Polynomial chaos basisΒΆ
This page introduces polynomial chaos expansion. We consider the notations introduced in Functional Chaos Expansion.
Let be the isoprobabilistic
transformation such that
has independent
marginals.
Therefore, the probability density function of the standard random vector
is the product of the marginal probability density functions:
for any where
is the joint PDF of the random vector
and
is the marginal PDF of the random variable
.
For any , let
be the family of univariate
polynomials of the
-th marginal orthogonal with respect to the
marginal distribution
, where
represents the
polynomial degree.
The multivariate polynomial basis can be built using the tensor product of
univariate polynomials which are orthonormal with respect to
.
The orthonormal polynomials are:
where is the multi-index
representing the marginal polynomial degrees and
is the
-th marginal univariate orthonormal polynomial of degree
.
The orthonormal polynomial basis with respect to the marginal
is
known for some distributions: see Orthogonal polynomials for more
details on classical orthonormal polynomial families.
For classical polynomials, the StandardDistributionPolynomialFactory
class implements the three-term recurrence.
If the family is not already known, the polynomials can be represented by
their three-term recurrence, using the adaptive Stieljes algorithm (see
AdaptiveStieltjesAlgorithm
).
Once the sequence of recurrence coefficients is known, the reverse Clenshaw
algorithm enables fast, stable evaluation of the polynomials
at any point (see OrthogonalUniVariatePolynomial
).