We introduce a spatial function-on-function regression model to capture spatial dependencies in functional data by integrating spatial autoregressive techniques with functional principal component analysis. The proposed model addresses a critical gap in functional regression by enabling the analysis of functional responses influenced by spatially correlated functional predictors, a common scenario in fields such as environmental sciences, epidemiology, and socio-economic studies. The model employs a spatial functional principal component decomposition on the response and a classical functional principal component decomposition on the predictor, transforming the functional data into a finite-dimensional multivariate spatial autoregressive framework. This transformation allows efficient estimation and robust handling of spatial dependencies through least squares methods. In a series of extensive simulations, the proposed model consistently demonstrated superior performance in estimating both spatial autocorrelation and regression coefficient functions compared to some favorably existing traditional approaches, particularly under moderate to strong spatial effects. Application of the proposed model to Brazilian COVID-19 data further underscored its practical utility, revealing critical spatial patterns in confirmed cases and death rates that align with known geographic and social interactions. An R package provides a comprehensive implementation of the proposed estimation method, offering a user-friendly and efficient tool for researchers and practitioners to apply the methodology in real-world scenarios.

翻译：本文提出了一种空间函数对函数回归模型，通过将空间自回归技术与函数主成分分析相结合，以捕捉函数型数据中的空间依赖性。该模型解决了函数回归中的一个关键空白，使得分析受空间相关函数型预测变量影响的函数型响应成为可能，这在环境科学、流行病学和社会经济学等领域是常见场景。模型对响应变量采用空间函数主成分分解，对预测变量采用经典函数主成分分解，从而将函数型数据转化为有限维多元空间自回归框架。这一转换通过最小二乘法实现了高效估计和对空间依赖性的稳健处理。在一系列广泛的模拟实验中，与现有的一些传统方法相比，所提模型在估计空间自相关和回归系数函数方面始终表现出更优的性能，尤其是在中等到强空间效应下。将该模型应用于巴西COVID-19数据进一步凸显了其实用价值，揭示了确诊病例和死亡率的显著空间模式，这些模式与已知的地理和社会交互作用一致。一个R软件包提供了所提估计方法的完整实现，为研究人员和实践者提供了一个用户友好且高效的工具，以便在实际场景中应用该方法。