratios))
def setup_sobol_g_function(nvars):
r"""
Setup the Sobol-G function benchmark
.. math:: f(z) = \prod_{i=1}^d\frac{\lvert 4z_i-2\rvert+a_i}{1+a_i}, \quad a_i=\frac{i-2}{2}
using
>>> from pyapprox.benchmarks.benchmarks import setup_benchmark
>>> benchmark=setup_benchmark('sobol_g',nvars=2)
>>> print(benchmark.keys())
dict_keys(['fun', 'mean', 'variance', 'main_effects', 'total_effects', 'variable'])
Parameters
----------
nvars : integer
The number of variables of the Sobol-G function
Returns
-------
benchmark : pya.Benchmark
Object containing the benchmark attributes
References
----------
.. [Saltelli1995] `Saltelli, A., & Sobol, I. M. About the use of rank transformation in sensitivity analysis of model output. Reliability Engineering & System Safety, 50(3), 225-239, 1995. <https://doi.org/10.1016/0951-8320(95)00099-2>`_
"""
univariate_variables = [stats.uniform(0, 1)] * nvars
variable = pya.IndependentMultivariateRandomVariable(univariate_variables)
a = (np.arange(1, nvars + 1) - 2) / 2
mean, variance, main_effects, total_effects = \
get_sobol_g_function_statistics(a)
return Benchmark({
'fun': partial(sobol_g_function, a),
'mean': mean,
'variance': variance,
'main_effects': main_effects,
'total_effects': total_effects,
'variable': variable
})
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)
poly.set_coefficients((np.random.normal(0, 1, poly.indices.shape[1]) /
(degrees + 1)**2)[:, np.newaxis])
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
})
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_cross_validate_pce_degree(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 = 3
poly.set_indices(pya.compute_hyperbolic_indices(num_vars, degree, 1.0))
num_samples = poly.num_terms() * 2
poly.set_coefficients(
np.random.normal(0, 1, (poly.indices.shape[1], 1)))
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
})
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)
poly, best_degree = cross_validate_pce_degree(poly, train_samples,
train_vals, 1,
degree + 2)
assert best_degree == degree
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())
def setup_rosenbrock_function(nvars):
r"""
Setup the Rosenbrock function benchmark
.. math:: f(z) = \sum_{i=1}^{d/2}\left[100(z_{2i-1}^{2}-z_{2i})^{2}+(z_{2i-1}-1)^{2}\right]
This benchmark can also be used to test Bayesian inference methods.
Specifically this benchmarks returns the log likelihood
.. math:: l(z) = -f(z)
which can be used to compute the posterior distribution
.. math:: \pi_{\text{post}}(\rv)=\frac{\pi(\V{y}|\rv)\pi(\rv)}{\int_{\rvdom} \pi(\V{y}|\rv)\pi(\rv)d\rv}
where the prior is the tensor product of :math:`d` independent and
identically distributed uniform variables on :math:`[-2,2]`, i.e.
:math:`\pi(\rv)=\frac{1}{4^d}`, and the likelihood is given by
.. math:: \pi(\V{y}|\rv)=\exp\left(l(\rv)\right)
Parameters
----------
nvars : integer
The number of variables of the Rosenbrock function
Returns
-------
benchmark : pya.Benchmark
Object containing the benchmark attributes documented below
fun : callable
The rosenbrock with signature
``fun(z) -> np.ndarray``
where ``z`` is a 2D np.ndarray with shape (nvars,nsamples) and the
output is a 2D np.ndarray with shape (nsamples,1)
jac : callable
The jacobian of ``fun`` with signature
``jac(z) -> np.ndarray``
where ``z`` is a 2D np.ndarray with shape (nvars,nsamples) and the
output is a 2D np.ndarray with shape (nvars,1)
hessp : callable
Hessian of ``fun`` times an arbitrary vector p with signature
``hessp(z, p) -> ndarray shape (nvars,1)``
where ``z`` is a 2D np.ndarray with shape (nvars,nsamples) and p is an
arbitraty vector with shape (nvars,1)
variable : pya.IndependentMultivariateRandomVariable
Object containing information of the joint density of the inputs z
which is the tensor product of independent and identically distributed
uniform variables on :math:`[-2,2]`.
