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Prediction of dynamical time series with additive noise using support vectormachines or kernel based regression has been proved to be consistent forcertain classes of discrete dynamical systems. Consistency implies that thesemethods are effective at computing the expected value of a point at a futuretime given the present coordinates. However, the present coordinates themselvesare noisy, and therefore, these methods are not necessarily effective atremoving noise. In this article, we consider denoising and prediction asseparate problems for flows, as opposed to discrete time dynamical systems, andshow that the use of smooth splines is more effective at removing noise.Combination of smooth splines and kernel based regression yields predictorsthat are more accurate on benchmarks typically by a factor of 2 or more. Weprove that kernel based regression in combination with smooth splines convergesto the exact predictor for time series extracted from any compact invariant setof any sufficiently smooth flow. As a consequence of convergence, one can findexamples where the combination of kernel based regression with smooth splinesis superior by even a factor of $100$. The predictors that we compute operateon delay coordinate data and not the full state vector, which is typically notobservable.

Prediction of Dynamical time Series Using Kernel Based Regression and Smooth Splines