def _set_points_and_weights(self):
"""
Private function that sets the quadrature points.
:param Poly self:
An instance of the Poly object.
"""
self.quadrature = Quadrature(parameters=self.parameters, basis=self.basis, \
points=self.inputs, mesh=self.mesh)
quadrature_points, quadrature_weights = self.quadrature.get_points_and_weights(
)
if self.subsampling_algorithm_name is not None:
P = self.get_poly(quadrature_points)
W = np.mat(np.diag(np.sqrt(quadrature_weights)))
A = W * P.T
self.A = A
mm, nn = A.shape
m_refined = int(np.round(self.sampling_ratio * nn))
z = self.subsampling_algorithm_function(A, m_refined)
self._quadrature_points = quadrature_points[z, :]
self._quadrature_weights = quadrature_weights[z] / np.sum(
quadrature_weights[z])
else:
self._quadrature_points = quadrature_points
self._quadrature_weights = quadrature_weights
P = self.get_poly(quadrature_points)
W = np.mat(np.diag(np.sqrt(quadrature_weights)))
A = W * P.T
self.A = A
def _set_statistics(self):
"""
Private method that is used withn the statistics routines.
"""
if self.statistics_object is None:
if self.method != 'numerical-integration' and self.dimensions <= 6 and self.highest_order <= MAXIMUM_ORDER_FOR_STATS:
quad = Quadrature(parameters=self.parameters, basis=Basis('tensor-grid', orders= np.array(self.parameters_order) + 1), \
mesh='tensor-grid', points=None)
quad_pts, quad_wts = quad.get_points_and_weights()
poly_vandermonde_matrix = self.get_poly(quad_pts)
else:
poly_vandermonde_matrix = self.get_poly(self._quadrature_points)
quad_pts, quad_wts = self.get_points_and_weights()
if self.highest_order <= MAXIMUM_ORDER_FOR_STATS:
self.statistics_object = Statistics(self.parameters, self.basis, self.coefficients, quad_pts, \
quad_wts, poly_vandermonde_matrix, max_sobol_order=self.highest_order)
else:
self.statistics_object = Statistics(self.parameters, self.basis, self.coefficients, quad_pts, \
quad_wts, poly_vandermonde_matrix, max_sobol_order=MAXIMUM_ORDER_FOR_STATS)
def _set_statistics(self):
"""
Private method that is used within the statistics routines.
"""
if self.statistics_object is None:
if hasattr(self, 'inv_R_Psi'):
# quad_pts, quad_wts = self.quadrature.get_points_and_weights()
N_quad = 20000
quad_pts = self.corr.get_correlated_samples(N=N_quad)
quad_wts = 1.0 / N_quad * np.ones(N_quad)
poly_vandermonde_matrix = self.get_poly(quad_pts)
elif self.method != 'numerical-integration' and self.dimensions <= 6 and self.highest_order <= MAXIMUM_ORDER_FOR_STATS:
quad = Quadrature(parameters=self.parameters, basis=Basis('tensor-grid', orders= np.array(self.parameters_order) + 1), \
mesh='tensor-grid', points=None)
quad_pts, quad_wts = quad.get_points_and_weights()
poly_vandermonde_matrix = self.get_poly(quad_pts)
elif self.mesh == 'monte-carlo':
quad = Quadrature(parameters=self.parameters,
basis=self.basis,
mesh=self.mesh,
points=None,
oversampling=10.0)
quad_pts, quad_wts = quad.get_points_and_weights()
N_quad = len(quad_wts)
quad_wts = 1.0 / N_quad * np.ones(N_quad)
poly_vandermonde_matrix = self.get_poly(quad_pts)
else:
poly_vandermonde_matrix = self.get_poly(
self._quadrature_points)
quad_pts, quad_wts = self.get_points_and_weights()
if self.highest_order <= MAXIMUM_ORDER_FOR_STATS and (
self.basis.basis_type.lower() == 'total-order'
or self.basis.basis_type.lower() == 'hyperbolic-basis'):
self.statistics_object = Statistics(self.parameters, self.basis, self.coefficients, quad_pts, \
quad_wts, poly_vandermonde_matrix, max_sobol_order=self.highest_order)
else:
self.statistics_object = Statistics(self.parameters, self.basis, self.coefficients, quad_pts, \
quad_wts, poly_vandermonde_matrix, max_sobol_order=MAXIMUM_ORDER_FOR_STATS)
class Poly(object):
"""
Definition of a polynomial object.
:param list parameters: A list of parameters, where each element of the list is an instance of the Parameter class.
