def _calculate_subspace(self, S, f):
parameters = [Parameter(distribution='uniform', lower=np.min(S[:,i]), upper=np.max(S[:,i]), order=1) for i in range(0, self.n)]
self.poly = Poly(parameters, basis=Basis('total-order'), method='least-squares', \
sampling_args={'sample-points': S, 'sample-outputs': f})
self.poly.set_model()
self.Subs = Subspaces(full_space_poly=self.poly, method='active-subspace', subspace_dimension=self.d)
if self.subspace_method == 'variable-projection':
U0 = self.Subs.get_subspace()[:,:self.d]
self.Subs = Subspaces(method='variable-projection', sample_points=S, sample_outputs=f, \
subspace_init=U0, subspace_dimension=self.d, polynomial_degree=2, max_iter=300, tol=1.0e-8)
self.U = self.Subs.get_subspace()[:, :self.d]
elif self.subspace_method == 'active-subspaces':
U0 = self.Subs.get_subspace()[:,1].reshape(-1,1)
U1 = null_space(U0.T)
self.U = U0
for i in range(self.d-1):
R = []
for j in range(U1.shape[1]):
U = np.hstack((self.U, U1[:, j].reshape(-1,1)))
Y = np.dot(S, U)
myParameters = [Parameter(distribution='uniform', lower=np.min(Y[:,k]), upper=np.max(Y[:,k]), \
order=2) for k in range(Y.shape[1])]
myBasis = Basis('total-order')
poly = Poly(myParameters, myBasis, method='least-squares', \
sampling_args={'sample-points':Y, 'sample-outputs':f})
poly.set_model()
f_eval = poly.get_polyfit(Y)
_,_,r,_,_ = linregress(f_eval.flatten(),f.flatten())
R.append(r**2)
index = np.argmax(R)
self.U = np.hstack((self.U, U1[:, index].reshape(-1,1)))
U1 = np.delete(U1, index, 1)
def _fit_poly(X, y):
try:
N, d = X.shape
myParameters = []
for dimension in range(d):
values = X[:,dimension]
values_min = np.amin(values)
values_max = np.amax(values)
if (values_min - values_max) ** 2 < 0.01:
myParameters.append(Parameter(distribution='Uniform', lower=values_min-0.01, upper=values_max+0.01, order=self.order))
else:
myParameters.append(Parameter(distribution='Uniform', lower=values_min, upper=values_max, order=self.order))
if self.basis == "hyperbolic-basis":
myBasis = Basis(self.basis, orders=[self.order for _ in range(d)], q=0.5)
else:
myBasis = Basis(self.basis, orders=[self.order for _ in range(d)])
container["index_node_global"] += 1
poly = Poly(myParameters, myBasis, method=self.poly_method, sampling_args={'sample-points':X, 'sample-outputs':y}, solver_args=self.poly_solver_args)
poly.set_model()
mse = np.linalg.norm(y - poly.get_polyfit(X).reshape(-1)) ** 2 / N
except Exception as e:
print("Warning fitting of Poly failed:", e)
print(d, values_min, values_max)
mse, poly = np.inf, None
return mse, poly
def test_multi_variate_sampling(self):
"""
test if the method returns a function object for sampling interface
"""
dimension = 3
sampling_ratio = 3
parameters = [Parameter(1, "gaussian")] * dimension
basis = Basis("total order", [5] * dimension)
induced_sampling = InducedSampling(parameters, basis, sampling_ratio,
"qr")
# Mock univariate sampling
def func(_input):
if isinstance(_input, tuple) and len(_input) == 3:
parameter = _input[0]
cdf = _input[1]
order = _input[2]
assert parameter.__class__ == Parameter
assert cdf.shape == (1, )
assert cdf < 1 and cdf > 0
# Check integer property
assert order - int(order) < 0.0001
assert order <= 5 and order >= 0
assert type(np.asscalar(order)) == float
return 1
else:
return 0
induced_sampling.univariate_sampling = func
quadrature_points = induced_sampling.samples()
true_array = np.ones((dimension * sampling_ratio, dimension))
assert_array_equal(quadrature_points, true_array)
def get_subspace_polynomial(self):
"""
Returns a polynomial defined over the dimension reducing subspace.
