def fit(self, X, y, **fit_params):
input_dimension = X.shape[1]
if self.distribution is None:
self.distribution = BuildDistribution(X)
if self.enumerate == 'linear':
enumerateFunction = ot.LinearEnumerateFunction(input_dimension)
elif self.enumerate == 'hyperbolic':
enumerateFunction = ot.HyperbolicAnisotropicEnumerateFunction(input_dimension, self.q)
else:
raise ValueError('enumerate should be "linear" or "hyperbolic"')
polynomials = [ot.StandardDistributionPolynomialFactory(self.distribution.getMarginal(i)) for i in range(input_dimension)]
productBasis = ot.OrthogonalProductPolynomialFactory(polynomials, enumerateFunction)
adaptiveStrategy = ot.FixedStrategy(productBasis, enumerateFunction.getStrataCumulatedCardinal(self.degree))
if self.sparse:
projectionStrategy = ot.LeastSquaresStrategy(ot.LeastSquaresMetaModelSelectionFactory(ot.LARS(), ot.CorrectedLeaveOneOut()))
else:
projectionStrategy = ot.LeastSquaresStrategy(X, y.reshape(-1, 1))
algo = ot.FunctionalChaosAlgorithm(X, y.reshape(-1, 1), self.distribution, adaptiveStrategy, projectionStrategy)
algo.run()
self._result = algo.getResult()
output_dimension = self._result.getMetaModel().getOutputDimension()
# sensitivity
si = ot.FunctionalChaosSobolIndices(self._result)
if output_dimension == 1:
self.feature_importances_ = [si.getSobolIndex(i) for i in range(input_dimension)]
else:
self.feature_importances_ = [[0.0] * input_dimension] * output_dimension
for k in range(output_dimension):
for i in range(input_dimension):
self.feature_importances_[k][i] = si.getSobolIndex(i, k)
self.feature_importances_ = np.array(self.feature_importances_)
return self
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 fit(self, X, y, **fit_params):
"""Fit Tensor regression model.
Parameters
----------
X : array-like, shape = (n_samples, n_features)
Training data.
y : array-like, shape = (n_samples, [n_output_dims])
Target values.
Returns
-------
self : returns an instance of self.
"""
if len(X) == 0:
raise ValueError(
"Can not perform a tensor approximation with empty sample")
# check data type is accurate
if (len(np.shape(X)) != 2):
raise ValueError("X has incorrect shape.")
input_dimension = len(X[1])
if (len(np.shape(y)) != 2):
raise ValueError("y has incorrect shape.")
if self.distribution is None:
self.distribution = BuildDistribution(X)
factoryCollection = [
ot.OrthogonalUniVariateFunctionFamily(
ot.OrthogonalUniVariatePolynomialFunctionFactory(
ot.StandardDistributionPolynomialFactory(
self.distribution.getMarginal(i))))
for i in range(input_dimension)
]
functionFactory = ot.OrthogonalProductFunctionFactory(
factoryCollection)
algo = ot.TensorApproximationAlgorithm(X, y, self.distribution,
functionFactory,
[self.nk] * input_dimension,
self.max_rank)
algo.run()
self.result_ = algo.getResult()
return self
# Create a distribution of dimension n
# for example n=3 with indpendent components
distribution = ot.ComposedDistribution(
[ot.Normal(), ot.Uniform(), ot.Gamma(2.75, 1.0), ot.Beta(2.5, 1.0, -1.0, 2.0)])
# %%
# Prepare the input/output samples
sampleSize = 250
X = distribution.getSample(sampleSize)
Y = myModel(X)
dimension = X.getDimension()
# %%
# build the orthogonal basis
coll = [ot.StandardDistributionPolynomialFactory(distribution.getMarginal(i)) for i in range(dimension)]
enumerateFunction = ot.LinearEnumerateFunction(dimension)
productBasis = ot.OrthogonalProductPolynomialFactory(coll, enumerateFunction)
# %%
# create the algorithm
degree = 6
adaptiveStrategy = ot.FixedStrategy(
productBasis, enumerateFunction.getStrataCumulatedCardinal(degree))
projectionStrategy = ot.LeastSquaresStrategy()
algo = ot.FunctionalChaosAlgorithm(X, Y, distribution, adaptiveStrategy, projectionStrategy)
algo.run()
# %%
# get the metamodel function
result = algo.getResult()
import openturns as ot
from openturns.viewer import View
dim = 1
f = ot.SymbolicFunction(['x'], ['x*sin(x)'])
uniform = ot.Uniform(0.0, 10.0)
distribution = ot.ComposedDistribution([uniform] * dim)
