.. _chaos_basis:

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 :math:`Y = h(\underline{X})`. However, the
  following derivations hold in case of a vector-valued response.
| Let us suppose that:

-  :math:`Y = h(\underline{X})` has a finite variance, i.e.
   :math:`\Var{h(\underline{X})} < \infty`;

-  :math:`\underline{X}` has independent components.

| Then it is shown that :math:`\underline{Y}` may be expanded onto the
  PC basis as follows:

  .. math::
    :label: PC

      Y \, \,  \equiv \, \,  h(\underline{X}) \, \, = \, \, \sum_{j=0}^{\infty} \; a_{j} \; \psi_{j}(\underline{X})

  where the :math:`\psi_{j}`\ ’s are multivariate polynomials that are
  orthonormal with respect to the joint PDF
  :math:`f_{\underline{X}}(\underline{x})`, that is:

  .. math:: \langle \psi_{j}(\underline{X}) \; , \; \psi_{k}(\underline{X}) \rangle \, \, \equiv \, \, \Expect{\psi_{j}(\underline{X}) \; \psi_{k}(\underline{X})} \, \, = \, \, \delta_{j,k}

  where :math:`\delta_{j,k} = 1` if :math:`j=k` and 0 otherwise, and
  the :math:`a_{j}`\ ’s are deterministic coefficients that fully
  characterize the response :math:`\underline{Y}`.

| **Building of the PC basis – independent random variables**

| We first consider the case of *independent* input random variables. In
  practice, the components :math:`X_i` of random vector
  :math:`\underline{X}` are rescaled using a specific mapping
  :math:`T_i`, usually referred to as an *isoprobabilistic
  transformation* (see ). The set of scaled random variables reads:

  .. math::
    :label: PC_isotransfo

      U_i \, \, = \, \, T_i(X_i) \quad \quad , \quad \, i=1,\dots,n

  Common choices for :math:`U_i` are standard distributions such as a
  standard normal distribution or a uniform distribution over
  :math:`[-1,1]`. For simplicity, it is assumed from now on that the
  components of the original input random vector :math:`\underline{X}`
  have been already scaled, i.e. :math:`X_i = U_i`.

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

  .. math::
    :label: 3.010qua

      f_{\vect{X}}(\vect{x}) \, = \, \prod_{i=1}^{n} f_{X_i}(x_{i})

  where :math:`f_{X_i}(x_{i})` is the marginal PDF of :math:`X_i`. Let
  us consider a family :math:`\{\pi^{(i)}_{j}, j \in \Nset\}` of
  orthonormal polynomials with respect to :math:`f_{X_i}`, :

  .. math::
    :label: 3.010cinq

      \langle \pi^{(i)}_{j}(X_{i}) \; , \; \pi^{(i)}_{k}(X_{i}) \rangle  \, \, \equiv \, \, \Expect{\pi^{(i)}_{j}(X_{i}) \;  \pi^{(i)}_{k}(X_{i})} \, \, = \, \, \delta_{j,k}

  The reader is referred to  for details on the selection of suitable
  families of orthogonal polynomials. It is assumed that the degree of
  :math:`\pi^{(i)}_{j}` is :math:`j` for :math:`j>0` and
  :math:`\pi^{(i)}_{0} \equiv 1` (:math:`i=1,\dots,n`). Upon tensorizing
  the :math:`n` resulting families of univariate polynomials, one gets a
  set of orthonormal multivariate polynomials
  :math:`\{\psi_{\idx}, \idx \in \Nset^n\}` defined by:

  .. math::
    :label: 3.010six

      \psi_{\idx}(\vect{x}) \, \, \equiv \,\, \pi^{(1)}_{\alpha_{1}}(x_{1}) \times \cdots \times \pi^{(n)}_{\alpha_{n}}(x_{n})

  where the multi-index notation
  :math:`\idx \equiv \{\alpha_{1},\dots,\alpha_{M}\}` 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:

  .. math::
    :label: 3.010seven

      \psi_{\idx}(\vect{x}) \, \, = \,\,  K(\underline{x}) \;\prod_{i=1}^M \pi^{(i)}_{\alpha_{i}}(x_{i})

  where :math:`K(\underline{x})` is a function of the copula of
  :math:`\vect{X}`. Note that such a basis is no longer polynomial. When
  dealing with independent random variables, one gets
  :math:`K(\underline{x})=1` and each basis element may be recast as in
  :eq:`3.010six`. Determining :math:`K(\underline{x})` is usually
  computionally expensive though, hence an alternative strategy for
  specific types of input random vectors.

| If :math:`\vect{X}` has an elliptical copula instead of an independent
  one, it may be recast as a random vector :math:`\vect{U}` with
  independent components using a suitable mapping
  :math:`T : \vect{X} \mapsto \vect{U}` such as the Nataf transformation.
  The so-called Rosenblatt transformation may also be applied in case
  of a Gaussian copula . Thus the model response :math:`Y` may be
  regarded as a function of :math:`\vect{U}` and expanded onto a
  *polynomial* chaos expansion with basis elements cast as in :eq:`3.010six`.

| **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 :math:`h` by a polynomial approximation over its
whole domain of definition as shown in . Actually this approach is
strictly equivalent to:

#. Regarding the input parameters as random uniform random variables

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


.. topic:: API:

    - See the available :ref:`orthogonal basis <orthogonal_basis>`.


.. topic:: Examples:

    - See :doc:`/examples/meta_modeling/functional_chaos`


.. topic:: References:

    - R. Ghanem and P. Spanos, 1991, "Stochastic finite elements -- A spectral approach", Springer Verlag. (Reedited by Dover Publications, 2003).

