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

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.

💡 Research Summary

The paper addresses a gap in functional data analysis for computer simulations where the response is a function defined over a spatial domain, and two distinct types of predictors are present: (i) functional predictors that are fixed across simulation runs (e.g., terrain elevation) and (ii) scalar “global” predictors that vary from run to run (e.g., hurricane intensity, track). Classical function‑on‑scalar (FOSR) and function‑on‑function (FoF) models cannot simultaneously accommodate this multi‑scale structure.

To solve this, the authors propose a Bayesian additive semiparametric regression model:

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