class Correlations(object):
"""
The class defines a Nataf transformation. The input correlated marginals are mapped from their physical space to a new
standard normal space, in which points are uncorrelated.
:param Poly poly: A polynomial object.
:param numpy.ndarray correlation_matrix: The correlation matrix associated with the joint distribution.
**References**
1. Melchers, R. E., (1945) Structural Reliability Analysis and Predictions. John Wiley and Sons, second edition.
"""
def __init__(self, poly, correlation_matrix, verbose=False):
self.poly = poly
D = self.poly.get_parameters()
self.D = D
self.R = correlation_matrix
self.std = Parameter(order=5,
distribution='normal',
shape_parameter_A=0.0,
shape_parameter_B=1.0)
inf_lim = -8.0
sup_lim = -inf_lim
p1 = Parameter(distribution='uniform',
lower=inf_lim,
upper=sup_lim,
order=31)
myBasis = Basis('tensor-grid')
Pols = Poly([p1, p1], myBasis, method='numerical-integration')
p = Pols.get_points()
w = Pols.get_weights() * (sup_lim - inf_lim)**2
p1 = p[:, 0]
p2 = p[:, 1]
R0 = np.eye((len(self.D)))
for i in range(len(self.D)):
for j in range(i + 1, len(self.D), 1):
if self.R[i, j] == 0:
R0[i, j] = 0.0
else:
tp11 = -(np.array(self.D[i].get_icdf(
self.std.get_cdf(points=p1))) -
self.D[i].mean) / np.sqrt(self.D[i].variance)
tp22 = -(np.array(self.D[j].get_icdf(
self.std.get_cdf(points=p2))) -
self.D[j].mean) / np.sqrt(self.D[j].variance)
rho_ij = self.R[i, j]
bivariateNormalPDF = (
1.0 / (2.0 * np.pi * np.sqrt(1.0 - rho_ij**2)) *
np.exp(-1.0 / (2.0 * (1.0 - rho_ij**2)) *
(p1**2 - 2.0 * rho_ij * p1 * p2 + p2**2)))
coefficientsIntegral = np.flipud(tp11 * tp22 * w)
def check_difference(rho_ij):
bivariateNormalPDF = (
1.0 / (2.0 * np.pi * np.sqrt(1.0 - rho_ij**2)) *
np.exp(-1.0 / (2.0 * (1.0 - rho_ij**2)) *
(p1**2 - 2.0 * rho_ij * p1 * p2 + p2**2)))
diff = np.dot(coefficientsIntegral, bivariateNormalPDF)
return diff - self.R[i, j]
if (self.D[i].name != 'custom') or (self.D[j].name !=
'custom'):
rho = optimize.newton(check_difference,
self.R[i, j],
maxiter=50)
else:
res = optimize.least_squares(check_difference,
R[i, j],
bounds=(-0.999, 0.999),
ftol=1.e-03)
rho = res.x
print('A Custom Marginal is present')
R0[i, j] = rho
R0[j, i] = R0[i, j]
self.A = np.linalg.cholesky(R0)
if verbose is True:
print('The Cholesky decomposition of fictive matrix R0 is:')
print(self.A)
print('The fictive matrix is:')
print(R0)
list_of_parameters = []
for i in range(0, len(self.D)):
standard_parameter = Parameter(order=self.D[i].order,
distribution='gaussian',
shape_parameter_A=0.,
shape_parameter_B=1.)
list_of_parameters.append(standard_parameter)
self.polystandard = deepcopy(self.poly)
self.polystandard._set_parameters(list_of_parameters)
self.standard_samples = self.polystandard.get_points()
self._points = self.get_correlated_from_uncorrelated(
self.standard_samples)
def get_points(self):
"""
Returns the correlated samples based on the quadrature rules used in poly.
:param Correlations self: An instance of the Correlations object.
:return:
**points**: A numpy.ndarray of sampled quadrature points with shape (number_of_samples, dimension).
"""
return self._points
def set_model(self, model=None, model_grads=None):
"""
Computes the coefficients of the polynomial.
:param Correlations self:
An instance of the Correlations 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.
"""
model_values = None
model_grads_values = None
if callable(model):
model_values = evaluate_model(self._points, model)
else:
model_values = model
if model_grads is not None:
if callable(model_grads):
model_grads_values = evaluate_model_gradients(
self._points, model_grads)
else:
model_grads_values = model_grads
self.polystandard.set_model(model_values, model_grads_values)
def get_transformed_poly(self):
"""
Returns the transformed polynomial.
:param Correlations self:
An instance of the Correlations class.
:return:
**poly**: An instance of the Poly class.
"""
return self.polystandard
def get_correlated_from_uncorrelated(self, X):
"""
Method for mapping uncorrelated variables from standard normal space to a new physical space in which variables are correlated.
:param Correlations self: An instance of the Correlations object.
