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.