:param Subspaces self:
An instance of the Subspaces object.
:return:
**subspacepoly**: A Poly object that defines a polynomial over the subspace. The distribution of parameters is
assumed to be uniform and the maximum and minimum bounds for each parameter are defined by the maximum and minimum values
of the project samples.
"""
active_subspace = self._subspace[:, 0:self.subspace_dimension]
projected_points = np.dot(self.sample_points, active_subspace)
myparameters = []
for i in range(0, self.subspace_dimension):
param = Parameter(distribution='uniform', lower=np.min(projected_points[:,i]), upper=np.max(projected_points[:,i]), \
order=self.polynomial_degree)
myparameters.append(param)
mybasis = Basis("total-order")
subspacepoly = Poly(myparameters, mybasis, method=self.poly_method, sampling_args={'sample-points':projected_points, \
'sample-outputs':self.sample_outputs},
solver_args=self.solver_args)
subspacepoly.set_model()
return subspacepoly
def convert2Full(poly_minimal):
"""
Converts a Poly_minimal object to a full Poly object, running through the constructor.
This gives some basic functionalities such as evaluating the polynomial fit and gradients.
Note: This method is not compliant with the EQ v8.0 philosophy, and is just temporary.
:param poly_minimal: A Poly_minimal object
:return: poly object
"""
basis = Basis(poly_minimal.basis_type, poly_minimal.basis_orders,
poly_minimal.basis_level, poly_minimal.basis_growth_rule,
poly_minimal.basis_q)
num_params = len(poly_minimal.orders)
parameters = []
for i in range(num_params):
parameters.append(
Parameter(poly_minimal.orders[i],
poly_minimal.distributions[i],
endpoints=poly_minimal.endpoints[i],
shape_parameter_A=poly_minimal.shape_parameter_As[i],
shape_parameter_B=poly_minimal.shape_parameter_Bs[i],
lower=poly_minimal.lowers[i],
upper=poly_minimal.uppers[i],
data=poly_minimal.datas[i]))
poly = Poly(parameters, basis)
if hasattr(poly_minimal, 'coefficients'):
poly.__setCoefficients__(poly_minimal.coefficients)
if hasattr(poly_minimal, 'quadraturePoints'):
poly.__setQuadrature__(poly_minimal.quadraturePoints,
poly_minimal.quadratureWeights)
return poly
def get_subspace_polynomial(self):
""" Returns a polynomial defined over the dimension reducing subspace.
Returns
-------
Poly
A Poly object that defines a polynomial over the subspace. The distribution of parameters
is assumed to be uniform and the maximum and minimum bounds for each parameter are defined by the maximum
and minimum values of the project samples.
