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.. _pce_cross_validation:
Cross validation of PCE models
------------------------------
Introduction
~~~~~~~~~~~~
The cross-validation of a polynomial chaos expansion
uses the theory presented in :ref:`cross_validation`.
In [blatman2009]_ page 84, the author applies the LOO equation to polynomial
chaos expansion
(see appendix D page 203 for a proof).
If the coefficients are estimated from integration, the same derivation cannot,
in theory, be applied.
Polynomial chaos expansion from linear least squares regression
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Let :math:`n \in \Nset` be an integer representing the number of
observations in the experimental design.
Let :math:`\set{D} \subseteq \set{X}` be a set of :math:`n` independent
observations of the random vector :math:`\vect{X}`:
.. math::
\set{D} = \left\{\vect{x}^{(1)}, ..., \vect{x}^{(n)}\right\}
Let :math:`P \in \Nset` be an integer representing the number of
coefficients in the polynomial chaos expansion.
The expansion is:
.. math::
\metaModel(\vect{x})
= \sum_{k = 0}^{P - 1} \widehat{a}_k \psi_k(\vect{x})
where :math:`(\widehat{a}_k)_{k = 0,..., P}`\ ’s is the vector of estimates of the
coefficients.
Assume that the coefficients are estimated using linear least squares.
The design matrix :math:`\boldsymbol{\Psi} \in \Rset^{n \times P}` is:
.. math::
\boldsymbol{\Psi}_{ik} = \psi_k\left(\vect{x}^{(i)}\right),
for :math:`i = 1, \dots, n` and :math:`k = 0, \dots, P-1`.
Cross-validation of a PCE
~~~~~~~~~~~~~~~~~~~~~~~~~
If the coefficients of the PCE are estimated using linear least squares,
then the fast methods presented in :ref:`cross_validation` can be applied:
- the fast leave-one-out cross-validation,
- the fast K-Fold cross-validation.
Fast methods are implemented in :class:`~openturns.experimental.FunctionalChaosValidation`.
.. topic:: API:
- See :class:`~openturns.experimental.FunctionalChaosValidation`
.. topic:: References:
- [blatman2009]_
|
