def __init__(self, beta_coeff, idx_set, jpdf):
self.beta_coeff = beta_coeff
self.idx_set = idx_set
self.jpdf = jpdf
self.N = jpdf.getDimension()
# get the distribution type of each random variable
dist_types = []
for i in range(self.N):
dist_type = self.jpdf.getMarginal(i).getName()
dist_types.append(dist_type)
# create orthogonal univariate bases
poly_collection = ot.PolynomialFamilyCollection(self.N)
for i in range(self.N):
pdf = jpdf.getDistributionCollection()[i]
algo = ot.AdaptiveStieltjesAlgorithm(pdf)
poly_collection[i] = ot.StandardDistributionPolynomialFactory(algo)
# create multivariate basis
multivariate_basis = ot.OrthogonalProductPolynomialFactory(
poly_collection, ot.EnumerateFunction(self.N))
# get enumerate function (multi-index handling)
enum_func = multivariate_basis.getEnumerateFunction()
# get epansion
self.expansion = multivariate_basis.getSubBasis(
transform_multi_index_set(idx_set, enum_func))
# create openturns surrogate model
sur_model = ot.FunctionCollection()
for i in range(len(self.expansion)):
multi = str(beta_coeff[i]) + '*x'
help_function = ot.SymbolicFunction(['x'], [multi])
sur_model.add(ot.ComposedFunction(help_function,
self.expansion[i]))
self.surrogate_model = np.sum(sur_model)
def _buildChaosAlgo(self, inputSample, outputSample):
"""
Build the functional chaos algorithm without running it.
"""
if self._distribution is None:
# create default distribution : Uniform between min and max of the
# input sample
inputSample = ot.NumericalSample(inputSample)
inputMin = inputSample.getMin()
inputMin[0] = np.min(self._defectSizes)
inputMax = inputSample.getMax()
inputMax[0] = np.max(self._defectSizes)
marginals = [
ot.Uniform(inputMin[i], inputMax[i]) for i in range(self._dim)
]
self._distribution = ot.ComposedDistribution(marginals)
# put description of the inputSample into decription of the distribution
self._distribution.setDescription(inputSample.getDescription())
if self._adaptiveStrategy is None:
# Create the adaptive strategy : default is fixed strategy of degree 5
# with linear enumerate function
polyCol = [0.] * self._dim
for i in range(self._dim):
polyCol[i] = ot.StandardDistributionPolynomialFactory(
self._distribution.getMarginal(i))
enumerateFunction = ot.EnumerateFunction(self._dim)
multivariateBasis = ot.OrthogonalProductPolynomialFactory(
polyCol, enumerateFunction)
# default degree is 3 (in __init__)
indexMax = enumerateFunction.getStrataCumulatedCardinal(
self._degree)
self._adaptiveStrategy = ot.FixedStrategy(multivariateBasis,
indexMax)
if self._projectionStrategy is None:
# sparse polynomial chaos
basis_sequence_factory = ot.LAR()
fitting_algorithm = ot.KFold()
approximation_algorithm = ot.LeastSquaresMetaModelSelectionFactory(
basis_sequence_factory, fitting_algorithm)
self._projectionStrategy = ot.LeastSquaresStrategy(
inputSample, outputSample, approximation_algorithm)
return ot.FunctionalChaosAlgorithm(inputSample, outputSample, \
self._distribution, self._adaptiveStrategy, self._projectionStrategy)
def dali_pce(func,
N,
jpdf_cp,
jpdf_ot,
tol=1e-12,
