We consider the problem of forecasting the next (observable) state of an unknown ergodic dynamical system from a noisy observation of the present state. Our main result shows that support vector machines (SVMs) using Gaussian RBF kernels can learn the best forecaster from a sequence of noisy observations if a) the unknown observational noise processes is bounded and has a summable $\alpha$-mixing rate and b) the unknown ergodic dynamical system is defined by a Lipschitz continuous function on some compact subset of $\R^d$ and has a summable decay of correlations for Lipschitz continuous functions. In order to prove this result we first establish a general learning theorem for SVMs and all stochastic processes that satisfy a mixing notion that is substantially weaker than $\alpha$-mixing.
I. Steinwart and M. Anghel, An SVM approach for forecasting the evolution of an unknown ergodic dynamical system from observations with unknown noise. Annals of Statistics, Vol. 37, pp. 841-875, 2009. Los Alamos National Laboratory Technical Report LA-UR-07-1829, 2007. [ Abstract | PDF (325 KB) ]