"""
# TODO: Try correlated poly here
active_subspace = self._subspace[:, 0:self.subspace_dimension]
projected_points = np.dot(self.std_sample_points, active_subspace)
myparameters = []
for i in range(0, self.subspace_dimension):
param = Parameter(distribution='uniform',
lower=np.min(projected_points[:, i]),
upper=np.max(projected_points[:, i]),
order=self.polynomial_degree)
myparameters.append(param)
mybasis = Basis("total-order")
subspacepoly = Poly(myparameters,
mybasis,
method='least-squares',
sampling_args={
'sample-points': projected_points,
'sample-outputs': self.sample_outputs
})
subspacepoly.set_model()
return subspacepoly
def vandermonde(eta, p):
"""
Internal function to variable_projection
Calculates the Vandermonde matrix using polynomial basis functions
:param eta: ndarray, the affine transformed projected values of inputs in active subspace
:param p: int, the maximum degree of polynomials
:return:
* **V (numpy array)**: The resulting Vandermode matrix
* **Polybasis (Poly object)**: An instance of Poly object containing the polynomial basis derived
"""
_, n = eta.shape
listing = []
for i in range(0, n):
listing.append(p)
Object = Basis('Total order', listing)
#Establish n Parameter objects
params = []
P = Parameter(order=p, lower=-1, upper=1, distribution='uniform')
for i in range(0, n):
params.append(P)
#Use the params list to establish the Poly object
Polybasis = Poly(params, Object)
V = Polybasis.getPolynomial(eta)
V = V.T
return V, Polybasis
def test_additive_mixture_sampling(self):
"""
test if the method returns a function object for sampling interface
"""
dimension = 3
sampling_ratio = 3
parameters = [Parameter(1, "gaussian")] * dimension
basis = Basis("total order", [5] * dimension)
induced_sampling = InducedSampling(parameters, basis, sampling_ratio,
"qr")
# Mock multi_variate_sampling
def func(sampled_cdf_values, index_set_used):
return sampled_cdf_values, index_set_used
induced_sampling.multi_variate_sampling = func
_placeholder = np.ones((dimension, 1))
cdf, index = induced_sampling.additive_mixture_sampling(_placeholder)
assert type(cdf) == np.ndarray
assert cdf.shape == (dimension, 1)
assert cdf.dtype == 'float64'
assert np.amax(cdf) < 1
assert np.amin(cdf) > 0
assert type(index) == np.ndarray
assert index.shape == (3, )
index_int = index.astype(int)
assert np.all(np.isclose(index, index_int, 0.0001))
# Check Total order
assert np.sum(index) <= 5
assert np.amax(index) <= 5
assert np.amin(index) >= 0
def __init__(self, method, full_space_poly=None, sample_points=None, sample_outputs=None, polynomial_degree=2, subspace_dimension=2, bootstrap=False, subspace_init=None, max_iter=1000, tol=None, poly_method='least-squares',solver_args=None):
self.full_space_poly = full_space_poly
self.sample_points = sample_points
self.Y = None # for the zonotope vertices
if self.sample_points is not None:
self.sample_points = standardise(sample_points)
self.sample_outputs = sample_outputs
self.method = method
self.subspace_dimension = subspace_dimension
self.polynomial_degree = polynomial_degree
self.bootstrap = bootstrap
self.poly_method = poly_method
self.solver_args = solver_args
if self.method.lower() == 'active-subspace' or self.method.lower() == 'active-subspaces':
self.method = 'active-subspace'
if self.full_space_poly is None:
N, d = self.sample_points.shape
param = Parameter(distribution='uniform', lower=-1, upper=1., order=self.polynomial_degree)
myparameters = [param for _ in range(d)]
mybasis = Basis("total-order")
mypoly = Poly(myparameters, mybasis, method=self.poly_method, sampling_args={'sample-points':self.sample_points, \
'sample-outputs':self.sample_outputs},
solver_args=self.solver_args)
mypoly.set_model()
self.full_space_poly = mypoly
self.sample_points = standardise(self.full_space_poly.get_points())
self.sample_outputs = self.full_space_poly.get_model_evaluations()
self._get_active_subspace()
elif self.method == 'variable-projection':
self._get_variable_projection(None,None,tol,max_iter,subspace_init,False)
def __init__(self, training_input, training_output, num_ridges, max_iters=1, learning_rate = 0.001,
W=None, coeffs=None, momentum_rate = .001, opt = 'sd', poly_deg = 2, verbose = False):
self.training_input = training_input
self.training_output = training_output
self.verbose = verbose
# network architecture params
if isinstance(num_ridges, int):
self.num_ridges = [num_ridges]
else:
self.num_ridges = num_ridges
# num_ridges is the number of hidden units at each hidden layer. Does not count the input layer
self.num_layers = len(self.num_ridges)
self.dims = training_input.shape[1]
# initialize network data structures
max_layer_size = max(self.num_ridges)
self.poly_array = np.empty(( self.num_layers, max_layer_size), dtype=object)
#TODO: not hardcode poly type? Have different ridges at different nodes?