max_fcalls=1000,
verbose=True,
interp_dict={}):
if not interp_dict: # if dictionary is empty --> cold-start
idx_act = [] # M_activated x N
idx_adm = [] # M_admissible x N
fevals_act = [] # M_activated x 1
fevals_adm = [] # M_admissible x 1
coeff_act = [] # M_activated x 1
coeff_adm = [] # M_admissible x 1
# start with 0 multi-index
knot0 = []
for n in range(N):
# get knots per dimension based on maximum index
kk, ww = seq_lj_1d(order=0, dist=jpdf_cp[n])
knot0.append(kk[0])
feval = func(knot0)
# update activated sets
idx_act.append([0] * N)
coeff_act.append(feval)
fevals_act.append(feval)
# local error indicators
local_error_indicators = np.abs(coeff_act)
# get the OT distribution type of each random variable
dist_types = []
for i in range(N):
dist_type = jpdf_ot.getMarginal(i).getName()
dist_types.append(dist_type)
# create orthogonal univariate bases
poly_collection = ot.PolynomialFamilyCollection(N)
for i in range(N):
if dist_types[i] == 'Uniform':
poly_collection[i] = ot.OrthogonalUniVariatePolynomialFamily(
ot.LegendreFactory())
elif dist_types[i] == 'Normal':
poly_collection[i] = ot.OrthogonalUniVariatePolynomialFamily(
ot.HermiteFactory())
elif dist_types[i] == 'Beta':
poly_collection[i] = ot.OrthogonalUniVariatePolynomialFamily(
ot.JacobiFactory())
elif dist_types[i] == 'Gamma':
poly_collection[i] = ot.OrthogonalUniVariatePolynomialFamily(
ot.LaguerreFactory())
else:
pdf = jpdf_ot.getDistributionCollection()[i]
algo = ot.AdaptiveStieltjesAlgorithm(pdf)
poly_collection[i] = ot.StandardDistributionPolynomialFactory(
algo)
# create multivariate basis
mv_basis = ot.OrthogonalProductPolynomialFactory(
poly_collection, ot.EnumerateFunction(N))
# get enumerate function (multi-index handling)
enum_func = mv_basis.getEnumerateFunction()
else:
idx_act = interp_dict['idx_act']
idx_adm = interp_dict['idx_adm']
coeff_act = interp_dict['coeff_act']
coeff_adm = interp_dict['coeff_adm']
fevals_act = interp_dict['fevals_act']
fevals_adm = interp_dict['fevals_adm']
mv_basis = interp_dict['mv_basis']
enum_func = interp_dict['enum_func']
# local error indicators
local_error_indicators = np.abs(coeff_adm)
# compute global error indicator
global_error_indicator = local_error_indicators.sum() # max or sum
# fcalls / M approx. terms up to now
fcalls = len(idx_act) + len(idx_adm) # fcalls = M --> approx. terms
# maximum index per dimension
max_idx_per_dim = np.max(idx_act + idx_adm, axis=0)
# univariate knots and polynomials per dimension
knots_per_dim = {}
for n in range(N):
kk, ww = seq_lj_1d(order=max_idx_per_dim[n], dist=jpdf_cp[n])
knots_per_dim[n] = kk
# start iterations
while global_error_indicator > tol and fcalls < max_fcalls:
if verbose:
print(fcalls)
print(global_error_indicator)
# the index added last to the activated set is the one to be refined
last_act_idx = idx_act[-1][:]
# compute the knot corresponding to the lastly added index
last_knot = [
knots_per_dim[n][i] for n, i in zip(range(N), last_act_idx)
]
# get admissible neighbors of the lastly added index
adm_neighbors = admissible_neighbors(last_act_idx, idx_act)