:param Basis basis: An instance of the Basis class corresponding to the multi-index set used.
:param str method: The method used for computing the coefficients. Should be one of: ``compressive-sensing``,
``numerical-integration``, ``least-squares``, ``least-squares-with-gradients``, ``minimum-norm``.
:param dict sampling_args: Optional arguments centered around the specific sampling strategy.
:string mesh: Avaliable options are: ``monte-carlo``, ``sparse-grid``, ``tensor-grid``, ``induced``, or ``user-defined``. Note that when the ``sparse-grid`` option is invoked, the sparse pseudospectral approximation method [1] is the adopted. One can think of this as being the correct way to use sparse grids in the context of polynomial chaos [2] techniques.
:string subsampling-algorithm: The ``subsampling-algorithm`` input refers to the optimisation technique for subsampling. In the aforementioned four sampling strategies, we generate a logarithm factor of samples above the required amount and prune down the samples using an optimisation technique (see [1]). Existing optimisation strategies include: ``qr``, ``lu``, ``svd``, ``newton``. These refer to QR with column pivoting [2], LU with row pivoting [3], singular value decomposition with subset selection [2] and a convex relaxation via Newton's method for determinant maximization [4]. Note that if the ``tensor-grid`` option is selected, then subsampling will depend on whether the Basis argument is a total order index set, hyperbolic basis or a tensor order index set.
:float sampling-ratio: Denotes the extent of undersampling or oversampling required. For values equal to unity (default), the number of rows and columns of the associated Vandermonde-type matrix are equal.
:numpy.ndarray sample-points: A numpy ndarray with shape (number_of_observations, dimensions) that corresponds to a set of sample points over the parameter space.
:numpy.ndarray sample-outputs: A numpy ndarray with shape (number_of_observations, 1) that corresponds to model evaluations at the sample points. Note that if ``sample-points`` is provided as an input, then the code expects ``sample-outputs`` too.
:numpy.ndarray sample-gradients: A numpy ndarray with shape (number_of_observations, dimensions) that corresponds to a set of sample gradient values over the parameter space.
:param dict solver_args: Optional arguments centered around the specific solver used for computing the coefficients.
:numpy.ndarray noise-level: The noise level to be used. Can take in both scalar- and vector-valued inputs.
:bool verbose: The default value is set to ``False``; when set to ``True`` details on the convergence of the solution will be provided. Note for direct methods, this will simply output the condition number of the matrix.
**Sample constructor initialisations**::
import numpy as np
from equadratures import *
# Subsampling from a tensor grid
param = Parameter(distribution='uniform', lower=-1., upper=1., order=3)
basis = Basis('total order')
poly = Poly(parameters=[param, param], basis=basis, method='least-squares' , sampling_args={'mesh':'tensor-grid', 'subsampling-algorithm':'svd', 'sampling-ratio':1.0})
# User-defined data with compressive sensing
X = np.loadtxt('inputs.txt')
y = np.loadtxt('outputs.txt')
param = Parameter(distribution='uniform', lower=-1., upper=1., order=3)
basis = Basis('total order')
poly = Poly([param, param], basis, method='compressive-sensing', sampling_args={'sample-points':X_red, \
'sample-outputs':Y_red})
# Using a sparse grid
param = Parameter(distribution='uniform', lower=-1., upper=1., order=3)
basis = Basis('sparse-grid', level=7, growth_rule='exponential')
poly = Poly(parameters=[param, param], basis=basis, method='numerical-integration')
**References**
1. Constantine, P. G., Eldred, M. S., Phipps, E. T., (2012) Sparse Pseudospectral Approximation Method. Computer Methods in Applied Mechanics and Engineering. 1-12. `Paper <https://www.sciencedirect.com/science/article/pii/S0045782512000953>`__
2. Xiu, D., Karniadakis, G. E., (2002) The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations. SIAM Journal on Scientific Computing, 24(2), `Paper <https://epubs.siam.org/doi/abs/10.1137/S1064827501387826?journalCode=sjoce3>`__
3. Seshadri, P., Iaccarino, G., Ghisu, T., (2018) Quadrature Strategies for Constructing Polynomial Approximations. Uncertainty Modeling for Engineering Applications. Springer, Cham, 2019. 1-25. `Preprint <https://arxiv.org/pdf/1805.07296.pdf>`__
4. Seshadri, P., Narayan, A., Sankaran M., (2017) Effectively Subsampled Quadratures for Least Squares Polynomial Approximations. SIAM/ASA Journal on Uncertainty Quantification, 5(1). `Paper <https://epubs.siam.org/doi/abs/10.1137/16M1057668>`__
5. Bos, L., De Marchi, S., Sommariva, A., Vianello, M., (2010) Computing Multivariate Fekete and Leja points by Numerical Linear Algebra. SIAM Journal on Numerical Analysis, 48(5). `Paper <https://epubs.siam.org/doi/abs/10.1137/090779024>`__
6. Joshi, S., Boyd, S., (2009) Sensor Selection via Convex Optimization. IEEE Transactions on Signal Processing, 57(2). `Paper <https://ieeexplore.ieee.org/document/4663892>`__
"""
def __init__(self,
parameters,
basis,
method=None,
sampling_args=None,
solver_args=None):
try:
len(parameters)
except TypeError:
parameters = [parameters]
self.parameters = parameters
self.basis = deepcopy(basis)
self.method = method
self.sampling_args = sampling_args
self.solver_args = solver_args
self.dimensions = len(parameters)
self.orders = []
self.gradient_flag = 0
for i in range(0, self.dimensions):
self.orders.append(self.parameters[i].order)
if not self.basis.orders:
self.basis.set_orders(self.orders)