:param numpy.ndarray X: Samples of uncorrelated points from the marginals; of shape (N,M)
:return:
**C**: A numpy.ndarray of shape (N, M), which contains the correlated samples.
"""
X = X.T
invA = np.linalg.inv(self.A)
Z = np.linalg.solve(invA, X)
Z = Z.T
xc = np.zeros((len(Z[:, 0]), len(self.D)))
for i in range(len(self.D)):
for j in range(len(Z[:, 0])):
xc[j, i] = self.std.get_cdf(points=Z[j, i])
Xc = np.zeros((len(Z[:, 0]), len(self.D)))
for i in range(len(self.D)):
for j in range(len(Z[:, 0])):
temporary = np.matrix(xc[j, i])
temp = self.D[i].get_icdf(temporary)
t = temp[0]
Xc[j, i] = t
return Xc
def get_correlated_samples(self, N=None):
"""
Method for generating correlated samples.
:param int N: Number of correlated samples required.
:param numpy.ndarray X: Samples of correlated points from the marginals; of shape (N,M)
:return:
**C**: A numpy.ndarray of shape (N, M), which contains the correlated samples.
"""
if N is not None:
distro = list()
for i in range(len(self.D)):
distro1 = self.std.get_samples(N)
# check dimensions ------------------#
distro1 = np.matrix(distro1)
dimension = np.shape(distro1)
if dimension[0] == N:
distro1 = distro1.T
#------------------------------------#
distro.append(distro1)
distro = np.reshape(distro, (len(self.D), N))
interm = np.dot(self.A, distro)
correlated = np.zeros((len(self.D), N))
for i in range(len(self.D)):
for j in range(N):
correlated[i, j] = self.D[i].get_icdf(
self.std.get_cdf(interm[i, j]))
correlated = correlated.T
return correlated
else:
raise (
ValueError,
'One input must be given to "get Correlated Samples" method: please choose between sampling N points or giving an array of uncorrelated data '
)
class Correlations(object):
"""
The class defines methods for polynomial approximations with correlated inputs, including the Nataf transform and Gram-Schmidt process.
:param numpy.ndarray correlation_matrix: The correlation matrix associated with the joint distribution.
:param Poly poly: Polynomial defined with parameters with marginal distributions in uncorrelated space.
:param list parameters: List of parameters with marginal distributions.
:param str method: `nataf-transform` or `gram-schmidt`.
:param bool verbose: Display Cholesky decomposition of the fictive matrix.
**References**
1. Melchers, R. E., (1945) Structural Reliability Analysis and Predictions. John Wiley and Sons, second edition.
2. Jakeman, J. D. et al., (2019) Polynomial chaos expansions for dependent random variables.
"""
def __init__(self,
correlation_matrix,
poly=None,
parameters=None,
method=None,
verbose=False):
if (poly is None) and (method is not None):
raise ValueError('Need to specify poly for probability transform.')
if poly is not None:
self.poly = poly
D = self.poly.get_parameters()
elif parameters is not None:
D = parameters
else:
raise ValueError('Need to specify either poly or parameters.')
self.D = D
self.R = correlation_matrix
self.std = Parameter(order=5,
distribution='normal',
shape_parameter_A=0.0,
shape_parameter_B=1.0)
inf_lim = -8.0
sup_lim = -inf_lim
p1 = Parameter(distribution='uniform',
lower=inf_lim,
upper=sup_lim,
order=31)
myBasis = Basis('tensor-grid')
self.Pols = Poly([p1, p1], myBasis, method='numerical-integration')
Pols = self.Pols
p = Pols.get_points()
# w = Pols.get_weights()
w = Pols.get_weights() * (sup_lim - inf_lim)**2
p1 = p[:, 0]
p2 = p[:, 1]
R0 = np.eye((len(self.D)))
for i in range(len(self.D)):
for j in range(i + 1, len(self.D), 1):
if self.R[i, j] == 0:
R0[i, j] = 0.0
else:
z1 = np.array(self.D[i].get_icdf(
self.std.get_cdf(points=p1)))
z2 = np.array(self.D[j].get_icdf(
self.std.get_cdf(points=p2)))
tp11 = (z1 - self.D[i].mean) / np.sqrt(self.D[i].variance)
tp22 = (z2 - self.D[j].mean) / np.sqrt(self.D[j].variance)
coefficientsIntegral = np.flipud(tp11 * tp22 * w)
def check_difference(rho_ij):
bivariateNormalPDF = (
1.0 / (2.0 * np.pi * np.sqrt(1.0 - rho_ij**2)) *
np.exp(-1.0 / (2.0 * (1.0 - rho_ij**2)) *
(p1**2 - 2.0 * rho_ij * p1 * p2 + p2**2)))
diff = np.dot(coefficientsIntegral, bivariateNormalPDF)
return diff - self.R[i, j]
# if (self.D[i].name!='custom') or (self.D[j].name!='custom'):
rho = optimize.newton(check_difference,
self.R[i, j],
maxiter=50)
# else:
# # ???
# res = optimize.least_squares(check_difference, self.R[i,j], bounds=(-0.999,0.999), ftol=1.e-03)
# rho = res.x
# print('A Custom Marginal is present')
R0[i, j] = rho
R0[j, i] = R0[i, j]
self.R0 = R0.copy()
self.A = np.linalg.cholesky(R0)
if verbose:
print('The Cholesky decomposition of fictive matrix R0 is:')
print(self.A)
print('The fictive matrix is:')
print(R0)
if method is None:
pass
elif method.lower() == 'nataf-transform':
list_of_parameters = []
for i in range(0, len(self.D)):
standard_parameter = Parameter(order=self.D[i].order,
distribution='gaussian',
shape_parameter_A=0.,
shape_parameter_B=1.)
