Consistency of support vector machines for forecasting the evolution of an unknown ergodic dynamical system from observations with unknown noise

TL;DR

This paper demonstrates that support vector machines with Gaussian kernels can effectively forecast the future states of an unknown ergodic dynamical system from noisy data, under certain mixing and decay conditions.

Contribution

It establishes the consistency of SVMs for forecasting ergodic dynamical systems from noisy observations, extending previous results to weaker mixing conditions.

Findings

SVMs can learn optimal forecasters from noisy data under specified conditions.

The paper proves a general consistency result for SVMs with weakly mixing processes.

Support vector machines are effective for forecasting in complex dynamical systems.

Abstract

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, for example, 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 process 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 $\mathbb{R}^d$ and has a summable decay of correlations for Lipschitz continuous functions. In order to prove this result we first establish a general consistency result for SVMs and all stochastic processes that satisfy a mixing notion that is substantially weaker than $\alpha$-mixing.