# Initialize some default values!
self.inputs = None
self.outputs = None
self.subsampling_algorithm_name = None
self.sampling_ratio = 1.0
self.statistics_object = None
self.parameters_order = [
parameter.order for parameter in self.parameters
]
self.highest_order = np.max(self.parameters_order)
if self.method is not None:
if self.method == 'numerical-integration' or self.method == 'integration':
self.mesh = self.basis.basis_type
elif self.method == 'least-squares':
self.mesh = 'tensor-grid'
elif self.method == 'least-squares-with-gradients':
self.gradient_flag = 1
self.mesh = 'tensor-grid'
elif self.method == 'compressed-sensing' or self.method == 'compressive-sensing':
self.mesh = 'monte-carlo'
elif self.method == 'minimum-norm':
self.mesh = 'monte-carlo'
# Now depending on user inputs, override these default values!
sampling_args_flag = 0
if self.sampling_args is not None:
if 'mesh' in sampling_args:
self.mesh = sampling_args.get('mesh')
sampling_args_flag = 1
if 'sampling-ratio' in sampling_args:
self.sampling_ratio = float(
sampling_args.get('sampling-ratio'))
sampling_args_flag = 1
if 'subsampling-algorithm' in sampling_args:
self.subsampling_algorithm_name = sampling_args.get(
'subsampling-algorithm')
sampling_args_flag = 1
if 'sample-points' in sampling_args:
self.inputs = sampling_args.get('sample-points')
sampling_args_flag = 1
self.mesh = 'user-defined'
if 'sample-outputs' in sampling_args:
self.outputs = sampling_args.get('sample-outputs')
sampling_args_flag = 1
if 'sample-gradients' in sampling_args:
self.gradients = sampling_args.get('sample-gradients')
sampling_args_flag = 1
elif sampling_args_flag == 0:
raise (
ValueError,
'An input value that you have specified is likely incorrect. Sampling arguments include: mesh, sampling-ratio, subsampling-algorithm, sample-points, sample-outputs and sample-gradients.'
)
self._set_solver()
self._set_subsampling_algorithm()
self._set_points_and_weights()
else:
print('WARNING: Method not declared.')
def _set_parameters(self, parameters):
"""
Private function that sets the parameters. Required by the Correlated class.
:param Poly self:
An instance of the Poly object.
"""
self.parameters = parameters
self._set_points_and_weights()
def get_parameters(self):
"""
Returns the list of parameters
:param Poly self:
An instance of the Poly object.
"""
return self.parameters
def get_summary(self, filename=None, tosay=False):
"""
A simple utility that returns file summarising what the polynomial approximation has determined.
:param Poly self:
An instance of the Poly object.
:param str filename:
The filename to write to.
:param bool tosay:
True will replace "-" signs with "minus" when writing to file for compatibility with os.say().
