Supervised Learning of Functional Outcomes with Predictors at Different Scales: A Functional Gaussian Process Approach

2026-02-10

AI Summary

This work proposes an additive nonlinear Bayesian regression framework to address the challenge of functional outputs in complex computer simulations that are jointly influenced by functional predictors defined over a fixed spatial domain and global scalar variables varying across simulation runs. The approach introduces a novel functional Gaussian process (fGP) prior that simultaneously models the unknown nonlinear effect of global variables across the entire spatial domain and captures spatially varying coefficients of local functional predictors through Gaussian processes. By explicitly encoding spatial dependence in the global effects, the fGP enables an interpretable decomposition of contributions from these two multiscale predictor types and provides principled uncertainty quantification. Experiments on synthetic data and the SLOSH hurricane storm surge model demonstrate that the method achieves high predictive accuracy alongside reliable uncertainty estimates.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Calibration & Uncertainty Quantification
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems
• Web Mining and Content Analysis: Web data generation and simulation

Abstract

The analysis of complex computer simulations, often involving functional data, presents unique statistical challenges. Conventional regression methods, such as function-on-function regression, typically associate functional outcomes with both scalar and functional predictors on a per-realization basis. However, simulation studies often demand a more nuanced approach to disentangle nonlinear relationships of functional outcome with predictors observed at multiple scales: domain-specific functional predictors that are fixed across simulation runs, and realization-specific global predictors that vary between runs. In this article, we develop a novel supervised learning framework tailored to this setting. We propose an additive nonlinear regression model that flexibly captures the influence of both predictor types. The effects of functional predictors are modeled through spatially-varying coefficients governed by a Gaussian process prior. Crucially, to capture the impact of global predictors on the functional outcome, we introduce a functional Gaussian process (fGP) prior. This new prior jointly models the entire collection of unknown, spatially-indexed nonlinear functions that encode the effects of the global predictors over the entire domain, explicitly accounting for their spatial dependence. This integrated architecture enables simultaneous learning from both predictor types, provides a principled strategies to quantify their respective contributions in predicting the functional outcome, and delivers rigorous uncertainty estimates for both model parameters and predictions. The utility and robustness of our approach are demonstrated through multiple synthetic datasets and a real-world application involving outputs from the Sea, Lake, and Overland Surges from Hurricanes (SLOSH) model.

Problem

• functional data
• multi-scale predictors
• function-on-function regression
• spatial dependence
• supervised learning

Innovation

• functional Gaussian process
• multi-scale predictors
• function-on-function regression
• spatially-varying coefficients
• Bayesian nonparametric modeling