for k in range(self.num_layers):
for j in range(self.num_ridges[k]):
self.poly_array[k,j] = Poly(Parameter(poly_deg, distribution='uniform', lower=-3, upper=3), Basis("total order"))
self.poly_card = self.poly_array[0,0].basis.cardinality
layer_sizes = [self.dims] + self.num_ridges
if W is None:
self.W = [np.random.randn(layer_sizes[k+1], layer_sizes[k]) for k in range(self.num_layers)]
else:
self.W = W
if coeffs is None:
self.coeffs = [np.random.randn(self.num_ridges[k], self.poly_card) for k in range(self.num_layers)]
else:
self.coeffs = coeffs
self.update_coeffs()
# Note: We will keep data for every input point in one array.
n_points = self.training_input.shape[0]
self.delta = []
for k in range(self.num_layers):
self.delta.append(np.zeros((self.num_ridges[k],n_points)))
self.act_mat = [] # Lambda
for k in range(self.num_layers):
self.act_mat.append(np.zeros((self.num_ridges[k], n_points)))
self.Z = [] # node value before activation
for k in range(self.num_layers):
self.Z.append(np.zeros((self.num_ridges[k],n_points)))
self.Y = [] # After activation
for k in range(self.num_layers):
self.Y.append(np.zeros((self.num_ridges[k],n_points)))
self.phi = [] # basis fn evaluations
for k in range(self.num_layers):
self.phi.append(np.zeros((self.num_ridges[k],n_points)))
self.evaluate_fit(self.training_input,train=True)
# optimization params
self.max_iters = max_iters
self.opt = opt
self.learning_rate = learning_rate
self.momentum_rate = momentum_rate
def _build_model(self, S, f):
"""
Constructs quadratic model for ``trust-region`` or ``omorf`` methods
"""
if self.method == 'trust-region':
myParameters = [Parameter(distribution='uniform', lower=np.min(S[:,i]), \
upper=np.max(S[:,i]), order=2) for i in range(self.n)]
myBasis = Basis('total-order')
my_poly = Poly(myParameters, myBasis, method='least-squares', \
sampling_args={'sample-points':S, 'sample-outputs':f})
elif self.method == 'omorf':
Y = np.dot(S, self.U)
myParameters = [Parameter(distribution='uniform', lower=np.min(Y[:,i]), \
upper=np.max(Y[:,i]), order=2) for i in range(self.d)]
myBasis = Basis('total-order')
my_poly = Poly(myParameters, myBasis, method='least-squares', \
sampling_args={'sample-points':Y, 'sample-outputs':f})
my_poly.set_model()
return my_poly
def _get_quadrature_points_and_weights(self, order):
param = Parameter(distribution='uniform',
lower=self.lower,
upper=self.upper,
order=order)
basis = Basis('univariate')
poly = Poly(method='numerical-integration',
parameters=param,
basis=basis)
points, weights = poly.get_points_and_weights()
return points, weights * (self.upper - self.lower)
def test_induced_sampling(self):
"""
An integration test for the whole routine
"""
dimension = 3
parameters = [Parameter(3, "Uniform", upper=1, lower=-1)]*dimension
basis = Basis("total-order", [3]*dimension)
induced_sampling = Induced(parameters, basis)
quadrature_points = induced_sampling.get_points()
assert quadrature_points.shape == (induced_sampling.samples_number, 3)
def test_generate_sampling_class(self):
"""
test if the method returns a function object for sampling interface
"""
parameters = [Parameter(1, "gaussian")] * 3
basis = Basis("total order")
generator_class = Sampling(parameters, basis,
('induced-sampling', {
"sampling-ratio": 2,
"subsampling-optimisation": 'qr'
}))
assert generator_class.sampling_class.__class__ == InducedSampling
def _fit_poly(X, y):
N, d = X.shape
myParameters = []
for dimension in range(d):
values = [X[i,dimension] for i in range(N)]
values_min = min(values)
values_max = max(values)
if (values_min - values_max) ** 2 < 0.01:
myParameters.append(Parameter(distribution='Uniform', lower=values_min-0.01, upper=values_max+0.01, order=self.order))
else:
myParameters.append(Parameter(distribution='Uniform', lower=values_min, upper=values_max, order=self.order))
