Partial Derivative Estimation for Underlying Functional‐Valued Process in a Unified Framework

We consider functional data analysis when the observations at each location are functional rather than scalar. When the dynamic of underlying functional‐valued process at each location is of interest, it is desirable to recover partial derivatives of a sample function, especially from sparse and noise‐contaminated measures. We propose a novel approach based on estimating derivatives of eigenfunctions of marginal kernels to obtain a representation for functional‐valued process and its partial derivatives in a unified framework in which the number of locations and number of observations at each location for each individual can be any rate relative to the sample size. We derive almost sure rates of convergence for the procedures and further establish consistency results for recovered partial derivatives.