factoryCollection = [
ot.OrthogonalUniVariateFunctionFamily(
ot.OrthogonalUniVariatePolynomialFunctionFactory(
ot.StandardDistributionPolynomialFactory(uniform)))
] * dim
functionFactory = ot.OrthogonalProductFunctionFactory(factoryCollection)
size = 10
sampleX = [[1.0], [2.0], [3.0], [4.0], [5.0], [6.0], [7.0], [8.0]]
sampleY = f(sampleX)
nk = [5] * dim
maxRank = 1
algo = ot.TensorApproximationAlgorithm(sampleX, sampleY, distribution,
functionFactory, nk, maxRank)
algo.run()
result = algo.getResult()
metamodel = result.getMetaModel()
graph = f.draw(0.0, 10.0)
graph.add(metamodel.draw(0.0, 10.0))
graph.add(ot.Cloud(sampleX, sampleY))
graph.setColors(['blue', 'red', 'black'])
graph.setLegends(['model', 'meta model', 'sample'])
graph.setLegendPosition('topleft')
graph.setTitle('y(x)=x*sin(x)')
# borehole
# dim = 8
# model = ot.SymbolicFunction(['rw', 'r', 'Tu', 'Hu', 'Tl', 'Hl', 'L', 'Kw'],
#['(2*_pi*Tu*(Hu-Hl))/(ln(r/rw)*(1+(2*L*Tu)/(ln(r/rw)*rw^2*Kw)+Tu/Tl))'])
# coll = [ot.Normal(0.1, 0.0161812),
# ot.LogNormal(7.71, 1.0056),
# ot.Uniform(63070.0, 115600.0),
# ot.Uniform(990.0, 1110.0),
# ot.Uniform(63.1, 116.0),
# ot.Uniform(700.0, 820.0),
# ot.Uniform(1120.0, 1680.0),
# ot.Uniform(9855.0, 12045.0)]
distribution = ot.ComposedDistribution(coll)
factoryCollection = [ot.OrthogonalUniVariateFunctionFamily(
ot.OrthogonalUniVariatePolynomialFunctionFactory(ot.StandardDistributionPolynomialFactory(dist))) for dist in coll]
functionFactory = ot.OrthogonalProductFunctionFactory(factoryCollection)
size = 1000
X = distribution.getSample(size)
Y = model(X)
# ot.ResourceMap.Set('TensorApproximationAlgorithm-Method', 'RankM')
# n-d
nk = [10] * dim
maxRank = 5
algo = ot.TensorApproximationAlgorithm(
# with:
#
# .. math::
# v_j^{(i)} (x_j) = \sum_{k=1}^{n_j} \beta_{j,k}^{(i)} \phi_{j,k} (x_j)
#
# We should define :
#
# - The family of univariate functions :math:`\phi_j`. We choose the orthogonal basis with respect to the marginal distribution measures.
# - The maximal rank :math:`m`. Here value is set to 5
# - The marginal degrees :math:`n_j`. Here we set the degrees to [4, 15, 3, 2]
#
# %%
factoryCollection = [
ot.OrthogonalUniVariatePolynomialFunctionFactory(
ot.StandardDistributionPolynomialFactory(_)) for _ in [E, F, L, I]
]
functionFactory = ot.OrthogonalProductFunctionFactory(factoryCollection)
nk = [4, 15, 3, 2]
maxRank = 1
# %%
# Finally we might launch the algorithm:
# %%
algo = ot.TensorApproximationAlgorithm(X_train, Y_train, myDistribution,
functionFactory, nk, maxRank)
algo.run()
result = algo.getResult()
metamodel = result.getMetaModel()
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
ot.Rayleigh(1.0),
ot.Student(22.0),
ot.Triangular(-1.0, 0.3, 1.0),
ot.Uniform(-1.0, 1.0),
ot.Uniform(-1.0, 3.0),
ot.Weibull(1.0, 3.0),
ot.Beta(1.0, 3.0, -1.0, 1.0),
ot.Beta(0.5, 1.0, -1.0, 1.0),
ot.Beta(0.5, 1.0, -2.0, 3.0),
ot.Gamma(1.0, 3.0),
ot.Arcsine()
]
for n in range(len(distributionCollection)):
distribution = distributionCollection[n]
name = distribution.getClassName()
polynomialFactory = ot.StandardDistributionPolynomialFactory(
ot.AdaptiveStieltjesAlgorithm(distribution))
print("polynomialFactory(", name, "=", polynomialFactory, ")")
for i in range(iMax):
print(name, " polynomial(", i, ")=", clean(polynomialFactory.build(i)))
roots = polynomialFactory.getRoots(iMax - 1)
print(name, " polynomial(", iMax - 1, ") roots=", roots)
nodes, weights = polynomialFactory.getNodesAndWeights(iMax - 1)
print(name, " polynomial(", iMax - 1, ") nodes=", nodes, " and weights=",
weights)
M = ot.SymmetricMatrix(iMax)
for i in range(iMax):
pI = polynomialFactory.build(i)
for j in range(i + 1):
pJ = polynomialFactory.build(j)
def kernel(x):
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()
max_fcalls = np.linspace(100, 1000, 10).tolist()
meanz = []
varz = []
cv_errz_rms = []
cv_errz_max = []
fcallz = []
# cross validation sample
np.random.seed(42)
def fit(self, X, y, **fit_params):
"""Fit PC regression model.
