An exact kernel framework for spatiotemporal dynamics
Abstract
A kernelbased framework for spatiotemporal data analysis is introduced that applies in situations when the underlying system dynamics are governed by a dynamic equation. The key ingredient is a representer theorem that involves timedependent kernels. Such kernels occur commonly in the expansion of solutions of partial differential equations. The representer theorem is applied to find among all solutions of a dynamic equation the one that minimizes the error with given spatiotemporal samples. This is motivated by the fact that very often a differential equation is given a priori (e.g.~by the laws of physics) and a practitioner seeks the best solution that is compatible with her noisy measurements. Our guiding example is the FokkerPlanck equation, which describes the evolution of density in stochastic diffusion processes. A regression and density estimation framework is introduced for spatiotemporal modeling under FokkerPlanck dynamics with initial and boundary conditions.
 Publication:

arXiv eprints
 Pub Date:
 November 2020
 arXiv:
 arXiv:2011.06848
 Bibcode:
 2020arXiv201106848S
 Keywords:

 Mathematics  Statistics Theory;
 Statistics  Machine Learning