"""
prec = '{:.3g}'
if self.dimensions == 1:
parameter_string = str('parameter.')
else:
parameter_string = str('parameters.')
introduction = str('Your problem has been defined by ' +
str(self.dimensions) + ' ' + parameter_string)
added = str('Their distributions are given as follows:')
for i in range(0, self.dimensions):
added_new = ('\nParameter ' + str(i + 1) + ' ' +
str(self.parameters[i].get_description()))
if i == 0:
added = introduction + added_new
else:
added = added + added_new
if self.statistics_object is not None:
mean_value, var_value = self.get_mean_and_variance()
X = self.get_points()
y_eval = self.get_polyfit(X)
y_valid = self._model_evaluations
a, b, r, _, _ = st.linregress(y_eval.flatten(), y_valid.flatten())
r2 = r**2
statistics = '\n \nA summary of computed output statistics is given below:\nThe mean is estimated to be '+ prec.format(mean_value) +\
' while the variance is ' + prec.format(var_value) +'.\nFor the data avaliable, the polynomial approximation had a r square value of '+prec.format(r2)+'.'
if self.dimensions > 1:
sobol_indices_array = np.argsort(
self.get_total_sobol_indices())
final_value = sobol_indices_array[-1] + 1
statistics_extra = str(
'\nAdditionally, the most important parameter--based on the total Sobol indices--was found to be parameter '
+ str(final_value) + '.')
statistics = statistics + statistics_extra
added = added + statistics
if (tosay is True):
added = added.replace('e-', 'e minus')
added = added.replace('minus0', 'minus')
if filename is None:
filename = 'effective-quadratures-output.txt'
output_file = open(filename, 'w')
output_file.write(added)
output_file.close()
def _set_subsampling_algorithm(self):
"""
Private function that sets the subsampling algorithm based on the user-defined method.
:param Poly self:
An instance of the Poly object.
"""
polysubsampling = Subsampling(self.subsampling_algorithm_name)
self.subsampling_algorithm_function = polysubsampling.get_subsampling_method(
)
def _set_solver(self):
"""
Private function that sets the solver depending on the user-defined method.
:param Poly self:
An instance of the Poly object.
"""
polysolver = Solver(self.method, self.solver_args)
self.solver = polysolver.get_solver()
def _set_points_and_weights(self):
"""
Private function that sets the quadrature points.
:param Poly self:
An instance of the Poly object.
"""
self.quadrature = Quadrature(parameters=self.parameters, basis=self.basis, \
points=self.inputs, mesh=self.mesh)
quadrature_points, quadrature_weights = self.quadrature.get_points_and_weights(
)
if self.subsampling_algorithm_name is not None:
P = self.get_poly(quadrature_points)
W = np.mat(np.diag(np.sqrt(quadrature_weights)))
A = W * P.T
self.A = A
mm, nn = A.shape
m_refined = int(np.round(self.sampling_ratio * nn))
z = self.subsampling_algorithm_function(A, m_refined)
self._quadrature_points = quadrature_points[z, :]
self._quadrature_weights = quadrature_weights[z] / np.sum(
quadrature_weights[z])
else:
self._quadrature_points = quadrature_points
self._quadrature_weights = quadrature_weights
P = self.get_poly(quadrature_points)
W = np.mat(np.diag(np.sqrt(quadrature_weights)))
A = W * P.T
self.A = A
def get_model_evaluations(self):
"""
Returns the points at which the model was evaluated at.
:param Poly self:
An instance of the Poly class.
"""
return self._model_evaluations
def get_mean_and_variance(self):
"""
Computes the mean and variance of the model.
:param Poly self:
An instance of the Poly class.
:return:
**mean**: The approximated mean of the polynomial fit; output as a float.
**variance**: The approximated variance of the polynomial fit; output as a float.
"""
self._set_statistics()
return self.statistics_object.get_mean(
), self.statistics_object.get_variance()
def get_skewness_and_kurtosis(self):
"""
Computes the skewness and kurtosis of the model.
:param Poly self:
An instance of the Poly class.
:return:
**skewness**: The approximated skewness of the polynomial fit; output as a float.
**kurtosis**: The approximated kurtosis of the polynomial fit; output as a float.
"""
self._set_statistics()
return self.statistics_object.get_skewness(
), self.statistics_object.get_kurtosis()
def _set_statistics(self):
"""
Private method that is used withn the statistics routines.
"""
if self.statistics_object is None:
if self.method != 'numerical-integration' and self.dimensions <= 6 and self.highest_order <= MAXIMUM_ORDER_FOR_STATS:
quad = Quadrature(parameters=self.parameters, basis=Basis('tensor-grid', orders= np.array(self.parameters_order) + 1), \
mesh='tensor-grid', points=None)
quad_pts, quad_wts = quad.get_points_and_weights()
poly_vandermonde_matrix = self.get_poly(quad_pts)
else:
poly_vandermonde_matrix = self.get_poly(
self._quadrature_points)
quad_pts, quad_wts = self.get_points_and_weights()
if self.highest_order <= MAXIMUM_ORDER_FOR_STATS:
self.statistics_object = Statistics(self.parameters, self.basis, self.coefficients, quad_pts, \
quad_wts, poly_vandermonde_matrix, max_sobol_order=self.highest_order)
else:
self.statistics_object = Statistics(self.parameters, self.basis, self.coefficients, quad_pts, \
quad_wts, poly_vandermonde_matrix, max_sobol_order=MAXIMUM_ORDER_FOR_STATS)
def get_sobol_indices(self, order):
"""
Computes the Sobol' indices.