myBasis = Basis('total-order')
y = np.reshape(y, (y.shape[0], 1))
poly = Poly(myParameters, myBasis, method='least-squares', sampling_args={'sample-points':X, 'sample-outputs':y})
poly.set_model()
mse = ((y-poly.get_polyfit(X))**2).mean()
return mse, poly
def test_sampling(self):
d = 4
order = 5
param = Parameter(distribution='uniform',
order=order,
lower=-1.0, upper=1.0)
myparameters = [param for _ in range(d)]
mybasis = Basis('total-order')
mypoly = Poly(myparameters, mybasis,
method='least-squares',
sampling_args={'mesh': 'induced',
'subsampling-algorithm': 'qr',
'sampling-ratio': 1})
assert mypoly._quadrature_points.shape == (mypoly.basis.cardinality, d)
def vandermonde(eta, p):
_, n = eta.shape
listing = []
for i in range(0, n):
listing.append(p)
Object = Basis('total-order', listing)
#Establish n Parameter objects
params = []
P = Parameter(order=p, lower=-1, upper=1, distribution='uniform')
for i in range(0, n):
params.append(P)
#Use the params list to establish the Poly object
Polybasis = Poly(params, Object, method='least-squares')
V = Polybasis.get_poly(eta)
V = V.T
return V, Polybasis
def _build_model(self, S, f, del_k):
"""
Constructs quadratic model for ``trust-region`` method
"""
myParameters = [
Parameter(distribution='uniform',
lower=S[0, i] - del_k,
upper=S[0, i] + del_k,
order=2) for i in range(S.shape[1])
]
myBasis = Basis('total-order')
my_poly = Poly(myParameters,
myBasis,
method='compressive-sensing',
sampling_args={
'sample-points': S,
'sample-outputs': f
})
my_poly.set_model()
return my_poly
def test_samples(self):
"""
test if the method returns a function object for sampling interface
"""
dimension = 3
sampling_ratio = 3
parameters = [Parameter(1, "gaussian")] * dimension
basis = Basis("total order", [5] * dimension)
induced_sampling = InducedSampling(parameters, basis, sampling_ratio,
"qr")
# Mock additive mixture sampling
def func(array_):
return np.array([1] * dimension, float)
induced_sampling.additive_mixture_sampling = func
quadrature_points = induced_sampling.samples()
true_array = np.ones((dimension * sampling_ratio, dimension))
assert_array_equal(quadrature_points, true_array)
def test_induced_jacobi_evaluation(self):
dimension = 3
parameters = [Parameter(1, "Uniform", upper=1, lower=-1)] * dimension
basis = Basis("total-order")
induced_sampling = Induced(parameters, basis)
parameter = parameters[0]
parameter.order = 3
cdf_value = induced_sampling.induced_jacobi_evaluation(
0, 0, 0, parameter)
np.testing.assert_allclose(cdf_value, 0.5, atol=0.00001)
cdf_value = induced_sampling.induced_jacobi_evaluation(
0, 0, 1, parameter)
assert cdf_value == 1
cdf_value = induced_sampling.induced_jacobi_evaluation(
0, 0, -1, parameter)
assert cdf_value == 0
cdf_value = induced_sampling.induced_jacobi_evaluation(
0, 0, 0.6, parameter)
np.testing.assert_allclose(cdf_value, 0.7462, atol=0.00005)
cdf_value = induced_sampling.induced_jacobi_evaluation(
0, 0, 0.999, parameter)
np.testing.assert_allclose(cdf_value, 0.99652, atol=0.000005)
def __init__(self,
method,
full_space_poly=None,
sample_points=None,
sample_outputs=None,
subspace_dimension=2,
polynomial_degree=2,
param_args=None,
poly_args=None,
dr_args=None):
self.full_space_poly = full_space_poly
self.sample_points = sample_points
self.Y = None # for the zonotope vertices
self.sample_outputs = sample_outputs
self.method = method
self.subspace_dimension = subspace_dimension
self.polynomial_degree = polynomial_degree
my_poly_args = {'method': 'least-squares', 'solver_args': {}}
if poly_args is not None:
my_poly_args.update(poly_args)
self.poly_args = my_poly_args
my_param_args = {
'distribution': 'uniform',
'order': self.polynomial_degree,
'lower': -1,
'upper': 1
}
if param_args is not None:
my_param_args.update(param_args)
