Kernel methods for nonparametric analog forecasting
Dimitris Giannakis, CIMS

Abstract: Analog forecasting is a nonparametric technique introduced by Lorenz in 1969 which predicts the evolution of observables of dynamical systems by following the evolution of samples in a historical record of observations of the system which most closely resemble the observations at forecast initialization. In this talk, we discuss a family of forecasting methods which improve traditional analog forecasting by combining ideas from kernel methods for machine learning and state-space reconstruction for dynamical systems. A key ingredient of our approach is to replace single-analog forecasting with weighted ensembles of analogs constructed using local similarity kernels. The kernels used here employ a number of dynamics-dependent features designed to improve forecast skill, including Takens' delay-coordinate maps (to recover information in the initial data lost through partial observations) and a directional dependence on the dynamical vector field generating the data. Mathematically, the approach is closely related to kernel methods for out-of-sample extension of functions, and we discuss alternative strategies based on the Nystrom method and the multiscale Laplacian pyramids technique. We illustrate these techniques in atmosphere ocean science applications.