Fast symmetric additive covariance smoothing

A fast bivariate smoothing approach for symmetric surfaces is proposed that has a wide range of applications. It is shown how it can be applied to estimate the covariance function in longitudinal data as well as multiple additive covariances in functional data with complex correlation structures. The proposed symmetric smoother can handle (possibly noisy) data sampled on a common, dense grid as well as irregularly or sparsely sampled data. Estimation is based on bivariate penalized spline smoothing using a mixed model representation and the symmetry is used to reduce computation time compared to the usual non-symmetric smoothers. The application of the approach in functional principal component analysis for very general functional linear mixed models is outlined and its practical value is demonstrated in two applications. The approach is evaluated in extensive simulations. Documented open source software is provided that implements the fast symmetric bivariate smoother building on established algorithms for additivemodels.