We introduce a new class of conditional autoregressive models for spatially dependent functional data, formulated through conditional means given neighboring functional observations and characterized by a covariance operator and a spatial dependence parameter. Our estimation strategy consists of three components: (i) estimating the covariance operator using conditionally centered data, (ii) estimating the spatial dependence parameter by maximizing the likelihood of projected observations, and (iii) applying a novel profile-based approach to obtain the final estimators. Under an expanding lattice framework, we establish two key theoretical results. First, we establish the consistency of the proposed covariance estimator, which is not attainable using naive methods based on marginally centered data. Second, we prove that the spatial dependence parameter estimator is superconsistent and asymptotically normal, where the latter property enables statistical inference for spatial dependence in functional data -- a contribution that is novel in the existing literature. Numerical studies support the theoretical results and demonstrate the computational efficiency of our method. Finally, we illustrate its practical utility by analyzing weekly PM$_{2.5}$ concentration trajectories in 2019 across counties in the Midwestern United States.

翻译：我们提出了一类新的用于空间相依函数型数据的条件自回归模型，该模型通过相邻函数型观测的条件均值进行建模，并以协方差算子和空间依赖参数为特征。我们的估计策略包括三个部分：(i) 使用条件中心化数据估计协方差算子，(ii) 通过最大化投影观测的似然函数来估计空间依赖参数，(iii) 采用一种新颖的基于轮廓的方法获得最终估计量。在扩展格点框架下，我们建立了两个关键的理论结果。首先，我们证明了所提出的协方差估计量的一致性，这是使用基于边缘中心化数据的朴素方法无法实现的。其次，我们证明了空间依赖参数估计量具有超一致性和渐近正态性，其中后者使得能够对函数型数据的空间依赖性进行统计推断——这是现有文献中的一项新颖贡献。数值研究支持了理论结果，并展示了我们方法的计算效率。最后，我们通过分析2019年美国中西部各县的每周PM$_{2.5}$浓度轨迹，说明了其实际效用。