A Functional Estimate of Covariation

The analysis of functional data calls for a bivariate functional covariance function σ(<i>s</i>, <i>t</i>) 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(<i>w</i>, <i>s</i>) to define σ(<i>s</i>, <i>t</i>) = ∫λ(<i>w</i>, <i>s</i>)λ(<i>w</i>, <i>t</i>) <i>dw</i>. 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(<i>s</i>, <i>t</i>) estimates the residual covariance structure in the presence of sampling points varying from one case to another. Supplementary materials are available online.