for an in adm_neighbors:
# update admissible index set
idx_adm.append(an)
# find which parameter/direction n (n=1,2,...,N) gets refined
n_ref = np.argmin(
[idx1 == idx2 for idx1, idx2 in zip(an, last_act_idx)])
# sequence of 1d Leja nodes/weights for the given refinement
knots_n, weights_n = seq_lj_1d(an[n_ref], jpdf_cp[int(n_ref)])
# update max_idx_per_dim, knots_per_dim, if necessary
if an[n_ref] > max_idx_per_dim[n_ref]:
max_idx_per_dim[n_ref] = an[n_ref]
knots_per_dim[n_ref] = knots_n
# find new_knot and compute function on new_knot
new_knot = last_knot[:]
new_knot[n_ref] = knots_n[-1]
feval = func(new_knot)
fevals_adm.append(feval)
fcalls += 1 # update function calls
# create PCE basis
idx_system = idx_act + idx_adm
idx_system_single = transform_multi_index_set(idx_system, enum_func)
system_basis = mv_basis.getSubBasis(idx_system_single)
# get corresponding evaluations
fevals_system = fevals_act + fevals_adm
# multi-dimensional knots
M = len(idx_system) # equations terms
knots_md = [[knots_per_dim[n][idx_system[m][n]] for m in range(M)]
for n in range(N)]
knots_md = np.array(knots_md).T
# design matrix
D = get_design_matrix(system_basis, knots_md)
# solve system of equaations
Q, R = scl.qr(D, mode='economic')
c = Q.T.dot(fevals_system)
coeff_system = scl.solve_triangular(R, c)
# find the multi-index with the largest contribution, add it to idx_act
# and delete it from idx_adm
coeff_act = coeff_system[:len(idx_act)].tolist()
coeff_adm = coeff_system[-len(idx_adm):].tolist()
help_idx = np.argmax(np.abs(coeff_adm))
idx_add = idx_adm.pop(help_idx)
pce_coeff_add = coeff_adm.pop(help_idx)
fevals_add = fevals_adm.pop(help_idx)
idx_act.append(idx_add)
coeff_act.append(pce_coeff_add)
fevals_act.append(fevals_add)
# re-compute coefficients of admissible multi-indices
# local error indicators
local_error_indicators = np.abs(coeff_adm)
# compute global error indicator
global_error_indicator = local_error_indicators.sum() # max or sum
# store expansion data in dictionary
interp_dict = {}
interp_dict['idx_act'] = idx_act
interp_dict['idx_adm'] = idx_adm
interp_dict['coeff_act'] = coeff_act
interp_dict['coeff_adm'] = coeff_adm
interp_dict['fevals_act'] = fevals_act
interp_dict['fevals_adm'] = fevals_adm
interp_dict['enum_func'] = enum_func
interp_dict['mv_basis'] = mv_basis
return interp_dict
def alsace(func,
N,
jpdf,
tol=1e-22,
sample_type='R',
limit_cond=5,
max_fcalls=1000,
seed=123,
ed_file=None,
ed_fevals_file=None,
verbose=True,
pce_dict={}):
"""
ALSACE - Approximations via Lower-Set and Least-Squares-based Adaptive Chaos Expansions
func: function to be approximated.
N: number of parameters.
jpdf: joint probability density function.
limit_cond: maximum allowed condition number of tr(inv(D.T*D))
sample_type: 'R'-random, 'L'-LHS
seed: sampling seed
tol, max_fcalls: exit criteria, self-explanatory.
ed_file, ed_fevals_file: experimental design and corresponding evaluations
'act': activated, i.e. already part of the approximation.
'adm': admissible, i.e. candidates for the approximation's expansion.
