An Efficient Nonparametric Estimate for Spatially Correlated Functional Data

Functional data are often generated by modern biomedical technologies where features related to the pathophysiology and pathogenesis of a disease are interrogated repeatedly over time and at multiple spatially interdependent regions. To reduce model complexity and simplify the resulting inference, possible spatial correlation among neighboring regions is often neglected. In this article, we propose a weighted kernel smoothing estimate of the mean function that leverages the spatial and temporal correlation. We also address the companion problem of developing a simultaneous prediction method for individual curves using discrete samples. We establish the asymptotic properties of the proposed estimate, including its unique maximum efficiency achieving minimum asymptotic variance. The proposed method improves estimation and prediction in the presence of sparse observations, and therefore, is advantageous to biomedical applications that utilize markers to identify features intrinsic to a particular disease at multiple interdependent sites within an organ. Our simulation and case studies show that the proposed method outperforms conventional approaches for characterizing the dynamic functional imaging data, with the maximum benefit achieved in the presence of a small number of repeated scans.