No abstract

Prediction of dynamical time series with additive noise using support vector machines or kernel based regression is consistent for certain classes of discrete dynamical systems. Consistency implies that these methods are effective at computing the expected value of a point at a future time given the present coordinates. However, the present coordinates themselves are noisy, and therefore, these methods are not necessarily effective at removing noise. In this article, we consider denoising and prediction as separate problems for flows, as opposed to discrete time dynamical systems, and show that the use of smooth splines is more effective at removing noise. Combination of smooth splines and kernel based regression yields predictors that are more accurate on benchmarks typically by a factor of 2 or more. We prove that kernel based regression in combination with smooth splines converges to the exact predictor for time series extracted from any compact invariant set of any sufficiently smooth flow. As a consequence of convergence, one can find examples where the combination of kernel based regression with smooth splines is superior by even a factor of 100. The predictors that we analyze and compute operate on delay coordinate data and not the full state vector, which is typically not observable.

No abstract

This article considers the numerical inversion of the power series p(x) = 1 + b 1 x + b 2 x 2 + · · · to compute the inverse series q(x) satisfying p(x)q(x) = 1. Numerical inversion is a special case of triangular back-substitution, which has been known for its beguiling numerical stability since the classic work of Wilkinson (1961). We prove the numerical stability of inversion of power series and obtain bounds on numerical error. A range of examples show these bounds to be quite good. When p(x) is a polynomial and x = a is a root with p(a) = 0, we show that root deflation via the simple division p(x)/(x − a) can trigger instabilities relevant to polynomial root finding and computation of finite-difference weights. When p(x) is a polynomial, the accuracy of the computed inverse q(x) is connected to the pseudozeros of p(x).