# I suppose we can detect if lower and upper is present to decide between these categories?
bounded_distrs = [
'analytical', 'beta', 'chebyshev', 'arcsine', 'truncated-gaussian',
'uniform'
]
unbounded_distrs = [
'gaussian', 'normal', 'gumbel', 'logistic', 'students-t',
'studentst'
]
semi_bounded_distrs = [
'chi', 'chi-squared', 'exponential', 'gamma', 'lognormal',
'log-normal', 'pareto', 'rayleigh', 'weibull'
]
if dr_args is not None:
if 'standardize' in dr_args:
dr_args['standardise'] = dr_args['standardize']
if self.method.lower() == 'active-subspace' or self.method.lower(
) == 'active-subspaces':
self.method = 'active-subspace'
if dr_args is not None:
self.standardise = getattr(dr_args, 'standardise', True)
else:
self.standardise = True
if self.full_space_poly is None:
# user provided input/output data
N, d = self.sample_points.shape
if self.standardise:
self.data_scaler = scaler_minmax()
self.data_scaler.fit(self.sample_points)
self.std_sample_points = self.data_scaler.transform(
self.sample_points)
else:
self.std_sample_points = self.sample_points.copy()
param = Parameter(**my_param_args)
if param_args is not None:
if (hasattr(dr_args, 'lower')
or hasattr(dr_args, 'upper')) and self.standardise:
warnings.warn(
'Points standardised but parameter range provided. Overriding default ([-1,1])...',
UserWarning)
myparameters = [param for _ in range(d)]
mybasis = Basis("total-order")
mypoly = Poly(myparameters,
mybasis,
sampling_args={
'sample-points': self.std_sample_points,
'sample-outputs': self.sample_outputs
},
**my_poly_args)
mypoly.set_model()
self.full_space_poly = mypoly
else:
# User provided polynomial
# Standardise according to distribution specified. Only care about the scaling (not shift)
# TODO: user provided callable with parameters?
user_params = self.full_space_poly.parameters
d = len(user_params)
self.sample_points = self.full_space_poly.get_points()
if self.standardise:
scale_factors = np.zeros(d)
centers = np.zeros(d)
for dd, p in enumerate(user_params):
if p.name.lower() in bounded_distrs:
scale_factors[dd] = (p.upper - p.lower) / 2.0
centers[dd] = (p.upper + p.lower) / 2.0
elif p.name.lower() in unbounded_distrs:
scale_factors[dd] = np.sqrt(p.variance)
centers[dd] = p.mean
else:
scale_factors[dd] = np.sqrt(p.variance)
centers[dd] = 0.0
self.param_scaler = scaler_custom(centers, scale_factors)
self.std_sample_points = self.param_scaler.transform(
self.sample_points)
else:
self.std_sample_points = self.sample_points.copy()
if not hasattr(self.full_space_poly, 'coefficients'):
raise ValueError('Please call set_model() first on poly.')
self.sample_outputs = self.full_space_poly.get_model_evaluations()
# TODO: use dr_args for resampling of gradient points
as_args = {'grad_points': None}
if dr_args is not None:
as_args.update(dr_args)
self._get_active_subspace(**as_args)
elif self.method == 'variable-projection':
self.data_scaler = scaler_minmax()
self.data_scaler.fit(self.sample_points)
self.std_sample_points = self.data_scaler.transform(
self.sample_points)
if dr_args is not None:
vp_args = {
'gamma': 0.1,
'beta': 1e-4,
'tol': 1e-7,
'maxiter': 1000,
'U0': None,
'verbose': False
}
vp_args.update(dr_args)
self._get_variable_projection(**vp_args)
else:
self._get_variable_projection()
class Nataf(object):
"""
The class defines a Nataf transformation.
