We review some recent developments on modeling and estimation of dynamic phenomena within the framework of Functional Data Analysis (FDA). The focus is on longitudinal data which correspond to sparsely and irregularly sampled repeated measurements that are contaminated with noise and are available for a sample of subjects. A main modeling assumption is that the data are generated by underlying but unobservable smooth trajectories that are realizations of a Gaussian process. In this setting, with only a few measurements available per subject, classical methods of Functional Data Analysis that are based on presmoothing individual trajectories will not work. We review the estimation of derivatives for sparse data, the PACE package to implement these procedures, and an empirically derived stochastic differential equation that the processes satisfy and that consists of a linear deterministic component and a drift process.
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