Spatial Functional Data Analysis

Spatial functional data has gained considerable attention due to its theoretical and ap-plicable value.In order to provide proper assessment of air pollutants and accurate spatial clustering outcomes,we propose a novel approach for modeling and clustering PM_2.5concen-trations across China.In our method,observed concentrations from monitoring stations are modeled as spatially dependent functional data.We assume latent emission processes originate from a functional mixture model with each component as a spatiotemporal process.Moreover,cluster memberships of cities or monitoring stations are assumed to follow a Markov random field model.The superior performance of our approach is demonstrated using extensive simu-lation studies.Our method is effective in dividing China and the Beijing-Tianjin-Hebei region into several regions based on PM_2.5concentrations,suggesting that separate local emission control policies are needed.Moreover,we study the validity of the generalized Karhunen-Loève expansion in the above spatio-temporal model.We introduce the concept of weak separability,and propose a formal testing procedure to examine its validity.The asymptotic distribution of the test statistic that adapts to potentially diverging ranks is derived by con-structing lag covariance estimation,which is easy to compute for practical implementation.We demonstrate the efficacy of the proposed test via simulations and illustrate its application in two examples:China's PM_2.5data and Harvard forest data.