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.
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
- 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
- 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
- get_transformed_poly()[source]¶
Returns the transformed polynomial.
- Returns
The transformed polynomial.
- Return type
- 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.