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

Abstract

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 $\mathbf{s}_{1},\mathbf{s}_{2},\ldots,\mathbf{s}_{N}$. 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 i.i.d. 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.