:param Poly self:
An instance of the Poly class.
:param int highest_sobol_order_to_compute:
The order of the Sobol' indices required.
:return:
**sobol_indices**: A dict comprising of Sobol' indices and constitutent mixed orders of the parameters.
"""
self._set_statistics()
return self.statistics_object.get_sobol(order)
def get_total_sobol_indices(self):
"""
Computes the total Sobol' indices.
:param Poly self:
An instance of the Poly class.
:return:
**total_sobol_indices**: Sobol
"""
self._set_statistics()
return self.statistics_object.get_sobol_total()
def get_conditional_skewness_indices(self, order):
"""
Computes the skewness indices.
:param Poly self:
An instance of the Poly class.
:param int order:
The highest order of the skewness indices required.
:return:
**skewness_indices**: A dict comprising of skewness indices and constitutent mixed orders of the parameters.
"""
self._set_statistics()
return self.statistics_object.get_conditional_skewness(order)
def get_conditional_kurtosis_indices(self, order):
"""
Computes the kurtosis indices.
:param Poly self:
An instance of the Poly class.
:param int order:
The highest order of the kurtosis indices required.
:return:
**kurtosis_indices**: A dict comprising of kurtosis indices and constitutent mixed orders of the parameters.
"""
self._set_statistics()
return self.statistics_object.get_conditional_kurtosis(order)
def set_model(self, model=None, model_grads=None):
"""
Computes the coefficients of the polynomial via the method selected.
:param Poly self:
An instance of the Poly class.
:param callable model:
The function that needs to be approximated. In the absence of a callable function, the input can be the function evaluated at the quadrature points.
:param callable model_grads:
The gradient of the function that needs to be approximated. In the absence of a callable gradient function, the input can be a matrix of gradient evaluations at the quadrature points.
"""
if (model is None) and (self.outputs is not None):
self._model_evaluations = self.outputs
else:
if callable(model):
y = evaluate_model(self._quadrature_points, model)
else:
y = model
assert (y.shape[0] == self._quadrature_points.shape[0])
if y.shape[1] != 1:
raise (ValueError, 'model values should be a column vector.')
self._model_evaluations = y
if self.gradient_flag == 1:
if (model_grads is None) and (self.gradients is not None):
grad_values = self.gradients
else:
if callable(model_grads):
grad_values = evaluate_model_gradients(
self._quadrature_points, model_grads, 'matrix')
else:
grad_values = model_grads
p, q = grad_values.shape
self._gradient_evaluations = np.zeros((p * q, 1))
W = np.diag(np.sqrt(self._quadrature_weights))
counter = 0
for j in range(0, q):
for i in range(0, p):
self._gradient_evaluations[counter] = W[
i, i] * grad_values[i, j]
counter = counter + 1
del grad_values
self.statistics_object = None
self._set_coefficients()
def _set_coefficients(self, user_defined_coefficients=None):
"""
Computes the polynomial approximation coefficients.
:param Poly self:
An instance of the Poly object.
:param numpy.ndarray user_defined_coefficients:
A numpy.ndarray of shape (N, 1) where N corresponds to the N coefficients provided by the user
"""
# Check to ensure that if there any NaNs, a different basis must be used and solver must be changed
# to least squares!
if user_defined_coefficients is not None:
self.coefficients = user_defined_coefficients
return
indices_with_nans = np.argwhere(np.isnan(self._model_evaluations))[:,
0]
if len(indices_with_nans) is not 0:
print(
'WARNING: One or more of your model evaluations have resulted in an NaN. We found '
+ str(len(indices_with_nans)) + ' NaNs out of ' +
str(len(self._model_evaluations)) + '.')
print(
'The code will now use a least-squares technique that will ignore input-output pairs of your model that have NaNs. This will likely compromise computed statistics.'