"""
if not pce_dict: # if pce_dict is empty --> cold-start
idx_act = []
idx_act.append([0] * N) # start with 0 multi-index
idx_adm = []
# set seed
ot.RandomGenerator.SetSeed(seed)
ed_size = 2 * N # initial number of samples
# initial experimental design and coresponding evaluations
ed, ed_fevals = get_ed(func,
jpdf,
ed_size,
sample_type=sample_type,
knots=[],
values=[],
ed_file=ed_file,
ed_fevals_file=ed_fevals_file)
global_error_indicator = 1.0 # give arbitrary sufficiently large value
# get the distribution type of each random variable
dist_types = []
for i in range(N):
dist_type = jpdf.getMarginal(i).getName()
dist_types.append(dist_type)
# create orthogonal univariate bases
poly_collection = ot.PolynomialFamilyCollection(N)
for i in range(N):
pdf = jpdf.getDistributionCollection()[i]
algo = ot.AdaptiveStieltjesAlgorithm(pdf)
poly_collection[i] = ot.StandardDistributionPolynomialFactory(algo)
# create multivariate basis
mv_basis = ot.OrthogonalProductPolynomialFactory(
poly_collection, ot.EnumerateFunction(N))
# get enumerate function (multi-index handling)
enum_func = mv_basis.getEnumerateFunction()
else: # get data from dictionary
idx_act = pce_dict['idx_act']
idx_adm = pce_dict['idx_adm']
pce_coeff_act = pce_dict['pce_coeff_act']
pce_coeff_adm = pce_dict['pce_coeff_adm']
ed = pce_dict['ed']
ed_fevals = pce_dict['ed_fevals']
ed_size = len(ed_fevals)
# compute local and global error indicators
global_error_indicator = np.sum(np.array(pce_coeff_adm)**2)
enum_func = pce_dict['enum_func']
mv_basis = pce_dict['mv_basis']
#
while ed_size < max_fcalls and global_error_indicator > tol:
# the index added last to the activated set is the one to be refined
last_act_idx = idx_act[-1][:]
# get admissible neighbors of the lastly added index
adm_neighbors = admissible_neighbors(last_act_idx, idx_act)
# update admissible indices
idx_adm = idx_adm + adm_neighbors
# get polynomial basis for the LS problem
idx_ls = idx_act + idx_adm
idx_ls_single = transform_multi_index_set(idx_ls, enum_func)
ls_basis = mv_basis.getSubBasis(idx_ls_single)
ls_basis_size = len(ls_basis)
# construct the design matrix D and compute its QR decomposition
D = get_design_matrix(ls_basis, ed)
Q, R = sp.qr(D, mode='economic')
# construct information matrix A= D^T*D
A = np.matmul(D.T, D) / ed_size
trAinv_test = np.sum(1. / np.linalg.eig(A)[0])
trAinv = np.trace(np.linalg.inv(A))
print('new trace ', trAinv_test)
print('old trace ', trAinv)
# If tr(A) becomes too large, enrich the ED until tr(A) becomes
# acceptable or until ed_size reaches max_fcalls
while (trAinv > limit_cond
and ed_size < max_fcalls) or ed_size < ls_basis_size:
# inform user
if verbose:
print('WARNING: tr(inv(A)) = ', trAinv)
print('WARNING: cond(D) = ', np.linalg.cond(D))
print("")
# select new size for the ED
if ls_basis_size > ed_size:
ed_size = ls_basis_size + N
elif ed_size + N > max_fcalls:
ed_size = max_fcalls
else:
ed_size = ed_size + N
# expand ED
ed, ed_fevals = get_ed(func,
jpdf,
ed_size,
sample_type=sample_type,
knots=ed,
values=ed_fevals,
ed_file=ed_file,
ed_fevals_file=ed_fevals_file)
# construct the design matrix D and compute its QR decomposition
D = get_design_matrix(ls_basis, ed)
Q, R = sp.qr(D, mode='economic')
# construct information matrix A= D^T*D
A = np.matmul(D.T, D) / ed_size
trAinv = np.trace(np.linalg.inv(A))
# solve LS problem
c = Q.T.dot(ed_fevals)
pce_coeff_ls = sp.solve_triangular(R, c)
# find the multi-index with the largest contribution, add it to idx_act
# and delete it from idx_adm
pce_coeff_act = pce_coeff_ls[:len(idx_act)].tolist()
pce_coeff_adm = pce_coeff_ls[-len(idx_adm):].tolist()
help_idx = np.argmax(np.abs(pce_coeff_adm))
idx_add = idx_adm.pop(help_idx)
pce_coeff_add = pce_coeff_adm.pop(help_idx)
idx_act.append(idx_add)
pce_coeff_act.append(pce_coeff_add)
# store expansion data in dictionary
pce_dict = {}
pce_dict['idx_act'] = idx_act
pce_dict['idx_adm'] = idx_adm
pce_dict['pce_coeff_act'] = pce_coeff_act
pce_dict['pce_coeff_adm'] = pce_coeff_adm
pce_dict['ed'] = ed
pce_dict['ed_fevals'] = ed_fevals
pce_dict['enum_func'] = enum_func
pce_dict['mv_basis'] = mv_basis
return pce_dict
def __init__(self, strategy, degree, distributions, N_quad=None, sample=None,
stieltjes=True, sparse_param={}):
"""Generate truncature and projection strategies.