Parameters
----------
X : array-like, shape = (n_samples, n_features)
Training data.
y : array-like, shape = (n_samples, [n_output_dims])
Target values.
Returns
-------
self : returns an instance of self.
"""
if len(X) == 0:
raise ValueError(
"Can not perform chaos expansion with empty sample")
# check data type is accurate
if (len(np.shape(X)) != 2):
raise ValueError("X has incorrect shape.")
input_dimension = len(X[1])
if (len(np.shape(y)) != 2):
raise ValueError("y has incorrect shape.")
if self.distribution is None:
self.distribution = ot.MetaModelAlgorithm.BuildDistribution(X)
if self.enumeratef == 'linear':
enumerateFunction = ot.LinearEnumerateFunction(input_dimension)
elif self.enumeratef == 'hyperbolic':
enumerateFunction = ot.HyperbolicAnisotropicEnumerateFunction(
input_dimension, self.q)
else:
raise ValueError('enumeratef should be "linear" or "hyperbolic"')
polynomials = [
ot.StandardDistributionPolynomialFactory(
self.distribution.getMarginal(i))
for i in range(input_dimension)
]
productBasis = ot.OrthogonalProductPolynomialFactory(
polynomials, enumerateFunction)
adaptiveStrategy = ot.FixedStrategy(
productBasis,
enumerateFunction.getStrataCumulatedCardinal(self.degree))
if self.sparse:
# Filter according to the sparse_fitting_algorithm key
if self.sparse_fitting_algorithm == "cloo":
fitting_algorithm = ot.CorrectedLeaveOneOut()
else:
fitting_algorithm = ot.KFold()
# Define the correspondinding projection strategy
projectionStrategy = ot.LeastSquaresStrategy(
ot.LeastSquaresMetaModelSelectionFactory(
ot.LARS(), fitting_algorithm))
else:
projectionStrategy = ot.LeastSquaresStrategy(X, y)
algo = ot.FunctionalChaosAlgorithm(X, y, self.distribution,
adaptiveStrategy,
projectionStrategy)
algo.run()
self.result_ = algo.getResult()
output_dimension = self.result_.getMetaModel().getOutputDimension()
# sensitivity
si = ot.FunctionalChaosSobolIndices(self.result_)
if output_dimension == 1:
self.feature_importances_ = [
si.getSobolIndex(i) for i in range(input_dimension)
]
else:
self.feature_importances_ = [[0.0] * input_dimension
] * output_dimension
for k in range(output_dimension):
for i in range(input_dimension):
self.feature_importances_[k][i] = si.getSobolIndex(i, k)
self.feature_importances_ = np.array(self.feature_importances_)
return self
# f(X_1, \dots, X_d) = \sum_{i=1}^m \prod_{j=1}^d v_j^{(i)} (x_j), \forall x \in \mathbb{R}^d
#
# with:
#
# .. math::
# v_j^{(i)} (x_j) = \sum_{k=1}^{n_j} \beta_{j,k}^{(i)} \phi_{j,k} (x_j)
#
# We should define :
#
# - The family of univariate functions :math:`\phi_j`. We choose the orthogonal basis with respect to the marginal distribution measures.
