Conformal prediction for full and sparse polynomial chaos expansions

TL;DR

This work advances uncertainty quantification for polynomial chaos expansions by introducing conformal prediction for both full and sparse PCEs. It develops efficient full conformal and Jackknife+ frameworks, with tailored adaptations for sparsity via Hybrid LARS and LASSO-homotopy techniques, yielding prediction intervals that consistently achieve nominal coverage and exhibit favorable stability versus bootstrap. Across Ishigami and Borehole benchmarks, the conformal methods provide well-calibrated local error estimates, with Jackknife+ offering robust, more homogeneous widths and full conformal delivering adaptive, potentially tighter intervals at higher computational cost. The results demonstrate practical benefits for data-scarce surrogate modeling and offer guidance on method selection and potential extensions to multi-fidelity and adaptive design workflows.

Abstract

Polynomial Chaos Expansions (PCEs) are widely recognized for their efficient computational performance in surrogate modeling. Yet, a robust framework to quantify local model errors is still lacking. While the local uncertainty of PCE prediction can be captured using bootstrap resampling, other methods offering more rigorous statistical guarantees are needed, especially in the context of small training datasets. Recently, conformal predictions have demonstrated strong potential in machine learning, providing statistically robust and model-agnostic prediction intervals. Due to its generality and versatility, conformal prediction is especially valuable, as it can be adapted to suit a variety of problems, making it a compelling choice for PCE-based surrogate models. In this contribution, we explore its application to PCE-based surrogate models. More precisely, we present the integration of two conformal prediction methods, namely the full conformal and the Jackknife+ approaches, into both full and sparse PCEs. For full PCEs, we introduce computational shortcuts inspired by the inherent structure of regression methods to optimize the implementation of both conformal methods. For sparse PCEs, we incorporate the two approaches with appropriate modifications to the inference strategy, thereby circumventing the non-symmetrical nature of the regression algorithm and ensuring valid prediction intervals. Our developments yield better-calibrated prediction intervals for both full and sparse PCEs, achieving superior coverage over existing approaches, such as the bootstrap, while maintaining a moderate computational cost.

Figures (13)

• Figure 1: Evaluation of prediction intervals built based on the full PCE surrogate of Ishigami function. Settings of the experiment: $n_{\textrm{ED}}=200$, $n_{\textrm{val}}=1{,}000$, PCE degree $p=5$, number of regressors $P=56$, $\varepsilon_{\textrm{val}}\approx 2\cdot 10^{-1}$, $n_{\textrm{R}}=100$.
• Figure 2: Comparison of the normalized width of prediction intervals at target coverage level $1-\alpha=0.9$ built based on the PCE surrogate of Ishigami function. Settings of the experiment: $n_{\textrm{ED}}=200$, $n_{\textrm{val}}=1{,}000$, PCE degree $p=5$, number of regressors $P=56$, $\varepsilon_{\textrm{val}}\approx 2\cdot 10^{-1}$.
• Figure 3: Evaluation of prediction intervals built based on the PCE surrogate of the Borehole function. Settings of the experiment: $n_{\textrm{ED}}=200$, $n_{\textrm{val}}=1{,}000$, PCE degree $p=2$, number of regressors $P=45$, $\varepsilon_{\textrm{val}}\approx 1\cdot 10^{-3}$, $n_{\textrm{R}}=100$.
• Figure 4: Comparison of the normalized width of prediction intervals at target coverage level $1-\alpha=0.9$ built based on the PCE surrogate of the Borehole function. Settings of the experiment: $n_{\textrm{ED}}=200$, $n_{\textrm{val}}=1{,}000$, PCE degree $p=2$, number of regressors $P=45$, $\varepsilon_{\textrm{val}}\approx 1\cdot 10^{-3}$.
• Figure 5: Performance of the naive implementation of conformal LARS predictions for the Ishigami function. Settings of the experiment: $n_{\textrm{ED}}=40$, $n_{\textrm{val}}=1{,}000$, PCE degree $p=6$, number of regressors $P\approx 23$, $\varepsilon_{\textrm{val}} \approx 1\cdot 10^{-1}$, $n_{\textrm{R}}=100$.