Differential equations (DIFE's) can represent the underlying processes giving rise to observed functional data, and as such can offer a number of potential advantages over parametric or nonparametric basis expansion models.
- DIFE's explicitly model the behavior of derivatives, and link this behavior to the observed function itself. Consequently, they model the rate of change in the data as well as their amplitudes.
- Solutions to a linear DIFE of order m span an m-dimensional space, and consequently have the capacity to model curve-to-curve variation as well as to fit the data.
- We can build known structural features into DIFE models more easily than is usually the case for conventional functional models.
- Derivative estimates based on DIFE's are usually superior to those derived from conventional data smoothers.
- And finally a DIFE offers a wider range of ways to introduce stochastic behavior into models.
In spite of the enormous importance of DIFE models in many areas of science and engineering, existing methods for actually identifying or estimating a differential equation from noisy data remain crude, inefficient, and unable to deliver estimates of sampling error.
I will discuss a technique for going directly from the discrete and noisy data to a DIFE that is based on the work of Heckman and Ramsay (2000). Some illustrations of its performance for simulated data will be offered as well as examples from chemical engineering and for medical data on treatment regimes for lupus.
Heckman, N. and Ramsay, J. O. (2000) Penalized regression with model-based penalties. The Canadian Journal of Statistics, 28, 241-258.