Learning Dynamics from Functional Data

Hans-Georg Müller
Department of Statistics
University of California, Davis

Longitudinal and functional data can often be viewed as a sample of realizations of an underlying stochastic process. Under weak conditions, the underlying process can be characterized by functional principal components and eigenfunctions. These turn out to be useful to represent not only the processes but also their dynamics.

For Gaussian processes, underlying dynamics are shown to always obey a first order stochastic linear differential equation with time-varying coefficients that includes a deterministic component and a smooth drift process. The resulting decomposition into population differential equation and drift process may be empirically obtained from longitudinal observations for a sample of subjects. This empirical learning approach is the opposite of the usual approach where one uses data to fit the coefficients of a pre-specified differential equation.

For the case of non-Gaussian processes, similar but  more complex decompositions hold and one may obtain data-adaptive dynamic equations from the observed data via a smoothing-based procedure. This approach is illustrated with an application to quantify the dynamics of human growth.