list_of_parameters.append(standard_parameter)
# have option so that we don't need to obtain
self.corrected_poly = deepcopy(self.poly)
if hasattr(self.corrected_poly, '_quadrature_points'):
self.corrected_poly._set_parameters(list_of_parameters)
self.standard_samples = self.corrected_poly._quadrature_points
self._points = self.get_correlated_samples(
X=self.standard_samples)
# self.corrected_poly._quadrature_points = self._points.copy()
elif method.lower() == 'gram-schmidt':
basis_card = poly.basis.cardinality
oversampling = 10
N_Psi = oversampling * basis_card
S_samples = self.get_correlated_samples(N=N_Psi)
w_weights = 1.0 / N_Psi * np.ones(N_Psi)
Psi = poly.get_poly(S_samples).T
WPsi = np.diag(np.sqrt(w_weights)) @ Psi
self.WPsi = WPsi
R_Psi = np.linalg.qr(WPsi)[1]
self.R_Psi = R_Psi
self.R_Psi[0, :] *= np.sign(self.R_Psi[0, 0])
self.corrected_poly = deepcopy(poly)
self.corrected_poly.inv_R_Psi = np.linalg.inv(self.R_Psi)
self.corrected_poly.corr = self
self.corrected_poly._set_points_and_weights()
P = self.corrected_poly.get_poly(
self.corrected_poly._quadrature_points)
W = np.mat(
np.diag(np.sqrt(self.corrected_poly._quadrature_weights)))
A = W * P.T
self.corrected_poly.A = A
self.corrected_poly.P = P
if hasattr(self.corrected_poly, '_quadrature_points'):
# TODO: Correlated quadrature points?
self._points = self.corrected_poly._quadrature_points
else:
raise ValueError('Invalid method for correlations.')
def get_points(self):
"""
Returns the correlated samples based on the quadrature rules used in poly.
:param Correlations self: An instance of the Correlations object.
:return:
**points**: A numpy.ndarray of sampled quadrature points with shape (number_of_samples, dimension).
"""
return self._points
def set_model(self, model=None, model_grads=None):
"""
Computes the coefficients of the polynomial.
:param Correlations self:
An instance of the Correlations 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.
"""
# Need to account for the nataf transform here?
model_values = None
model_grads_values = None
if callable(model):
model_values = evaluate_model(self._points, model)
else:
model_values = model
if model_grads is not None:
if callable(model_grads):
model_grads_values = evaluate_model_gradients(
self._points, model_grads)
else:
model_grads_values = model_grads
self.corrected_poly.set_model(model_values, model_grads_values)
def get_transformed_poly(self):
"""
Returns the transformed polynomial.
:param Correlations self:
An instance of the Correlations class.
:return:
**poly**: An instance of the Poly class.
"""
return self.corrected_poly
def get_correlated_samples(self, N=None, X=None):
"""
Method for generating correlated samples.
:param int N: Number of correlated samples required.
:param ndarray X: (Optional) Points in the uncorrelated space to map to the correlated space.
:return:
**C**: A numpy.ndarray of shape (N, M), which contains the correlated samples.
"""
d = len(self.D)
if X is None:
if N is None:
raise ValueError(
'Need to specify number of points to generate.')
X = np.random.multivariate_normal(np.zeros(d), np.eye(d), size=N)
X_test = X @ self.A.T
U_test = np.zeros(X_test.shape)
for i in range(d):
U_test[:, i] = self.std.get_cdf(X_test[:, i])
Z_test = np.zeros(U_test.shape)
for i in range(d):
Z_test[:, i] = self.D[i].get_icdf(U_test[:, i])
return Z_test
def get_pdf(self, X):
"""
Evaluate PDF at the sample points.
:param numpy.ndarray X: Sample points (Number of points by dimensions)
:return:
**C**: A numpy.ndarray of shape (N,) with evaluations of the PDF.
"""
parameters = self.D
if len(X.shape) == 1:
X = X.reshape(-1, 1)
d = X.shape[1]
U = np.zeros(X.shape)
for i in range(d):
U[:, i] = norm.ppf(parameters[i].get_cdf(X[:, i]))
cop_num = multivariate_normal(mean=np.zeros(d), cov=self.R0).pdf(U)
cop_den = np.prod(np.array([norm.pdf(U[:, i]) for i in range(d)]),
axis=0)
marginal_prod = np.prod(np.array(
[parameters[i].get_pdf(X[:, i]) for i in range(d)]),
axis=0)
return cop_num / cop_den * marginal_prod