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 be an integer representing the number of observations in the experimental design. Let be a set of independent observations of the random vector :
Let be an integer representing the number of coefficients in the polynomial chaos expansion. The expansion is:
where ’s is the vector of estimates of the coefficients. Assume that the coefficients are estimated using linear least squares. The design matrix is:
for and .
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.