Allong with the strategies the sample is storred as an attribute.
:attr:`sample` as well as corresponding weights: :attr:`weights`.
:param str strategy: Least square or Quadrature ['LS', 'Quad', 'SparseLS'].
:param int degree: Polynomial degree.
:param distributions: Distributions of each input parameter.
:type distributions: lst(:class:`openturns.Distribution`)
:param array_like sample: Samples for least square
(n_samples, n_features).
:param bool stieltjes: Wether to use Stieltjes algorithm for the basis.
:param dict sparse_param: Parameters for the Sparse Cleaning Truncation
Strategy and/or hyperbolic truncation of the initial basis.
- **max_considered_terms** (int) -- Maximum Considered Terms,
- **most_significant** (int), Most Siginificant number to retain,
- **significance_factor** (float), Significance Factor,
- **hyper_factor** (float), factor for hyperbolic truncation
strategy.
"""
# distributions
self.in_dim = len(distributions)
self.dist = ot.ComposedDistribution(distributions)
self.sparse_param = sparse_param
if 'hyper_factor' in self.sparse_param:
enumerateFunction = ot.EnumerateFunction(self.in_dim, self.sparse_param['hyper_factor'])
else:
enumerateFunction = ot.EnumerateFunction(self.in_dim)
if stieltjes:
# Tend to result in performance issue
self.basis = ot.OrthogonalProductPolynomialFactory(
[ot.StandardDistributionPolynomialFactory(
ot.AdaptiveStieltjesAlgorithm(marginal))
for marginal in distributions], enumerateFunction)
else:
self.basis = ot.OrthogonalProductPolynomialFactory(
[ot.StandardDistributionPolynomialFactory(margin)
for margin in distributions], enumerateFunction)
self.n_basis = enumerateFunction.getStrataCumulatedCardinal(degree)
# Strategy choice for expansion coefficient determination
self.strategy = strategy
if self.strategy == 'LS' or self.strategy == 'SparseLS': # least-squares method
self.sample = sample
else: # integration method
# redefinition of sample size
# n_samples = (degree + 1) ** self.in_dim
# marginal degree definition
# by default: the marginal degree for each input random
# variable is set to the total polynomial degree 'degree'+1
measure = self.basis.getMeasure()
if N_quad is not None:
degrees = [int(N_quad ** 0.25)] * self.in_dim
else:
degrees = [degree + 1] * self.in_dim
self.proj_strategy = ot.IntegrationStrategy(
ot.GaussProductExperiment(measure, degrees))
self.sample, self.weights = self.proj_strategy.getExperiment().generateWithWeights()
if not stieltjes:
transformation = ot.Function(ot.MarginalTransformationEvaluation(
[measure.getMarginal(i) for i in range(self.in_dim)],
distributions, False))
self.sample = transformation(self.sample)
self.pc = None
self.pc_result = None
#! /usr/bin/env python
import openturns as ot
from math import log
Sample = ot.NumericalSample.ImportFromTextFile("DATA_test2.csv", ",")
# Sample.setDescription(["BIO","SAL","pH","K","Na","Zn"])
print(Sample)
X = Sample[:, 1:6]
Y = Sample[:, 0]
################################################################################################
# Build a model BIO~SAL+pH+K+Na+Zn
dim = X.getDimension()
enumerateFunction = ot.EnumerateFunction(dim)
factory = ot.OrthogonalProductPolynomialFactory([ot.MonomialFactory()]*dim, enumerateFunction)
