The paper introduces a multivariate functional areal spatial principal component analysis (mfasPCA) framework, together with multivariate functional Moran's I statistics, to enable the assessment of spatial autocorrelation and dimension reduction for multivariate functional data observed over areal units. The proposed framework is spatial-functional in scope: the functional argument may represent time, age, wavelength, or another ordered continuum, while spatial dependence is introduced across areal units through a spatial weight matrix. The principal component method is defined through a Moran-type spatially weighted criterion. We propose eigenvalue-based permutation tests to assess the significance of spatially structured components. The testing framework includes omnibus tests, componentwise tests with Holm adjustment, and sequential rank-wise tests based on tail sums of eigenvalues. Simulation studies show that mfasPCA captures positive and negative spatial-functional structures and concentrates them in the leading components under the respective autocorrelation regimes. A real-data application illustrates how mfasPCA identifies spatially structured modes of multivariate functional variation.

翻译：本文提出了多变量函数型面元空间主成分分析（mfasPCA）框架，结合多变量函数型莫兰 I 统计量，用于评估面元观测的多变量函数型数据的空间自相关性并实现降维。该框架具有空间-函数双重特征：函数参数可表示时间、年龄、波长或其他有序连续变量，而跨面元的空间依赖性通过空间权重矩阵引入。主成分方法基于莫兰型空间加权准则进行定义。我们提出基于特征值的置换检验以评估空间结构化成分的显著性，检验框架包括全局检验、经霍尔姆校正的分量检验以及基于特征值尾部和的序贯秩检验。模拟研究表明，mfasPCA 能够捕捉正负空间-函数结构，并将其分别集中于相应自相关模式下的主导成分中。实际数据应用案例展示了 mfasPCA 如何识别多变量函数型变异的空间结构化模式。