)
self.inputs = np.delete(self._quadrature_points,
indices_with_nans,
axis=0)
self.outputs = np.delete(self._model_evaluations,
indices_with_nans,
axis=0)
self.subsampling_algorithm_name = None
number_of_basis_to_prune_down = self.basis.cardinality - len(
self.outputs)
if number_of_basis_to_prune_down > 0:
self.basis.prune(
number_of_basis_to_prune_down +
self.dimensions) # To make it an over-determined system!
self.method = 'least-squares'
self.mesh = 'user-defined'
self._set_solver()
self._set_points_and_weights()
self.set_model(self.outputs)
if self.mesh == 'sparse-grid':
counter = 0
multi_index = []
coefficients = np.empty([1])
multindices = np.empty([1, self.dimensions])
for tensor in self.quadrature.list:
P = self.get_poly(tensor.points, tensor.basis.elements)
W = np.diag(np.sqrt(tensor.weights))
A = np.dot(W, P.T)
_, _, counts = np.unique(np.vstack(
[tensor.points, self._quadrature_points]),
axis=0,
return_index=True,
return_counts=True)
indices = [i for i in range(0, len(counts)) if counts[i] == 2]
b = np.dot(W, self._model_evaluations[indices])
del counts, indices
coefficients_i = self.solver(
A, b) * self.quadrature.sparse_weights[counter]
multindices_i = tensor.basis.elements
coefficients = np.vstack([coefficients_i, coefficients])
multindices = np.vstack([multindices_i, multindices])
counter = counter + 1
multindices = np.delete(multindices, multindices.shape[0] - 1, 0)
coefficients = np.delete(coefficients, coefficients.shape[0] - 1)
unique_indices, indices, counts = np.unique(multindices,
axis=0,
return_index=True,
return_counts=True)
coefficients_final = np.zeros((unique_indices.shape[0], 1))
for i in range(0, unique_indices.shape[0]):
for j in range(0, multindices.shape[0]):
if np.array_equiv(unique_indices[i, :], multindices[j, :]):
coefficients_final[
i] = coefficients_final[i] + coefficients[j]
self.coefficients = coefficients_final
self.basis.elements = unique_indices
else:
P = self.get_poly(self._quadrature_points)
W = np.diag(np.sqrt(self._quadrature_weights))
A = np.dot(W, P.T)
b = np.dot(W, self._model_evaluations)
if self.gradient_flag == 1:
# Now, we can reduce the number of rows!
dP = self.get_poly_grad(self._quadrature_points)
C = cell2matrix(dP, W)
G = np.vstack([A, C])
r = np.linalg.matrix_rank(G)
m, n = A.shape
print(
'Gradient computation: The rank of the stacked matrix is '
+ str(r) + '.')
print('The number of unknown basis terms is ' + str(n))
if n > r:
print(
'WARNING: Please increase the number of samples; one way to do this would be to increase the sampling-ratio.'
)
self.coefficients = self.solver(A, b, C,
self._gradient_evaluations)
else:
self.coefficients = self.solver(A, b)
def get_multi_index(self):
"""
Returns the multi-index set of the basis.
:param Poly self:
An instance of the Poly object.
:return:
**multi_indices**: A numpy.ndarray of the coefficients with size (cardinality_of_basis, dimensions).
"""
return self.basis.elements
def get_coefficients(self):
"""
Returns the coefficients of the polynomial approximation.
:param Poly self:
An instance of the Poly object.
:return:
**coefficients**: A numpy.ndarray of the coefficients with size (number_of_coefficients, 1).
"""
return self.coefficients
def get_points(self):
"""
Returns the samples based on the sampling strategy.
:param Poly self:
An instance of the Poly object.
:return:
**points**: A numpy.ndarray of sampled quadrature points with shape (number_of_samples, dimension).
"""
return self._quadrature_points
def get_weights(self):
"""
Computes quadrature weights.
:param Poly self:
An instance of the Poly class.
:return:
**weights**: A numpy.ndarray of the corresponding quadrature weights with shape (number_of_samples, 1).
"""
return self._quadrature_weights
def get_points_and_weights(self):
"""
Returns the samples and weights based on the sampling strategy.
:param Poly self:
An instance of the Poly object.
:return:
**x**: A numpy.ndarray of sampled quadrature points with shape (number_of_samples, dimension).
**w**: A numpy.ndarray of the corresponding quadrature weights with shape (number_of_samples, 1).
"""
return self._quadrature_points, self._quadrature_weights
def get_polyfit(self, stack_of_points):
"""
Evaluates the the polynomial approximation of a function (or model data) at prescribed points.
:param Poly self:
An instance of the Poly class.