# Build 'interactions' as a list of list [a1,a2,a3,a4,a5], and we will generate tensorized
# polynomials SAL^a1*pH^a2*K^a3*Na^a4*Zn^a5.
# BIO~SAL+pH+K+Na+Zn
interactions = []
interactions.append([0]*dim)
for i in xrange(dim):
indices = [0]*dim
indices[i] = 1
# Y ~ I(Xi)^1
interactions.append(indices[:])
basis = ot.Basis([factory.build(enumerateFunction.inverse(indices)) for indices in interactions])
# Get input & output sample
lhs = ot.LHSExperiment(distribution, samplingSize)
inputSample = lhs.generate()
outputSample = model(inputSample)
# Validation of results on independent samples
validationSize = 10
inputValidation = distribution.getSample(validationSize)
outputValidation = model(inputValidation)
# 1) SPC algorithm
# Create the orthogonal basis
polynomialCollection = [ot.LegendreFactory()] * dimension
enumerateFunction = ot.EnumerateFunction(dimension)
productBasis = ot.OrthogonalProductPolynomialFactory(
polynomialCollection, enumerateFunction)
# Create the adaptive strategy
degree = 8
basisSize = enumerateFunction.getStrataCumulatedCardinal(degree)
adaptiveStrategy = ot.FixedStrategy(productBasis, basisSize)
# Select the fitting algorithm
fittingAlgorithm = ot.KFold()
leastSquaresFactory = ot.LeastSquaresMetaModelSelectionFactory(
ot.LARS(), fittingAlgorithm)
# Projection strategy
projectionStrategy = ot.LeastSquaresStrategy(inputSample, outputSample,
poly_collection[i] = ot.OrthogonalUniVariatePolynomialFamily(
ot.HermiteFactory())
elif dist_types[i] == 'Beta':
poly_collection[i] = ot.OrthogonalUniVariatePolynomialFamily(
ot.JacobiFactory())
elif dist_types[i] == 'Gamma':
poly_collection[i] = ot.OrthogonalUniVariatePolynomialFamily(
ot.LaguerreFactory())
else:
pdf = jpdf_ot.getDistributionCollection()[i]
algo = ot.AdaptiveStieltjesAlgorithm(pdf)
poly_collection[i] = ot.StandardDistributionPolynomialFactory(algo)
# create multivariate basis
mv_basis = ot.OrthogonalProductPolynomialFactory(poly_collection,
ot.EnumerateFunction(N))
# get enumerate function (multi-index handling)
enum_func = mv_basis.getEnumerateFunction()
max_fcalls = np.linspace(100, 1000, 10).tolist()
meanz = []
varz = []
cv_errz_rms = []
cv_errz_max = []
fcallz = []
# cross validation sample
np.random.seed(42)
nsamples = 1000
cv_samples_in = jpdf.sample(nsamples).T
cv_samples_out = [meroND(sample) for sample in cv_samples_in]
for mfc in max_fcalls:
def _legendre_galerkin_scheme(self,
legendre_galerkin_order=10,
legendre_quadrature_order=None):
# Input checks
if legendre_galerkin_order <= 0:
raise ValueError('legendre_galerkin_order must be a positive ' +
'integer!')
if legendre_quadrature_order is not None:
if legendre_quadrature_order <= 0:
raise ValueError('legendre_quadrature_order must be a ' +
'positive integer!')