mean : float
The mean of the rosenbrock function with respect to the pdf of variable.
loglike : callable
The log likelihood of the Bayesian inference problem for inferring z
given the uniform prior specified by variable and the negative
log likelihood given by the Rosenbrock function. loglike has the
signature
``loglike(z) -> np.ndarray``
where ``z`` is a 2D np.ndarray with shape (nvars,nsamples) and the
output is a 2D np.ndarray with shape (nsamples,1)
loglike_grad : callable
The gradient of the ``loglike`` with the signature
``loglike_grad(z) -> np.ndarray``
where ``z`` is a 2D np.ndarray with shape (nvars,nsamples) and the
output is a 2D np.ndarray with shape (nsamples,1)
References
----------
.. [DixonSzego1990] `Dixon, L. C. W.; Mills, D. J. "Effect of Rounding Errors on the Variable Metric Method". Journal of Optimization Theory and Applications. 80: 175–179. 1994 <https://doi.org/10.1007%2FBF02196600>`_
Examples
--------
>>> from pyapprox.benchmarks.benchmarks import setup_benchmark
>>> benchmark=setup_benchmark('rosenbrock',nvars=2)
>>> print(benchmark.keys())
dict_keys(['fun', 'jac', 'hessp', 'variable', 'mean', 'loglike', 'loglike_grad'])
"""
univariate_variables = [stats.uniform(-2, 4)] * nvars
variable = pya.IndependentMultivariateRandomVariable(univariate_variables)
benchmark = Benchmark({
'fun': rosenbrock_function,
'jac': rosenbrock_function_jacobian,
'hessp': rosenbrock_function_hessian_prod,
'variable': variable,
'mean': rosenbrock_function_mean(nvars)
})
benchmark.update({
'loglike': lambda x: -benchmark['fun'](x),
'loglike_grad': lambda x: -benchmark['jac'](x)
})
return benchmark
where :math:`\rv_1,\rv_2\sim\mathcal{U}(-1,1)` and all :math:`A` and :math:`\theta` coefficients are real. We choose to set :math:`A=\sqrt{11}`, :math:`A_1=\sqrt{7}` and :math:`A_2=\sqrt{3}` to obtain unitary variance for each model. The parameters :math:`s_1,s_2` control the bias between the models. Here we set :math:`s_1=1/10,s_2=1/5`. Similarly we can change the correlation between the models in a systematic way (by varying :math:`\theta_1`. We will levarage this later in the tutorial.
"""
#%%
# Lets setup the problem
import pyapprox as pya
import numpy as np
import matplotlib.pyplot as plt
from pyapprox.tests.test_control_variate_monte_carlo import TunableModelEnsemble
from scipy.stats import uniform
np.random.seed(1)
univariate_variables = [uniform(-1, 2), uniform(-1, 2)]
variable = pya.IndependentMultivariateRandomVariable(univariate_variables)
print(variable)
shifts = [.1, .2]
model = TunableModelEnsemble(np.pi / 2 * .95, shifts=shifts)
#%%
# Now let us compute the mean of :math:`f_1` using Monte Carlo
nsamples = int(1e3)
samples = pya.generate_independent_random_samples(variable, nsamples)
values = model.m1(samples)
pya.print_statistics(samples, values)
#%%
# We can compute the exact mean using sympy and compute the MC MSE
import sympy as sp
z1, z2 = sp.Symbol('z1'), sp.Symbol('z2')
def setup_genz_function(nvars, test_name, coefficients=None):
r"""
Setup the Genz Benchmarks.
