Abstract
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.
Highlights
With the rapid advance in computational and analytical technology, many time-dynamic processes are monitored and recoded continuously during a time interval or intermittently at several discrete time points
All technique lemmas and all proofs are included in Appendix
For lack of a better approach, we suggest picking the bandwidths by minimizing the integrated mean square error (IMSE). at is, for each function above, one calculated the IMSE over a range of h and selected the one that minimizes the IMSE
Summary
With the rapid advance in computational and analytical technology, many time-dynamic processes are monitored and recoded continuously during a time interval or intermittently at several discrete time points. Traditional functional data typically consist of a random sample of independent real-valued functions, which can be viewed as the realization of a one-dimensional stochastic process. In this field of research, a general introduction of the available methods can be found in Ramsay and Silverman [1] and Wang et al [2]. Since currently available statistical methods for estimating derivatives require densely observed data, it is quiet challenging to recover derivatives from sparse functional data with noise-contaminated measurements. All technique lemmas and all proofs are included in Appendix
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