Principal components analysis for sparsely observed correlated functional data using a kernel smoothing approach

Abstract

We consider the problem of functional principal component analysis for correlated functional data. In particular, we focus on a separable covariance structure and consider irregularly and possibly sparsely observed sample trajectories. By observing that under the sparse measurements setting, the empirical covariance of pre-smoothed sample trajectories is a highly biased estimator along the diagonal, we propose to modify the empirical covariance by estimating the diagonal and off-diagonal parts of the covariance kernel separately. We prove that under a separable covariance structure, this method can consistently estimate the eigenfunctions of the covariance kernel. We also quantify the role of the correlation in the L² risk of the estimator, and show that under a weak correlation regime, the risk achieves the optimal nonparametric rate when the number of measurements per curve is bounded.