# - The maximal rank :math:`m`. Here value is set to 5
# - The marginal degrees :math:`n_j`. Here we set the degrees to [4, 15, 3, 2]
#
# %%
factoryCollection = [ot.OrthogonalUniVariatePolynomialFunctionFactory(ot.StandardDistributionPolynomialFactory(_)) for _ in [E,F,L,I]]
functionFactory = ot.OrthogonalProductFunctionFactory(factoryCollection)
nk = [4, 15, 3, 2]
maxRank = 1
# %%
# Finally we might launch the algorithm:
# %%
algo = ot.TensorApproximationAlgorithm(X_train, Y_train, myDistribution, functionFactory, nk, maxRank)
algo.run()
result = algo.getResult()
metamodel = result.getMetaModel()
# %%
# The `run` method has optimized the hyperparameters of the metamodel (:math:`\beta` coefficients).
def train(self):
self.input_dim = self.training_points[None][0][0].shape[1]
x_train = ot.Sample(self.training_points[None][0][0])
y_train = ot.Sample(self.training_points[None][0][1])
# Distribution choice of the inputs to Create the input distribution
distributions = []
dist_specs = self.options["uncertainty_specs"]
if dist_specs:
if len(dist_specs) != self.input_dim:
raise SurrogateOpenturnsException(
"Number of distributions should be equal to input \
dimensions. Should be {}, got {}".format(
self.input_dim, len(dist_specs)
)
)
for ds in dist_specs:
dist_klass = getattr(sys.modules["openturns"], ds["name"])
args = [ds["kwargs"][name] for name in DISTRIBUTION_SIGNATURES[ds["name"]]]
distributions.append(dist_klass(*args))
else:
for i in range(self.input_dim):
mean = np.mean(x_train[:, i])
lower, upper = 0.95 * mean, 1.05 * mean
if mean < 0:
lower, upper = upper, lower
distributions.append(ot.Uniform(lower, upper))
distribution = ot.ComposedDistribution(distributions)
# Polynomial basis
# step 1 - Construction of the multivariate orthonormal basis:
# Build orthonormal or orthogonal univariate polynomial families
# (associated to associated input distribution)
polynoms = [0.0] * self.input_dim
for i in range(distribution.getDimension()):
polynoms[i] = ot.StandardDistributionPolynomialFactory(
distribution.getMarginal(i)
)
enumerateFunction = ot.LinearEnumerateFunction(self.input_dim)
productBasis = ot.OrthogonalProductPolynomialFactory(
polynoms, enumerateFunction
)
# step 2 - Truncation strategy of the multivariate orthonormal basis:
# a strategy must be chosen for the selection of the different terms
# of the multivariate basis.
# Truncature strategy of the multivariate orthonormal basis
# We choose all the polynomials of degree <= degree
degree = self.options["pce_degree"]
index_max = enumerateFunction.getStrataCumulatedCardinal(degree)
adaptive_strategy = ot.FixedStrategy(productBasis, index_max)
basis_sequenceFactory = ot.LARS()
fitting_algorithm = ot.CorrectedLeaveOneOut()
approximation_algorithm = ot.LeastSquaresMetaModelSelectionFactory(
basis_sequenceFactory, fitting_algorithm
)
projection_strategy = ot.LeastSquaresStrategy(
x_train, y_train, approximation_algorithm
)
algo = ot.FunctionalChaosAlgorithm(
x_train, y_train, distribution, adaptive_strategy, projection_strategy
)
# algo = ot.FunctionalChaosAlgorithm(X_train_NS, Y_train_NS)
algo.run()
self._pce_result = algo.getResult()
# model = ot.SymbolicFunction(['rw', 'r', 'Tu', 'Hu', 'Tl', 'Hl', 'L', 'Kw'],
# ['(2*pi_*Tu*(Hu-Hl))/(ln(r/rw)*(1+(2*L*Tu)/(ln(r/rw)*rw^2*Kw)+Tu/Tl))'])
# coll = [ot.Normal(0.1, 0.0161812),
# ot.LogNormal(7.71, 1.0056),
# ot.Uniform(63070.0, 115600.0),
# ot.Uniform(990.0, 1110.0),
# ot.Uniform(63.1, 116.0),
# ot.Uniform(700.0, 820.0),
# ot.Uniform(1120.0, 1680.0),
# ot.Uniform(9855.0, 12045.0)]
distribution = ot.ComposedDistribution(coll)
factoryCollection = [
ot.OrthogonalUniVariateFunctionFamily(
ot.OrthogonalUniVariatePolynomialFunctionFactory(
ot.StandardDistributionPolynomialFactory(dist))) for dist in coll
]
functionFactory = ot.OrthogonalProductFunctionFactory(factoryCollection)
size = 1000
X = distribution.getSample(size)
Y = model(X)
# ot.ResourceMap.Set('TensorApproximationAlgorithm-Method', 'RankM')
# n-d
nk = [10] * dim
maxRank = 5
algo = ot.TensorApproximationAlgorithm(X, Y, distribution, functionFactory, nk,
maxRank)
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
def fit(self, X, y, **fit_params):
input_dimension = X.shape[1]
if self.distribution is None:
self.distribution = BuildDistribution(X)
factoryCollection = [ot.OrthogonalUniVariateFunctionFamily( ot.OrthogonalUniVariatePolynomialFunctionFactory(ot.StandardDistributionPolynomialFactory(self.distribution.getMarginal(i)))) for i in range(input_dimension)]
functionFactory = ot.OrthogonalProductFunctionFactory(factoryCollection)
algo = ot.TensorApproximationAlgorithm(X, y.reshape(-1, 1), self.distribution, functionFactory, [self.nk]*input_dimension, self.max_rank)
algo.run()
self._result = algo.getResult()
return self
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
polyColl = ot.PolynomialFamilyCollection(inputDimension)
