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

TL;DR

This paper establishes conditions under which the spatial average of functional data and their principal components are consistent estimators of their population counterparts, considering dependence and spatial distribution.

Contribution

It provides a theoretical framework for the consistency of spatially distributed functional data estimators, accounting for dependence and spatial arrangement.

Findings

Consistency of the spatial mean estimator established

Consistency of the empirical covariance operator proven

Convergence rates depend on dependence strength and spatial distances

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.