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 \texttt{robflreg} package on CRAN.

翻译：函数对函数回归模型是分析函数型协变量与响应变量之间关系的基础方法。然而，现有的大多数函数对函数回归方法均假设观测值之间相互独立，这对于具有空间结构的函数型数据往往不符合实际。为克服这一局限，本文提出了一种新颖的惩罚性空间函数对函数回归模型。该方法将广义空间两阶段最小二乘估计量扩展至函数型数据，同时利用B样条张量积对回归系数函数施加粗糙度惩罚。这种惩罚机制能确保最优平滑度，有效缓解过拟合问题并提升模型可解释性。所提出的惩罚性空间两阶段最小二乘估计量能够充分考虑空间依赖性，显著提高估计精度与预测性能。我们在温和的正则性条件下建立了估计量的渐近性质，证明了其具有√n相合性与渐近正态性。大量蒙特卡洛模拟实验表明，本方法在中等至强空间依赖条件下，相较于现有非惩罚估计量具有显著优越性。此外，通过对北达科他州气象数据的实证分析，展示了本方法在建模空间相关气象变量方面的实际应用价值。研究结果凸显了惩罚机制在提升空间函数对函数回归模型鲁棒性与效率方面的关键作用。本方法通过CRAN平台的\\texttt{robflreg}软件包实现。