Penalized spatial function-on-function regression

2025-11-28

AI Summary

To address the neglect of spatial dependence among observations in functional data, this paper proposes a penalized spatial function-on-function regression model. Methodologically, it extends spatial two-stage least squares to the functional framework for the first time, represents the coefficient function via tensor-product B-spline bases, and incorporates a roughness penalty to ensure both smoothness and interpretability. Theoretically, the estimator is proven to be √n-consistent and asymptotically normal. Simulation studies demonstrate substantial improvements over non-penalized benchmarks under moderate-to-strong spatial dependence. Applied to meteorological data from North Dakota, the model successfully captures the spatial correlation structure between temperature and precipitation curves, yielding enhanced predictive accuracy and robustness. The core contribution lies in unifying the modeling of functional responses, functional covariates, and spatial autocorrelation, while establishing a theoretically grounded penalized estimation framework with provable statistical properties.

Technology Category

• Data Mining & Knowledge Management: Mining of Spatial, Temporal or Spatio-Temporal Data
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Security and Privacy: Large-scale security measurements

Abstract

The function-on-function regression model is fundamental for analyzing relationships between functional covariates and responses. However, most existing function-on-function regression methodologies assume independence between observations, which is often unrealistic for spatially structured functional data. We propose a novel penalized spatial function-on-function regression model to address this limitation. Our approach extends the generalized spatial two-stage least-squares estimator to functional data, while incorporating a roughness penalty on the regression coefficient function using a tensor product of B-splines. This penalization ensures optimal smoothness, mitigating overfitting, and improving interpretability. The proposed penalized spatial two-stage least-squares estimator effectively accounts for spatial dependencies, significantly improving estimation accuracy and predictive performance. We establish the asymptotic properties of our estimator, proving its $sqrt{n}$-consistency and asymptotic normality under mild regularity conditions. Extensive Monte Carlo simulations demonstrate the superiority of our method over existing non-penalized estimators, particularly under moderate to strong spatial dependence. In addition, an application to North Dakota weather data illustrates the practical utility of our approach in modeling spatially correlated meteorological variables. Our findings highlight the critical role of penalization in enhancing robustness and efficiency in spatial function-on-function regression models. To implement our method we used the exttt{robflreg} package on CRAN.

Problem

• Develops penalized spatial function-on-function regression for dependent data
• Incorporates roughness penalty via B-splines to prevent overfitting
• Addresses spatial dependencies to improve estimation and prediction accuracy

Innovation

• Penalized spatial two-stage least-squares estimator for functional data
• Tensor product B-splines with roughness penalty for smoothness
• Accounting for spatial dependencies to improve estimation accuracy