References for theory:
Melchers, R., E. (Robert E.), 1945- Structural reliability analysis
and predictions - 2nd edition - John Wiley & Sons Ltd.
The input correlated marginals are mapped from their physical space to a new
standard normal space, in which points are uncorrelated.
Attributes of the class:
:param list D:
List of parameters (distributions), interpreted here as the marginals.
:param numpy-matrix R:
The correlation matrix associated with the joint distribution.
:param object std:
A standard normal distribution
:param numpy-matrix A:
The Cholesky decomposition of Fictive matrix R0,
associated with the set of normal intermediate
correlated distributions.
"""
def __init__(self, D=None, R=None):
if D is None:
raise(ValueError, 'Distributions must be given')
else:
self.D = D
if R is None:
raise(ValueError, 'Correlation matrix must be specified')
else:
self.R = R
self.std = Parameter(order=5, distribution='normal',shape_parameter_A = 0.0, shape_parameter_B = 1.0)
#
# R0 = fictive matrix of correlated normal intermediate variables
#
# 1) Check the type of correlated marginals
# 2) Use Effective Quadrature for solving Legendre
# 3) Calculate the fictive matrix
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 = Polyint([p1, p1], myBasis)
p = Pols.quadraturePoints
w = Pols.quadratureWeights * (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].getiCDF(self.std.getCDF(points=p1))) - self.D[i].mean ) / np.sqrt( self.D[i].variance )
tp22 = -(np.array(self.D[j].getiCDF(self.std.getCDF(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)
print('The Cholesky decomposition of fictive matrix R0 is:')
print(self.A)
print('The fictive matrix is:')
print(R0)
def C2U(self, X):
""" Method for mapping correlated variables to a new standard space.
The imput matrix must have [Nxm] dimension, where m is the number
of correlated marginals.
:param numpy-matrix X:
A N-by-M Matrix where input marginals are organized along columns
M represents the number of correlated marginals
:return:
A N-by-M Matrix which contains standardized uncorrelated data.
The transformation of each i-th input marginal is stored along
the i-th column of the output matrix.
"""
c = X[:,0]
w1 = np.zeros((len(c),len(self.D)))
for i in range(len(self.D)):
for j in range(len(c)):
w1[j,i] = self.D[i].getCDF(points=X[j,i])
if (w1[j,i] >= 1.0):
w1[j,i] = 1.0 - 10**(-10)
elif (w1[j,i] <= 0.0):
w1[j,i] = 0.0 + 10**(-10)
#-----------------------------------------------#
#plt.figure()
#plt.grid(linewidth=0.5, color='k')
#plt.plot(X[:,0], w1[:,0], 'ro', label='first')
#plt.plot(X[:,1], w1[:,1], 'bx', label='second')
#plt.title('from nataf class: w1 VS X input')
#plt.legend(loc='upper left')
#plt.show()
#-----------------------------------------------#
sU = np.zeros((len(c),len(self.D)))
for i in range(len(self.D)):
for j in range(len(c)):
sU[j,i] = self.std.getiCDF(w1[j,i])
sU = np.array(sU)
sU = sU.T
xu = np.linalg.solve(self.A,sU)
xu = np.array(xu)
xu = xu.T
return xu
def U2C(self, X):
""" Method for mapping uncorrelated variables from standard normal space
to a new physical space in which variables are correlated.
Input matrix must have [mxN] dimension, where m is the number of input marginals.
:param numpy-matrix X:
A Matrix of M-by-N dimensions, in which uncorrelated marginals
are organized along rows.