:param numpy.ndarray stack_of_points:
An ndarray with shape (number_of_observations, dimensions) at which the polynomial fit must be evaluated at.
:return:
**p**: A numpy.ndarray of shape (1, number_of_observations) corresponding to the polynomial approximation of the model.
"""
N = len(self.coefficients)
return np.dot(
self.get_poly(stack_of_points).T, self.coefficients.reshape(N, 1))
def get_polyfit_grad(self, stack_of_points, dim_index=None):
"""
Evaluates the gradient of the polynomial approximation of a function (or model data) at prescribed points.
:param Poly self:
An instance of the Poly class.
:param numpy.ndarray stack_of_points:
An ndarray with shape (number_of_observations, dimensions) at which the polynomial fit approximation's
gradient must be evaluated at.
:return:
**p**: A numpy.ndarray of shape (dimensions, number_of_observations) corresponding to the polynomial gradient approximation of the model.
"""
N = len(self.coefficients)
if stack_of_points.ndim == 1:
no_of_points = 1
else:
no_of_points, _ = stack_of_points.shape
H = self.get_poly_grad(stack_of_points, dim_index=dim_index)
grads = np.zeros((self.dimensions, no_of_points))
if self.dimensions == 1:
return np.dot(self.coefficients.reshape(N, ), H)
for i in range(0, self.dimensions):
grads[i, :] = np.dot(self.coefficients.reshape(N, ), H[i])
return grads
def get_polyfit_hess(self, stack_of_points):
"""
Evaluates the hessian of the polynomial approximation of a function (or model data) at prescribed points.
:param Poly self:
An instance of the Poly class.
:param numpy.ndarray stack_of_points:
An ndarray with shape (number_of_observations, dimensions) at which the polynomial fit approximation's
Hessian must be evaluated at.
:return:
**h**: A numpy.ndarray of shape (dimensions, dimensions, number_of_observations) corresponding to the polynomial Hessian approximation of the model.
"""
if stack_of_points.ndim == 1:
no_of_points = 1
else:
no_of_points, _ = stack_of_points.shape
H = self.get_poly_hess(stack_of_points)
if self.dimensions == 1:
return np.dot(self.coefficients.T, H)
hess = np.zeros((self.dimensions, self.dimensions, no_of_points))
for i in range(0, self.dimensions):
for j in range(0, self.dimensions):
hess[i, j, :] = np.dot(self.coefficients.T,
H[i * self.dimensions + j])
return hess
def get_polyfit_function(self):
"""
Returns a callable polynomial approximation of a function (or model data).
:param Poly self:
An instance of the Poly class.
:return:
A callable function.
"""
N = len(self.coefficients)
return lambda x: np.dot(
self.get_poly(x).T, self.coefficients.reshape(N, 1))
def get_polyfit_grad_function(self):
"""
Returns a callable for the gradients of the polynomial approximation of a function (or model data).
:param Poly self:
An instance of the Poly class.
:return:
A callable function.
"""
return lambda x: self.get_polyfit_grad(x)
def get_polyfit_hess_function(self):
"""
Returns a callable for the hessian of the polynomial approximation of a function (or model data).
:param Poly self:
An instance of the Poly class.
:return:
A callable function.
"""
return lambda x: self.get_polyfit_hess(x)
def get_poly(self, stack_of_points, custom_multi_index=None):
"""
Evaluates the value of each polynomial basis function at a set of points.
:param Poly self:
An instance of the Poly class.
:param numpy.ndarray stack_of_points:
An ndarray with shape (number of observations, dimensions) at which the polynomial must be evaluated.
:return:
**polynomial**: A numpy.ndarray of shape (cardinality, number_of_observations) corresponding to the polynomial basis function evaluations
at the stack_of_points.
"""
if custom_multi_index is None:
basis = self.basis.elements
else:
basis = custom_multi_index
basis_entries, dimensions = basis.shape
if stack_of_points.ndim == 1:
no_of_points = 1
else:
no_of_points, _ = stack_of_points.shape
p = {}
# Save time by returning if univariate!
if dimensions == 1:
poly, _, _ = self.parameters[0]._get_orthogonal_polynomial(
stack_of_points, int(np.max(basis)))
return poly
else:
for i in range(0, dimensions):
if len(stack_of_points.shape) == 1:
stack_of_points = np.array([stack_of_points])
p[i], _, _ = self.parameters[i]._get_orthogonal_polynomial(
stack_of_points[:, i], int(np.max(basis[:, i])))
# One loop for polynomials
polynomial = np.ones((basis_entries, no_of_points))
for k in range(dimensions):
basis_entries_this_dim = basis[:, k].astype(int)
polynomial *= p[k][basis_entries_this_dim]
return polynomial
def get_poly_grad(self, stack_of_points, dim_index=None):
"""
Evaluates the gradient for each of the polynomial basis functions at a set of points,
with respect to each input variable.
