chaospy.generate_expansion

chaospy.generate_expansion(order, dist, rule='three_terms_recurrence', normed=False, graded=True, reverse=True, cross_truncation=1.0, **kws)

Create orthogonal polynomial expansion.

This function is a frontend wrapper for the three methods for creating orthogonal polynomials:

Algorithm	Description
three_terms_recurrence	Three terms recurrence coefficients generated using Stieltjes and Golub-Welsch method. The most stable of the methods, but do not work on dependent distributions.
gram_schmidt	Gram-Schmidt orthogonalization method applied on polynomial expansions. Know for being numerically unstable.
cholesky	Orthogonalization through decorrelation of the covariance matrix. Uses Gill-King’s Cholesky decomposition method for higher numerical stability. Still not scalable to high number of dimensions.

Args:

order (int):

Order of polynomial expansion.

dist (Distribution):

Distribution space where polynomials are orthogonal. If the method dist._ttr exists, it will be used.

rule (str):

The orthogonalization method used.

normed (bool):

If True orthonormal polynomials will be used.

graded (bool):

Graded sorting, meaning the indices are always sorted by the index sum. E.g. q0**2*q1**2*q2**2 has an exponent sum of 6, and will therefore be consider larger than both q0**2*q1*q2, q0*q1**2*q2 and q0*q1*q2**2, which all have exponent sum of 5.

reverse (bool):

Reverse lexicographical sorting meaning that q0*q1**3 is considered bigger than q0**3*q1, instead of the opposite.

retall (bool):

If true return numerical stabilized norms as well. Roughly the same as cp.E(orth**2, dist).

cross_truncation (float):

Use hyperbolic cross truncation scheme to reduce the number of terms in expansion. only include terms where the exponents K satisfied the equation order >= sum(K**(1/cross_truncation))**cross_truncation.

Returns:

(numpoly.ndpoly, numpy.ndarray):

Orthogonal polynomial expansion. norms of the orthogonal expansion on the form E(orth**2, dist). Calculated using recurrence coefficients for stability.

Examples:

>>> distribution = chaospy.Normal()
>>> expansion, norms = generate_expansion(
...     3, distribution, retall=True)
>>> expansion
polynomial([1.0, q0, q0**2-1.0, q0**3-3.0*q0])
>>> norms
array([1., 1., 2., 6.])