Parameter¶
Definition of a univariate parameter.
- class equadratures.parameter.Parameter(order=1, distribution='Uniform', endpoints=None, shape_parameter_A=None, shape_parameter_B=None, variable='parameter', lower=None, upper=None, weight_function=None)[source]¶
This class defines a univariate parameter.
- Parameters
lower (float, optional) – Lower bound for the parameter.
upper (float, optional) – Upper bound for the parameter.
order (int, optional) – Order of the parameter.
param_type (str, optional) – The type of distribution that characterizes the parameter (see [1, 2]). Options include chebyshev (arcsine), gaussian, truncated-gaussian, beta, cauchy, exponential, uniform, triangular, gamma, weibull, rayleigh, pareto, lognormal, students-t, logistic, gumbel, chi and chi-squared. If no string is provided, a
uniform
distribution is assumed. Data-driven and custom analytical parameters can also be constructed by setting this option todata
andanalytical
and providing a weight_function (see examples).shape_parameter_A (float, optional) – Most of the aforementioned distributions are characterized by two shape parameters. For instance, in the case of a
gaussian
(ortruncated-gaussian
), this represents the mean. In the case of a beta distribution this represents the alpha value. For auniform
distribution this input is not required.shape_parameter_B (float, optional) – This is the second shape parameter that characterizes the distribution selected. In the case of a
gaussian
ortruncated-gaussian
, this is the variance.data (numpy.ndarray, optional) – A data-set with shape (number_of_data_points, 2), where the first column comprises of parameter values, while the second column corresponds to the data observations. This input should only be used with the
Analytical
distribution.endpoints (str, optional) – If set to
both
, then the quadrature points and weights will have end-points, based on Gauss-Lobatto quadrature rules. If set toupper
orlower
a Gauss-Radau rule is used to compute one end-point at either the upper or lower bound.weight_function (Weight, optional) – An instance of Weight, which contains a bespoke analytical or data-driven weight (probability density) function.
Examples
- A uniform parameter
>>> param = eq.Parameter(distribution='uniform', lower=-2, upper=2., order=3)
- A beta parameter
>>> param = eq.Parameter(distribution='beta', lower=-2., upper=15., order=4, >>> shape_parameter_A=3.2, shape_parameter_B=1.7)
- A data-driven parameter
>>> pdf = eq.Weight( stats.gaussian_kde(data, bw_method='silverman'), >>> support=[-3, 3.2]) >>> param = eq.Parameter(distribution='analytical', >>> weight_function=pdf, order=2)
References
Xiu, D., Karniadakis, G. E., (2002) The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations. SIAM Journal on Scientific Computing, 24(2), Paper
Gautschi, W., (1985) Orthogonal Polynomials-Constructive Theory and Applications. Journal of Computational and Applied Mathematics 12 (1985), pp. 61-76. Paper
- get_cdf(points=None)[source]¶
Computes the cumulative density function associated with the Parameter.
- Parameters
points (numpy.ndarray, optional) – Values of the parameter at which the CDF must be evaluated.
- Returns
numpy.ndarray – If
points!=None
. ndarray containing the cumulative density function evaluated at the points inpoints
.tuple – If
points=None
. A tuple (x, cdf), where cdf is the cumulative density function evaluated at the points in x.
- get_description()[source]¶
Provides a description of the Parameter.
- Returns
A description of the parameter.
- Return type
- get_icdf(cdf_values)[source]¶
Computes the inverse cumulative density function associated with the Parameter.
- Parameters
cdf_values (numpy.ndarray) – Values of the cumulative density function for which its inverse needs to be computed.
- Returns
The inverse cumulative density function.
- Return type
- get_jacobi_eigenvectors(order=None)[source]¶
Computes the eigenvectors of the Jacobi matrix.
- Parameters
order (int) – Order of the recurrence coefficients.
- Returns
Array of eigenvectors.
- Return type
- get_jacobi_matrix(order=None, ab=None)[source]¶
Computes the Jacobi matrix—a tridiagonal matrix of the recurrence coefficients.
- Parameters
order (int) – Order of the recurrence coefficients.
- Returns
2D array containing the Jacobi matrix.
- Return type
- get_pdf(points=None)[source]¶
Computes the probability density function associated with the Parameter.
- Parameters
points (numpy.ndarray, optional) – Values of the parameter at which the PDF must be evaluated.
- Returns
numpy.ndarray – If
points!=None
. ndarray containing the probability density function evaluated at the points inpoints
.tuple – If
points=None
. A tuple (x, pdf), where pdf is the probability density function evaluated at the points in x.
- get_recurrence_coefficients(order=None)[source]¶
Generates the recurrence coefficients.
- Parameters
order (int, optional) – Order of the recurrence coefficients.
- Returns
Array of recurrence coefficients.
- Return type
- get_samples(number_of_samples_required)[source]¶
Generates samples from the distribution associated with the Parameter.
- Parameters
number_of_samples_required (int) – Number of samples that are required.
- Returns
The generated samples.
- Return type
- plot_orthogonal_polynomials(ax=None, order_limit=None, number_of_points=200, show=True)[source]¶
Plots the first few orthogonal polynomials. See
plot_orthogonal_polynomials()
for full description.
- plot_pdf(ax=None, data=None, show=True)[source]¶
Plots the probability density function for a Parameter. See
plot_pdf()
for full description.