Correlations#

Utilities for dealing with correlated inputs.

class equadratures.correlations.Correlations(correlation_matrix, poly=None, parameters=None, method=None, verbose=False)[source]#

The class defines methods for polynomial approximations with correlated inputs, including the Nataf transform and Gram-Schmidt process.

Parameters

correlation_matrix (numpy.ndarray) – The correlation matrix associated with the joint distribution.
poly (Poly, optional) – Polynomial defined with parameters with marginal distributions in uncorrelated space.
parameters (list, optional) – List of parameters with marginal distributions.
method (str, optional) – nataf-transform or gram-schmidt.
verbose (bool, optional) – Display Cholesky decomposition of the fictive matrix.

Example

>>> def func(x):
>>>     s1 = s[0]
>>>     s2 = s[1]
>>>     return s1**2 - s2**3 - 2. * s1 - 0.5 * s2
>>>
>>> parameters = [Parameter(distribution='beta', shape_parameter_A=5.0, shape_parameter_B=2.0, lower=0.0, upper=1.0, order=15) for _ in range(2)]
>>> basis = Basis('total-order')
>>> uncorr_poly = Poly(parameters, basis, method='least-squares',
>>>                     sampling_args={'mesh': 'monte-carlo',
>>>                                     'subsampling-algorithm': 'lu'})
>>>
>>> corr_mat = np.array([[1.0, -0.855],
>>>                     [-0.855, 1.0]])
>>>
>>> corr_gs = Correlations(corr_mat, poly=uncorr_poly, method='gram-schmidt')
>>> corr_gs.set_model(func)
>>> corrected_poly_gs = corr_gs.get_transformed_poly()
>>> print(corrected_poly_gs.get_mean_and_variance())

References

Melchers, R. E., (1945) Structural Reliability Analysis and Predictions. John Wiley and Sons, second edition.
Jakeman, J. D. et al., (2019) Polynomial chaos expansions for dependent random variables.

get_correlated_samples(N=None, X=None)[source]#

Method for generating correlated samples.

Parameters

N (int) – Number of correlated samples required.
X (numpy.ndarray, optional) – Points in the uncorrelated space to map to the correlated space.

Returns

Array with shape (N, M), which contains the correlated samples.

Return type

numpy.ndarray

get_pdf(X)[source]#

Evaluate PDF at the sample points.

Parameters: X (numpy.ndarray) – Array of sample points with shape (number_of_points, dimensions).
Returns: Array with shape (N,) containing evaluations of the PDF.
Return type: numpy.ndarray

get_points()[source]#

Returns the quadrature points accounting for correlations.

For Nataf transform, returns points in the correlated standard normal space. For Gram-Schmidt, returns points according to the GS polynomial basis.

Returns: Array of quadrature points with shape (number_of_samples, dimension).
Return type: numpy.ndarray

get_transformed_poly()[source]#

Returns the transformed polynomial.

Returns: The transformed polynomial.
Return type: Poly

set_model(model=None, model_grads=None)[source]#

Computes the coefficients of transformed polynomial (equivalent to calling self.get_transformed_poly().set_model(...))

Parameters

model (Callable,numpy.ndarray, optional) – 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.
model_grads (Callable,numpy.ndarray, optional) – 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.