# Polynomial chaos basis¶

The current section is focused on a specific kind of functional chaos representation that has been implemented, namely polynomial chaos expansions.
Mathematical framework
Throughout this section, the model response is assumed to be a scalar random variable . However, the following derivations hold in case of a vector-valued response.
Let us suppose that:
• has a finite variance, i.e. ;

• has independent components.

Then it is shown that may be expanded onto the PC basis as follows:

(1)

where the ’s are multivariate polynomials that are orthonormal with respect to the joint PDF , that is:

where if and 0 otherwise, and the ’s are deterministic coefficients that fully characterize the response .

Building of the PC basis – independent random variables
We first consider the case of independent input random variables. In practice, the components of random vector are rescaled using a specific mapping , usually referred to as an isoprobabilistic transformation (see ). The set of scaled random variables reads:

(2)

Common choices for are standard distributions such as a standard normal distribution or a uniform distribution over . For simplicity, it is assumed from now on that the components of the original input random vector have been already scaled, i.e. .

Let us first notice that due to the independence of the input random variables, the input joint PDF may be cast as:

(3)

where is the marginal PDF of . Let us consider a family of orthonormal polynomials with respect to , :

(4)

The reader is referred to  for details on the selection of suitable families of orthogonal polynomials. It is assumed that the degree of is for and (). Upon tensorizing the resulting families of univariate polynomials, one gets a set of orthonormal multivariate polynomials defined by:

(5)

where the multi-index notation has been introduced.

Building of the PC basis – dependent random variables

In case of dependent variables, it is possible to build up an orthonormal basis as follows:

(6)

where is a function of the copula of . Note that such a basis is no longer polynomial. When dealing with independent random variables, one gets and each basis element may be recast as in (5). Determining is usually computationally expensive though, hence an alternative strategy for specific types of input random vectors.

If has an elliptical copula instead of an independent one, it may be recast as a random vector with independent components using a suitable mapping such as the Nataf transformation. The so-called Rosenblatt transformation may also be applied in case of a Gaussian copula . Thus the model response may be regarded as a function of and expanded onto a polynomial chaos expansion with basis elements cast as in (5).
Link with classical deterministic polynomial approximation

In a deterministic setting (i.e. when the input parameters are considered to be deterministic), it is of common practice to substitute the model function by a polynomial approximation over its whole domain of definition as shown in . Actually this approach is strictly equivalent to:

1. Regarding the input parameters as random uniform random variables

2. Expanding any quantity of interest provided by the model onto a PC expansion made of Legendre polynomials