Spatial Scalar-on-Function Quantile Regression Model

2025-10-18

AI Summary

This paper addresses the limitations of classical scalar-on-function regression models arising from spatial dependence and heterogeneity in functional covariates. We propose the first spatial scalar-on-function quantile regression model, which incorporates a spatial lag of the response to capture spatial autocorrelation and models nonlinear scalar–functional relationships at each quantile level. Innovatively integrating spatial autoregression with functional quantile regression, we develop two instrumental variable estimation procedures to resolve endogeneity induced by the spatial lag term. We establish √n-consistency and asymptotic normality of the estimators and implement the methodology in the R package *ssofqrm*. Simulation studies demonstrate that our method substantially outperforms mean regression and existing robust alternatives under strong spatial dependence and outlier contamination. An empirical application to predicting PM₂.₅ concentrations from ozone curves in Lombardy, Italy, confirms significant improvements in predictive accuracy, robustness, and interpretability.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Intelligent Robots: State Estimation

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

This paper introduces a novel spatial scalar-on-function quantile regression model that extends classical scalar-on-function models to account for spatial dependence and heterogeneous conditional distributions. The proposed model incorporates spatial autocorrelation through a spatially lagged response and characterizes the entire conditional distribution of a scalar outcome given a functional predictor. To address the endogeneity induced by the spatial lag term, we develop two robust estimation procedures based on instrumental variable strategies. $sqrt{n}$-consistency and asymptotic normality of the proposed estimators are established under mild regularity conditions. We demonstrate through extensive Monte Carlo simulations that the proposed estimators outperform existing mean-based and robust alternatives, particularly in settings with strong spatial dependence and outlier contamination. We apply our method to high-resolution environmental data from the Lombardy region in Italy, using daily ozone trajectories to predict daily mean particulate matter with a diameter of less than 2.5 micrometers concentrations. The empirical results confirm the superiority of our approach in predictive accuracy, robustness, and interpretability across various quantile levels. Our method has been implemented in the exttt{ssofqrm} R package.

Problem

• Extends scalar-on-function regression to incorporate spatial dependence
• Addresses endogeneity in spatial lag terms via instrumental variables
• Models heterogeneous conditional distributions using quantile regression

Innovation

• Spatial scalar-on-function quantile regression model with autocorrelation
• Instrumental variable strategies address spatial lag endogeneity
• Robust estimators outperform alternatives in contaminated spatial data