# %%
# We could use the Krawtchouk and Charlier families (for discrete distributions).
# %%
polyColl[0] = ot.KrawtchoukFactory()
polyColl[1] = ot.CharlierFactory()
# %%
# We could also use the automatic selection of the polynomial which corresponds to the distribution: this is done with the `StandardDistributionPolynomialFactory` class.
# %%
for i in range(inputDimension):
marginal = distribution.getMarginal(i)
polyColl[i] = ot.StandardDistributionPolynomialFactory(marginal)
# %%
# In our specific case, we use specific polynomial factories.
# %%
polyColl[0] = ot.HermiteFactory()
polyColl[1] = ot.LegendreFactory()
polyColl[2] = ot.LaguerreFactory(2.75)
# Parameter for the Jacobi factory : 'Probabilty' encoded with 1
polyColl[3] = ot.JacobiFactory(2.5, 3.5, 1)
# %%
# Create the enumeration function.
# %%
ot.Uniform(),
ot.Gamma(2.75, 1.0),
ot.Beta(2.5, 1.0, -1.0, 2.0)
])
# %%
# Prepare the input/output samples
sampleSize = 250
X = distribution.getSample(sampleSize)
Y = myModel(X)
dimension = X.getDimension()
# %%
# build the orthogonal basis
coll = [
ot.StandardDistributionPolynomialFactory(distribution.getMarginal(i))
for i in range(dimension)
]
enumerateFunction = ot.LinearEnumerateFunction(dimension)
productBasis = ot.OrthogonalProductPolynomialFactory(coll, enumerateFunction)
# %%
# create the algorithm
degree = 6
adaptiveStrategy = ot.FixedStrategy(
productBasis, enumerateFunction.getStrataCumulatedCardinal(degree))
projectionStrategy = ot.LeastSquaresStrategy()
algo = ot.FunctionalChaosAlgorithm(X, Y, distribution, adaptiveStrategy,
projectionStrategy)
algo.run()
#['(2*_pi*Tu*(Hu-Hl))/(ln(r/rw)*(1+(2*L*Tu)/(ln(r/rw)*rw^2*Kw)+Tu/Tl))'])
# coll = [ot.Normal(0.1, 0.0161812),
# ot.LogNormal(7.71, 1.0056),
# ot.Uniform(63070.0, 115600.0),
# ot.Uniform(990.0, 1110.0),
# ot.Uniform(63.1, 116.0),
# ot.Uniform(700.0, 820.0),
# ot.Uniform(1120.0, 1680.0),
# ot.Uniform(9855.0, 12045.0)]
distribution = ot.ComposedDistribution(coll)
factoryCollection = list(
map(
lambda dist: ot.OrthogonalUniVariateFunctionFamily(
ot.OrthogonalUniVariatePolynomialFunctionFactory(
ot.StandardDistributionPolynomialFactory(dist))), coll))
functionFactory = ot.OrthogonalProductFunctionFactory(factoryCollection)
size = 1000
X = distribution.getSample(size)
Y = model(X)
# ot.ResourceMap.Set('TensorApproximationAlgorithm-Method', 'RankM')
# n-d
nk = [10] * dim
maxRank = 5
algo = ot.TensorApproximationAlgorithm(X, Y, distribution, functionFactory, nk,
maxRank)
algo.run()
