Inferring stochastic dynamics from functional data

In most current data modelling for time-dynamic systems, one works with a prespecified differential equation and attempts to estimate its parameters. In contrast, we demonstrate that in the case of functional data, the equation itself can be inferred. Assuming only that the dynamics are described by a first-order nonlinear differential equation with a random component, we obtain data-adaptive dynamic equations from the observed data via a simple smoothing-based procedure. We prove consistency and introduce diagnostics to ascertain the fraction of variance that is explained by the deterministic part of the equation. This approach is shown to yield useful insights into the time-dynamic nature of human growth. Copyright 2012, Oxford University Press.