:param Poly self:
An instance of the Poly class.
:param numpy.ndarray stack_of_points:
An ndarray with shape (number_of_observations, dimensions) at which the gradient must be evaluated.
:return:
**Gradients**: A list with d elements, where d corresponds to the dimension of the problem. Each element is a numpy.ndarray of shape
(cardinality, number_of_observations) corresponding to the gradient polynomial evaluations at the stack_of_points.
"""
# "Unpack" parameters from "self"
basis = self.basis.elements
basis_entries, dimensions = basis.shape
if len(stack_of_points.shape) == 1:
if dimensions == 1:
# a 1d array of inputs, and each input is 1d
stack_of_points = np.reshape(stack_of_points,
(len(stack_of_points), 1))
else:
# a 1d array representing 1 point, in multiple dimensions!
stack_of_points = np.array([stack_of_points])
no_of_points, _ = stack_of_points.shape
p = {}
dp = {}
# Save time by returning if univariate!
if dimensions == 1:
_, dpoly, _ = self.parameters[0]._get_orthogonal_polynomial(
stack_of_points, int(np.max(basis)))
return dpoly
else:
for i in range(0, dimensions):
if len(stack_of_points.shape) == 1:
stack_of_points = np.array([stack_of_points])
p[i], dp[i], _ = self.parameters[i]._get_orthogonal_polynomial(
stack_of_points[:, i], int(np.max(basis[:, i])))
# One loop for polynomials
R = []
if dim_index is None:
dim_index = range(dimensions)
for v in range(dimensions):
if not (v in dim_index):
R.append(np.zeros((basis_entries, no_of_points)))
else:
polynomialgradient = np.ones((basis_entries, no_of_points))
for k in range(dimensions):
basis_entries_this_dim = basis[:, k].astype(int)
if k == v:
polynomialgradient *= dp[k][basis_entries_this_dim]
else:
polynomialgradient *= p[k][basis_entries_this_dim]
R.append(polynomialgradient)
return R
def get_poly_hess(self, stack_of_points):
"""
Evaluates the Hessian for each of the polynomial basis functions at a set of points,
with respect to each input variable.
:param Poly self:
An instance of the Poly class.
:param numpy.ndarray stack_of_points:
An ndarray with shape (number_of_observations, dimensions) at which the Hessian must be evaluated.
:return:
**Hessian**: A list with d^2 elements, where d corresponds to the dimension of the model. Each element is a numpy.ndarray of shape
(cardinality, number_of_observations) corresponding to the hessian polynomial evaluations at the stack_of_points.
"""
# "Unpack" parameters from "self"
basis = self.basis.elements
basis_entries, dimensions = basis.shape
if stack_of_points.ndim == 1:
no_of_points = 1
else:
no_of_points, _ = stack_of_points.shape
p = {}
dp = {}
d2p = {}
# Save time by returning if univariate!
if dimensions == 1:
_, _, d2poly = self.parameters[0]._get_orthogonal_polynomial(
stack_of_points, int(np.max(basis)))
return d2poly
else:
for i in range(0, dimensions):
if len(stack_of_points.shape) == 1:
stack_of_points = np.array([stack_of_points])
p[i], dp[i], d2p[i] = self.parameters[
i]._get_orthogonal_polynomial(stack_of_points[:, i],
int(np.max(basis[:, i]) + 1))
H = []
for w in range(0, dimensions):
gradDirection1 = w
for v in range(0, dimensions):
gradDirection2 = v
polynomialhessian = np.zeros((basis_entries, no_of_points))
for i in range(0, basis_entries):
temp = np.ones((1, no_of_points))
for k in range(0, dimensions):
if k == gradDirection1 == gradDirection2:
polynomialhessian[i, :] = d2p[k][int(
basis[i, k])] * temp
elif k == gradDirection1:
polynomialhessian[i, :] = dp[k][int(
basis[i, k])] * temp
elif k == gradDirection2:
polynomialhessian[i, :] = dp[k][int(
basis[i, k])] * temp
else:
polynomialhessian[i, :] = p[k][int(
basis[i, k])] * temp
temp = polynomialhessian[i, :]
H.append(polynomialhessian)
return H
