This study presents a Bayesian hierarchical model for analyzing spatially correlated functional data and handling irregularly spaced observations. The model uses Bernstein polynomial (BP) bases combined with autoregressive random effects, allowing for nuanced modeling of spatial correlations between sites and dependencies of observations within curves. Moreover, the proposed procedure introduces a distinct structure for the random effect component compared to previous works. Simulation studies conducted under various challenging scenarios verify the model's robustness, demonstrating its capacity to accurately recover spatially dependent curves and predict observations at unmonitored locations. The model's performance is further supported by its application to real-world data, specifically PM$_{10}$ particulate matter measurements from a monitoring network in Mexico City. This application is of practical importance, as particles can penetrate the respiratory system and aggravate various health conditions. The model effectively predicts concentrations at unmonitored sites, with uncertainty estimates that reflect spatial variability across the domain. This new methodology provides a flexible framework for the FDA in spatial contexts and addresses challenges in analyzing irregular domains with potential applications in environmental monitoring.

翻译：本研究提出了一种贝叶斯分层模型，用于分析空间相关的函数型数据并处理不规则分布的观测值。该模型采用伯恩斯坦多项式基与自回归随机效应相结合的方法，能够精细建模站点间的空间相关性以及曲线内部观测值的依赖性。此外，与先前研究相比，所提出的方法为随机效应分量引入了独特的结构。在不同挑战性场景下进行的模拟研究验证了模型的稳健性，证明其能够准确还原空间相关曲线并预测未监测位置的观测值。模型在实际数据中的应用进一步支持了其性能，具体案例为墨西哥城监测网络的PM$_{10}$颗粒物测量数据。该应用具有重要现实意义，因为颗粒物可侵入呼吸系统并加剧多种健康问题。模型能有效预测未监测站点的浓度，其不确定性估计反映了研究域内的空间变异性。这一新方法为空间背景下的函数型数据分析提供了灵活框架，解决了非规则域分析中的挑战，在环境监测等领域具有潜在应用价值。