Analytical Extraction of Conditional Sobol' Indices via Basis Decomposition of Polynomial Chaos Expansions

2026-04-22 · Source: stat.ML updates on arXiv.org · Field: Science & Research — Mathematics & Computational Sciences, Engineering & Applied Sciences, Artificial Intelligence & Machine Learning · Depth: Expert, long

Summary

A new framework has been developed for the analytical extraction of conditional Sobol’ indices from a global Polynomial Chaos Expansion (PCE) model. This method addresses the computational expense and inconsistency of traditional point-wise modeling approaches for evaluating sensitivity measures under specific conditions, such as spatial fields or varying operating conditions. The framework reformulates a pre-trained global PCE model by leveraging the tensor-product property of PCE bases, transforming it into a set of analytical coefficient fields dependent on conditioning variables. This allows for the derivation of closed-form expressions for conditional variances and Sobol’ indices, bypassing repetitive modeling or additional sampling. Numerical benchmarks demonstrate that this approach ensures physical coherence, superior numerical robustness, and computational efficiency compared to conventional Monte Carlo and point-wise PCE methods, even when introducing spatially varying Gaussian noise.

Key takeaway

For AI Scientists and Research Scientists working with uncertainty quantification in complex systems, this analytical conditional PCE framework offers a robust alternative to traditional point-wise methods. You should consider adopting this approach to derive conditional Sobol' indices, especially for parameterized responses like spatial fields, as it provides physically coherent and computationally efficient sensitivity maps, enhancing the reliability of your model analysis.

Key insights

Conditional Sobol' indices can be analytically extracted from a global PCE model via basis decomposition, ensuring physical coherence.

Principles

• PCE basis factorization encapsulates conditioning variable influence.
• Orthonormality of polynomial basis is preserved under conditional probability.
• Analytical continuity provides global filtering against noise.

Method

A global PCE model is reformulated by partitioning multi-indices and factorizing tensor-product bases. This yields parametric coefficient fields from which conditional variances and Sobol' indices are algebraically derived.

In practice

• Analyze spatially distributed stochastic fields efficiently.
• Reduce computational cost by avoiding repetitive local modeling.
• Improve robustness of sensitivity analysis against sampling noise.

Topics

• Polynomial Chaos Expansion
• Conditional Sobol' Indices
• Uncertainty Quantification
• Basis Decomposition
• Sensitivity Analysis

Best for: AI Scientist, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.