Spatio-temporal data are common in practice. Existing methods for analyzing such data often employ parametric modelling with different sets of model assumptions. However, spatio-temporal data in practice often have complicated structures, including complex spatial and temporal data variation, latent spatio-temporal data correlation, and unknown data distribution. Because such data structures reflect the complicated impact of confounding variables, such as weather, demographic variables, life styles, and other cultural and environmental factors, they are usually too complicated to describe by parametric models. In this paper, we suggest a general modelling framework for estimating the mean and covariance functions of spatio-temporal data using a three-step local smoothing procedure. The suggested method can well accommodate the complicated structure of real spatio-temporal data. Under some regularity conditions, the consistency of the proposed estimators is established. Both simulation studies and a real-data application show that our proposed method could work well in practice.

时空数据在实践中很常见。用于分析此类数据的现有方法经常采用具有不同模型假设集的参数建模。但是，实践中的时空数据通常具有复杂的结构，包括复杂的时空数据变化，潜在的时空数据相关性和未知的数据分布。由于此类数据结构反映了混杂变量（如天气，人口统计学变量，生活方式以及其他文化和环境因素）的复杂影响，因此它们通常过于复杂，无法用参数模型来描述。在本文中，我们建议使用三步局部平滑过程来估计时空数据的均值和协方差函数的通用建模框架。所提出的方法可以很好地适应实际时空数据的复杂结构。在某些规律性条件下，建立了估计量的一致性。仿真研究和实际数据应用均表明，我们提出的方法可以在实践中很好地工作。