Cross validation of PCE models

Introduction

The cross-validation of a polynomial chaos expansion uses the theory presented in Validation and cross validation of metamodels. 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 n \in \Nset be an integer representing the number of observations in the experimental design. Let \set{D} \subseteq \set{X} be a set of n independent observations of the random vector \vect{X}:

\set{D} = \left\{\vect{x}^{(1)}, ..., \vect{x}^{(n)}\right\}

Let P \in \Nset be an integer representing the number of coefficients in the polynomial chaos expansion. The expansion is:

\metaModel(\vect{x}) = \sum_{k = 0}^{P - 1} \widehat{a}_k \psi_k(\vect{x})

where (\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 \boldsymbol{\Psi} \in \Rset^{n \times P} is:

\boldsymbol{\Psi}_{ik} = \psi_k\left(\vect{x}^{(i)}\right),

for i = 1, \dots, n and 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 Validation and cross validation of metamodels can be applied:

  • the fast leave-one-out cross-validation,

  • the fast K-Fold cross-validation.