# Settings
dimension = self._lower_bound.size
truncation_order = self._truncation_order
galerkin_size = ot.EnumerateFunction(
dimension).getStrataCumulatedCardinal(legendre_galerkin_order)
if legendre_quadrature_order is None:
legendre_quadrature_order = 2 * legendre_galerkin_order + 1
# Check if the current settings are compatible
if truncation_order > galerkin_size:
raise ValueError(
'The truncation order must be less than or ' +
'equal to the size of the functional basis in the chosen ' +
'Legendre Galerkin scheme. Current size of the galerkin basis '
+ 'only allows to get %d terms in the KL expansion.' %
galerkin_size)
# Construction of the Galerkin basis: tensorized Legendre polynomials
tensorized_legendre_polynomial_factory = \
ot.PolynomialFamilyCollection([ot.LegendreFactory()] * dimension)
tensorized_legendre_polynomial_factory = \
ot.OrthogonalProductPolynomialFactory(
tensorized_legendre_polynomial_factory)
tensorized_legendre_polynomials = \
[tensorized_legendre_polynomial_factory.build(i)
for i in range(galerkin_size)]
# Compute matrix C coefficients using Gauss-Legendre quadrature
polyColl = ot.PolynomialFamilyCollection([ot.LegendreFactory()] *
dimension * 2)
polynoms = ot.OrthogonalProductPolynomialFactory(polyColl)
U, W = polynoms.getNodesAndWeights(
ot.Indices([legendre_quadrature_order] * dimension * 2))
W = np.ravel(W)
scale = (self._upper_bound - self._lower_bound) / 2.
shift = (self._upper_bound + self._lower_bound) / 2.
U = np.array(U)
X = np.repeat(scale, 2) * U + np.repeat(shift, 2)
if self.verbose:
print('Computing matrix C...')
try:
available_memory = int(.9 * get_available_memory())
except:
if self.verbose:
print('WRN: Available memory estimation failed! '
'Assuming 1Gb is available (first guess).')
available_memory = 1024**3
max_size = int(available_memory / 8 / galerkin_size**2)
batch_size = min(W.size, max_size)
if self.verbose and batch_size < W.size:
print('RAM: %d Mb available' % (available_memory / 1024**2))
print('RAM: %d allocable terms / %d total terms' %
(max_size, W.size))
print('RAM: %d loops required' % np.ceil(float(W.size) / max_size))
while True:
C = np.zeros((galerkin_size, galerkin_size))
try:
n_done = 0
while n_done < W.size:
covariance_at_X = self._covariance(X[n_done:(n_done +
batch_size)])
H1 = np.vstack([
np.ravel(tensorized_legendre_polynomials[i](
U[n_done:(n_done + batch_size), :dimension]))
for i in range(galerkin_size)
])
H2 = np.vstack([
np.ravel(tensorized_legendre_polynomials[i](
U[n_done:(n_done + batch_size), dimension:]))
for i in range(galerkin_size)
])
C += np.sum(W[np.newaxis, np.newaxis,
n_done:(n_done + batch_size)] *
covariance_at_X[np.newaxis, np.newaxis, :] *
H1[np.newaxis, :, :] * H2[:, np.newaxis, :],
axis=-1)
del covariance_at_X, H1, H2
n_done += batch_size
break
except MemoryError:
batch_size /= 2
C *= np.prod(self._upper_bound - self._lower_bound)**2.
# Matrix B is orthonormal up to some constant
B = np.diag(
np.repeat(np.prod(self._upper_bound - self._lower_bound),
galerkin_size))
# Solve the generalized eigenvalue problem C D = L B D in L, D
if self.verbose:
print('Solving generalized eigenvalue problem...')
eigenvalues, eigenvectors = linalg.eigh(C, b=B, lower=True)
eigenvalues, eigenvectors = eigenvalues.real, eigenvectors.real
# Sort the eigensolutions in the descending order of eigenvalues
order = eigenvalues.argsort()[::-1]
eigenvalues = eigenvalues[order]
eigenvectors = eigenvectors[:, order]
# Truncate the expansion
eigenvalues = eigenvalues[:truncation_order]
eigenvectors = eigenvectors[:, :truncation_order]
# Eliminate unsignificant negative eigenvalues
if eigenvalues.min() <= 0.:
if eigenvalues.min() > .01 * eigenvalues.max():
raise Exception(
'The smallest significant eigenvalue seems ' +
'to be negative... Check the positive definiteness of the '
+ 'covariance function.')
