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.
"Principal components analysis for sparsely observed correlated functional data using a kernel smoothing approach." Electron. J. Statist. 5 1960 - 2003, 2011. https://doi.org/10.1214/11-EJS662