Spatial Principal Component Analysis and Moran Statistics for Multivariate Functional Areal Data

Abstract

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.