A Functional Estimate of Covariation

The analysis of functional data calls for a bivariate functional covariance function σ(s, t) that may be evaluated at any discrete set of points to define a variance-covariance matrix Σ. This article uses finite element methodology to construct a representation of a functional Choleski factor λ(w, s) to define σ(s, t) = ∫λ(w, s)λ(w, t) dw. An estimate of Σ-1 is especially important for applications and, where the eigenstructure of the covariance permits, this is readily available since the resulting Σ is almost always positive definite. A simulation study compares the performance of estimates of Σ and Σ-1 to those from the classic covariance matrix estimate and an estimate using glasso package in R. The method’s capability of constraining estimates of Σ-1 to be strongly band-structured resulted in superior estimates. The real data application is to the smoothing of the Fels female growth data where σ(s, t) estimates the residual covariance structure in the presence of sampling points varying from one case to another. Supplementary materials are available online.