Fast symmetric additive covariance smoothing

TL;DR

This paper introduces a fast symmetric bivariate smoothing method for covariance estimation in functional and longitudinal data, improving computational efficiency and applicability to various sampling schemes.

Contribution

It presents a novel symmetric bivariate penalized spline smoothing approach that reduces computation time and handles complex correlation structures in functional data.

Findings

Effective in estimating covariance functions from dense and sparse data

Reduces computation time compared to non-symmetric smoothers

Demonstrates practical utility in functional principal component analysis

Abstract

We propose a fast bivariate smoothing approach for symmetric surfaces that has a wide range of applications. We show 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. Our 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. We outline the application of our approach in functional principal component analysis and demonstrate its practical value in two applications. The approach is evaluated in extensive simulations. We provide documented open source software implementing our fast symmetric bivariate smoother building on established algorithms for additive models.