Estimating a Variance-Covariance Surface for Functional Data using Finite Elements

In functional data analysis, as in its multivariate counterpart, estimates of the bivariate covariance kernel ?(s,t) and its inverse are useful for many things, and we need the inverse of a covariance matrix or kernel especially often. However, the dimensionality of functional observations often exceeds the sample size available to estimate ?(s,t). Then the analogue of the multivariate sample estimate is singular and non-invertible. Even when this is not the case, the high dimensionality of the usual estimate often implies unacceptable sample variability and loss of degrees of freedom for model fitting. The common practice of employing low-dimensional principal component approximations to ?(s,t) to achieve invertibility also raises serious issues.

This talk describes a functional and nonsingular estimate of ?(s,t) and its inverse defined by an expansion in terms of finite element basis functions that permits the user to control the resolution of the estimate as well as the time lag over which covariance may be nonzero. This estimate also permits the estimation of covariances and correlations at observed pairs of sampling points, and therefore has applications to many classical statistical problems, such as discrete but unequally spaced time and spatial series.