else:
truncation_order = np.nonzero(eigenvalues <= 0)[0][0]
eigenvalues = eigenvalues[:truncation_order]
eigenvectors = eigenvectors[:, :truncation_order]
self._truncation_order = truncation_order
print('WRN: truncation_order was too large.')
print('It has been reset to: %d' % truncation_order)
# Define eigenfunctions
class LegendrePolynomialsBasedEigenFunction():
def __init__(self, vector):
self._vector = vector
def __call__(self, x):
x = np.asanyarray(x)
if x.ndim <= 1:
x = np.atleast_2d(x).T
u = (x - shift) / scale
return np.sum([
np.ravel(tensorized_legendre_polynomials[i](u)) *
self._vector[i] for i in range(truncation_order)
],
axis=0)
# Set attributes
self._eigenvalues = eigenvalues
self._eigenfunctions = [
LegendrePolynomialsBasedEigenFunction(vector)
for vector in eigenvectors.T
]
self._legendre_galerkin_order = legendre_galerkin_order
self._legendre_quadrature_order = legendre_quadrature_order
graph2 = metaModel(validationInputSample).drawMarginal(0)
graph2.setColors(['blue'])
graph.add(graph2)
graph.setTitle('Comparaison modele/meta-modele')
graph.setXTitle(r'$t$')
graph.setYTitle(r'$z$')
otv.View(graph)
# Second, using a more evolved interface
basis = ot.OrthogonalProductPolynomialFactory([
ot.StandardDistributionPolynomialFactory(distX.getMarginal(i))
for i in range(distX.getDimension())
])
adaptiveStrategy = ot.FixedStrategy(
basis,
ot.EnumerateFunction(distX.getDimension()).getStrataCumulatedCardinal(6))
projectionStrategy = ot.LeastSquaresStrategy(
ot.LeastSquaresMetaModelSelectionFactory(ot.LARS(),
ot.CorrectedLeaveOneOut()))
algo = ot.FunctionalChaosAlgorithm(inputSample, outputSampleChaos, distX,
adaptiveStrategy, projectionStrategy)
algo.run()
metaModel = ot.PointToFieldConnection(postProcessing,
algo.getResult().getMetaModel())
graph = validationOutputSample.drawMarginal(0)
graph.setColors(['red'])
graph2 = metaModel(validationInputSample).drawMarginal(0)
graph2.setColors(['blue'])
graph.add(graph2)
graph.setTitle('Comparaison modele/meta-modele')
# Input distribution
distribution = ot.ComposedDistribution([ot.Normal()] * inputDimension)
# Correlated input distribution
S = ot.CorrelationMatrix(inputDimension)
S[1, 0] = 0.3
R = ot.NormalCopula().GetCorrelationFromSpearmanCorrelation(S)
myCopula = ot.NormalCopula(R)
myCorrelatedInputDistribution = ot.ComposedDistribution(
[ot.Normal()] * inputDimension, myCopula)
sample = myCorrelatedInputDistribution.getSample(2000)
# Orthogonal basis
enumerateFunction = ot.EnumerateFunction(inputDimension)
productBasis = ot.OrthogonalProductPolynomialFactory(
[ot.HermiteFactory()] * inputDimension, enumerateFunction)
# Adaptive strategy
adaptiveStrategy = ot.FixedStrategy(
productBasis, enumerateFunction.getStrataCumulatedCardinal(4))
# Projection strategy
samplingSize = 250
projectionStrategy = ot.LeastSquaresStrategy(
ot.MonteCarloExperiment(samplingSize))
# Polynomial chaos algorithm
algo = ot.FunctionalChaosAlgorithm(model, distribution, adaptiveStrategy,
projectionStrategy)
algo.run()