For example, the two-dimensional oscillatory Genz problem can be defined
using
>>> from pyapprox.benchmarks.benchmarks import setup_benchmark
>>> benchmark=setup_benchmark('genz',nvars=2,test_name='oscillatory')
>>> print(benchmark.keys())
dict_keys(['fun', 'mean', 'variable'])
Parameters
----------
nvars : integer
The number of variables of the Genz function
test_name : string
The test_name of the specific Genz function. See notes
for options the string needed is given in brackets
e.g. ('oscillatory')
coefficients : tuple (ndarray (nvars), ndarray (nvars))
The coefficients :math:`c_i` and :math:`w_i`
If None (default) then
:math:`c_j = \hat{c}_j\left(\sum_{i=1}^d \hat{c}_i\right)^{-1}` where
:math:`\hat{c}_i=(10^{-15\left(\frac{i}{d}\right)^2)})`
Returns
-------
benchmark : pya.Benchmark
Object containing the benchmark attributes
References
----------
.. [Genz1984] `Genz, A. Testing multidimensional integration routines. In Proc. of international conference on Tools, methods and languages for scientific and engineering computation (pp. 81-94), 1984 <https://dl.acm.org/doi/10.5555/2837.2842>`_
Notes
-----
Corner Peak ('corner-peak')
.. math:: f(z)=\left( 1+\sum_{i=1}^d c_iz_i\right)^{-(d+1)}
Oscillatory ('oscillatory')
.. math:: f(z) = \cos\left(2\pi w_1 + \sum_{i=1}^d c_iz_i\right)
Gaussian Peak ('gaussian-peak')
.. math:: f(z) = \exp\left( -\sum_{i=1}^d c_i^2(z_i-w_i)^2\right)
Continuous ('continuous')
.. math:: f(z) = \exp\left( -\sum_{i=1}^d c_i\lvert z_i-w_i\rvert\right)
Product Peak ('product-peak')
.. math:: f(z) = \prod_{i=1}^d \left(c_i^{-2}+(z_i-w_i)^2\right)^{-1}
Discontinuous ('discontinuous')
.. math:: f(z) = \begin{cases}0 & x_1>u_1 \;\mathrm{or}\; x_2>u_2\\\exp\left(\sum_{i=1}^d c_iz_i\right) & \mathrm{otherwise}\end{cases}
"""
genz = GenzFunction(test_name, nvars)
univariate_variables = [stats.uniform(0, 1)] * nvars
variable = pya.IndependentMultivariateRandomVariable(univariate_variables)
if coefficients is None:
genz.set_coefficients(1, 'squared-exponential-decay', 0)
else:
genz.c, genz.w = coefficients
attributes = {'fun': genz, 'mean': genz.integrate(), 'variable': variable}
if test_name == 'corner-peak':
attributes['variance'] = genz.variance()
from scipy.optimize import OptimizeResult
return Benchmark(attributes)
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))
Lets first consider a simple 1D example. The following sets up the problem
"""
import pyapprox as pya
from scipy.stats import uniform
from pyapprox.examples.multi_index_advection_diffusion import *
from pyapprox.models.wrappers import MultiLevelWrapper
nmodels = 3
num_vars = 1
max_eval_concurrency = 1
base_model = setup_model(num_vars, max_eval_concurrency)
multilevel_model = MultiLevelWrapper(base_model,
base_model.base_model.num_config_vars,
base_model.cost_function)
variable = pya.IndependentMultivariateRandomVariable(
[uniform(-np.sqrt(3), 2 * np.sqrt(3))], [np.arange(num_vars)])
#%%
#Now lets us plot each model as a function of the random variable
lb, ub = variable.get_statistics('interval', alpha=1)[0]
nsamples = 10
random_samples = np.linspace(lb, ub, nsamples)[np.newaxis, :]
config_vars = np.arange(nmodels)[np.newaxis, :]
samples = pya.get_all_sample_combinations(random_samples, config_vars)
values = multilevel_model(samples)
values = np.reshape(values, (nsamples, nmodels))
import dolfin as dl
plt.figure(figsize=(nmodels * 8, 2 * 6))
config_samples = multilevel_model.map_to_multidimensional_index(config_vars)
for ii in range(nmodels):
nx, ny, dt = base_model.base_model.get_degrees_of_freedom_and_timestep(
