Consistency of the mean and the principal components of spatially distributed functional data

This paper develops a framework for the estimation of the functional mean and the functional principal components when the functions form a random field. More specifically, the data we study consist of curves $X(\mathbf{s}_k; t), \ t \in [0, T],$ observed at spatial points $\mathbf{s}_1, \mathbf{s}_2,..., \mathbf{s}_N$. We establish conditions for the sample average (in space) of the $X(\mathbf{s}_k)$ to be a consistent estimator of the population mean function, and for the usual empirical covariance operator to be a consistent estimator of the population covariance operator. These conditions involve an interplay of the assumptions on an appropriately defined dependence between the functions $X(\mathbf{s}_k)$ and the assumptions on the spatial distribution of the points $\mathbf{s}_k$. The rates of convergence may be the same as for iid functional samples, but generally depend on the strength of dependence and appropriately quantified distances between the points $\mathbf{s}_k$. We also formulate conditions for the lack of consistency. The general results are specialized to functional spatial models of practical interest. They are established using an appropriate quadratic loss function which we can bound by terms that reflect the assumptions on the spatial dependence and the distribution of the points. This technique is broadly applicable to all statistics obtained by simple averaging of functional data at spatial locations.