:return:
A N-by-M matrix in which the result of the inverse transformation
applied to the i-th marginal is stored along the i-th column
of the ouput matrix.
"""
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.getCDF(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].getiCDF(temporary)
t = temp[0]
Xc[j,i] = t
return Xc
def getUncorrelatedSamples(self, N=None):
""" Method for sampling uncorrelated data:
:param integer N:
represents the number of the samples inside a range
:return:
A N-by-M matrix, each i-th column contains the points
which belong to the i-th distribution stored into list D.
"""
if N is not None:
distro = list()
for i in range(len(self.D)):
distro1 = self.D[i].getSamples(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))
distro = distro.T
else:
raise(ValueError, 'One input must be given to "get Correlated Samples" method')
return distro
def getCorrelatedSamples(self, N=None):
""" Method for sampling correlated data:
:param integer N:
represents the number of the samples inside a range
points represents the array we want to correlate.
:return:
A N-by-M matrix in which correlated samples are organized
along columns: the result of the run of the present method
for the i-th marginal into the input matrix is stored
along the i-th column of the output matrix.
"""
if N is not None:
distro = list()
for i in range(len(self.D)):
distro1 = self.std.getSamples(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].getiCDF(self.std.getCDF(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 ')
def CorrelationMatrix(self, X):
""" The following calculations check the correlation
matrix of input arrays and determine the covariance
matrix: The input matrix mush have [Nxm] dimensions where
m is the number of the marginals.
:param X:
Matrix of correlated data
:param D:
diagonal matrix which cointains the variances
:param S:
covariance matrix
:return:
A correlation matrix R
"""
N = len(X)
D = np.zeros((len(self.D),len(self.D)))
for i in range(len(self.D)):
for j in range(len(self.D)):
if i==j:
D[i,j] = np.sqrt(self.D[i].variance)
else:
D[i,j] = 0
diff1 = np.zeros((N, len(self.D))) # (x_j - mu_j)
diff2 = np.zeros((N, len(self.D))) # (x_k - mu_k)
prod_n = np.zeros(N)
prod_square1 = np.zeros(N)
prod_square2 = np.zeros(N)
R = np.zeros((len(self.D),len(self.D)))
for j in range(len(self.D)):
for k in range(len(self.D)):
if j==k:
R[j,k] = 1.0
else:
for i in range(N):
diff1[i,j] = (X[i,j] - self.D[j].mean)
diff2[i,k] = (X[i,k] - self.D[k].mean)
prod_n[i] = 1.0*(diff1[i,j]*diff2[i,k])
prod_square1[i] = (diff1[i,j])**2
prod_square2[i] = (diff2[i,k])**2
den1 = np.sum(prod_square1)
den2 = np.sum(prod_square2)
den11 = np.sqrt(den1)
den22 = np.sqrt(den2)
R[j,k] = np.sum(prod_n)/(den11*den22)
return R
def __init__(self, D=None, R=None):
if D is None:
raise(ValueError, 'Distributions must be given')
else:
self.D = D
if R is None:
raise(ValueError, 'Correlation matrix must be specified')
else:
self.R = R
self.std = Parameter(order=5, distribution='normal',shape_parameter_A = 0.0, shape_parameter_B = 1.0)
#
# R0 = fictive matrix of correlated normal intermediate variables
#
# 1) Check the type of correlated marginals
# 2) Use Effective Quadrature for solving Legendre
# 3) Calculate the fictive matrix
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 = Polyint([p1, p1], myBasis)
p = Pols.quadraturePoints
w = Pols.quadratureWeights * (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].getiCDF(self.std.getCDF(points=p1))) - self.D[i].mean ) / np.sqrt( self.D[i].variance )
tp22 = -(np.array(self.D[j].getiCDF(self.std.getCDF(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)
print('The Cholesky decomposition of fictive matrix R0 is:')
print(self.A)
print('The fictive matrix is:')
print(R0)
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)
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
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.')