Polynomial-chaos surrogates get locally adaptive intervals: high-noise regions stop being under-covered, from a single fit
Synopsis
Polynomial chaos expansions have long offered only global error metrics, leaving the reliability of an individual prediction unstated; normalizing jackknife+ conformal nonconformity scores with an auxiliary polynomial chaos model makes interval width follow local input uncertainty. For fixed-basis, least-squares-fitted expansions every quantity is available in closed form, so the finite-sample jackknife+ coverage guarantee is retained without retraining; on a cantilever-beam benchmark with known heteroscedastic noise and on steam-generator clogging simulation data, the adaptive intervals reach target coverage and clearly outperform non-adaptive ones in high-uncertainty regions.
Interpretation
Predictions from a polynomial chaos surrogate can now carry prediction intervals that vary with input location, rather than only a global error number for the whole input space. Earlier conformal-prediction combinations with polynomial chaos used absolute residuals as nonconformity scores, producing intervals of nearly constant width across the input space that cannot reflect input-dependent surrogate error. On the cantilever-beam benchmark with known heteroscedastic noise, adaptive jackknife+ reduces mean interval width by roughly 4%, 13% and 25% and the Winkler score by roughly 20%, 24% and 31% relative to its non-adaptive counterpart at the three significance levels.
Local uncertainty is estimated by an auxiliary polynomial chaos model trained on the logarithms of the base model's leave-one-out residuals, then exponentiated to scale the intervals. Normalized nonconformity scores require such a local uncertainty estimator, while the jackknife+ coverage guarantee requires each leave-one-out auxiliary model to depend only on data with that point removed. To meet that requirement the auxiliary models are trained on nested leave-one-out residuals, omitting two training points at a time; for fixed-basis, least-squares-fitted polynomial chaos these quantities follow in closed form from the Sherman–Morrison–Woodbury formula, needing only one base-model fit.
For sparse polynomial chaos, where basis selection depends on the training data, the closed forms no longer apply, so a heuristic reuses the leave-one-out residuals and limits adaptivity overhead to sparse fits of the auxiliary model. The guarantee-preserving route would need on the order of n-squared sparse fits of the base model, which is computationally prohibitive. In the small-sample sparse case the heuristic still reaches target coverage and lowers the Winkler score by roughly 13%, 17% and 23% relative to the non-adaptive version, but the formal coverage guarantee no longer holds.
Perspective
The demonstrated scope is scalar-output regression: a cantilever-beam benchmark with known heteroscedastic noise and a steam-generator clogging engineering simulation dataset, both implemented with OpenTURNS and run on a standard laptop. The intended readers are uncertainty quantification and engineering simulation practitioners who need prediction intervals for polynomial chaos surrogates.
The coverage provided is marginal, and adaptivity improves conditional coverage only to the extent that the auxiliary model captures local prediction uncertainty; with small training sets and high target coverage, jackknife+ intervals are dominated by extreme nonconformity scores and become wide and variable. The guarantee-preserving approach is computationally prohibitive for sparse polynomial chaos, so coverage of those adaptive intervals rests on numerical evidence only.