[Paper Review] Functional principal component analysis for global sensitivity analysis of model with spatial output

Tran Vi-Vi Élodie Perrin, Olivier Roustant+7 more|arXiv (Cornell University)|May 20, 2020

Probabilistic and Robust Engineering Design29 references4 citations

TL;DR

This paper proposes a functional principal component analysis (FPCA)-based methodology to enable global sensitivity analysis (GSA) for computationally expensive simulators with high-dimensional spatial outputs, such as coastal flood models. By combining functional basis decomposition (wavelets or B-splines), basis coefficient selection via energy criteria or penalized regression, and FPCA with a Gram matrix metric, the method reduces output dimension while preserving spatial structure, enabling accurate metamodeling and analytical variance-based sensitivity indices even for non-stationary, locally varying outputs.

ABSTRACT

Motivated by risk assessment of coastal flooding, we consider time-consuming simulators with a spatial output. The aim is to perform sensitivity analysis (SA), quantifying the influence of input parameters on the output. There are three main issues. First, due to computational time, standard SA techniques cannot be directly applied on the simulator. Second, the output is infinite dimensional, or at least high dimensional if the output is discretized. Third, the spatial output is non-stationary and exhibits strong local variations. We show that all these issues can be addressed all together by using functional PCA (FPCA). We first specify a functional basis, such as wavelets or B-splines, designed to handle local variations. Secondly, we select the most influential basis terms, either with an energy criterion after basis orthonormalization, or directly on the original basis with a penalized regression approach. Then FPCA further reduces dimension by doing PCA on the most influential basis coefficients, with an ad-hoc metric. Finally, fast-to-evaluate metamodels are built on the few selected principal components. They provide a proxy on which SA can be done. As a by-product, we obtain analytical formulas for variance-based sensitivity indices, generalizing known formula assuming orthonormality of basis functions.

Motivation & Objective

• To address the challenge of performing global sensitivity analysis on time-consuming simulators with spatial outputs that exhibit strong local variations.
• To overcome the limitations of standard PCA and functional basis methods in handling non-stationary, high-dimensional spatial outputs with discontinuities.
• To develop a dimension reduction framework that preserves spatial dependence and enables accurate, fast-to-evaluate metamodels.
• To derive analytical formulas for variance-based sensitivity indices valid for arbitrary (non-orthonormal) basis functions.
• To demonstrate the method’s superiority over classical PCA in capturing sharp spatial features in coastal flood modeling.

Proposed method

• Apply a functional basis decomposition using wavelets or B-splines to represent the spatial output as a set of coefficients.
• Perform a preliminary selection of basis terms using either an energy criterion after orthonormalization or a penalized regression approach (e.g., Lasso) on the original basis.
• Apply functional PCA (FPCA) on the selected coefficients using a metric defined by the Gram matrix of the basis functions, accounting for non-orthogonality.
• Construct a Gaussian process metamodel on the few most influential principal components to enable fast evaluation and sensitivity analysis.
• Derive generalized variance-based sensitivity indices using the FPCA framework, valid without assuming orthonormality of the basis functions.
• Use the resulting metamodel to compute both main and total effect sensitivity indices efficiently.

Experimental results

Research questions

• RQ1 Can functional PCA with basis selection outperform standard PCA in capturing spatial features with strong local variations in high-dimensional outputs?
• RQ2 How can sensitivity analysis be efficiently performed on spatial simulators when the output is high-dimensional and computationally expensive?
• RQ3 What is the impact of using non-orthonormal bases (e.g., B-splines) on the accuracy and interpretability of sensitivity indices in a functional PCA framework?
• RQ4 Can a two-stage dimension reduction—basis selection followed by FPCA—improve metamodel accuracy while reducing computational cost?
• RQ5 How do the proposed sensitivity indices compare to classical ones when the basis functions are not orthonormal?

Key findings

• FPCA with basis selection significantly improves metamodel accuracy over standard PCA in regions with sharp spatial discontinuities, such as flood boundaries.
• The energy criterion after orthonormalization of the basis proved more effective than penalized regression in terms of both accuracy and computational cost in the authors' experiments.
• The method successfully reduced the dimensionality of a 256×256 spatial output (65,536 pixels) to a manageable number of principal components while preserving critical spatial features.
• Analytical formulas for variance-based sensitivity indices were derived that are valid for any basis, removing the need for orthonormality assumptions.
• In the real coastal flooding case study, the sensitivity indices correctly identified inputs related to overflow processes, such as surge peak and duration.
• The approach remains feasible for higher-resolution grids where standard PCA becomes computationally intractable.