Functional Variance Processes

We introduce the notion of a functional variance process to quantify variation in functional data. The functional data are modeled as samples of smooth random trajectories observed under additive noise. The noise is assumed to be composed of white noise and a smooth random process—the functional variance process—which gives rise to smooth random trajectories of variance. The functional variance process is a tool for analyzing stochastic time trends in noise variance. As a smooth random process, it can be characterized by the eigenfunctions and eigenvalues of its autocovariance operator. We develop methods to estimate these characteristics from the data, applying concepts from functional data analysis to the residuals obtained after an initial smoothing step. Asymptotic justifications for the proposed estimates are provided. The proposed functional variance process extends the concept of a variance function, an established tool in nonparametric and semiparametric regression analysis, to the case of functional data. We demonstrate that functional variance processes offer a novel data analysis technique that leads to relevant findings in applications, ranging from a seismic discrimination problem to the analysis of noisy reproductive trajectories in evolutionary biology.