Local-geometric-projection method for noise reduction in chaotic maps and flows.

Abstract

We describe a method for noise reduction in chaotic systems that is based on projection of the set of points comprising an embedded noisy orbit in ${\mathit{openR}}^{\mathit{d}}$ toward a finite patchwork of best-fit local approximations to an m-dimensional surface M'\ensuremath{\subset}${\mathit{openR}}^{\mathit{d}}$, m\ensuremath{\le}d. We generate the orbits by the delay coordinate construction of Ruelle and Takens [N. H. Packard et al., Phys. Rev. Lett. 45, 712 (1980); F. Takens, in Dynamical Systems and Turbulence, Warwick, 1980, edited by D. A. Rand and L.-S. Young (Springer, Berlin, 1981)] from time series v(t), which in an experimental situation we would assume to have come, together with additional high-dimensional background noise, from an underlying dynamical system ${\mathit{f}}^{\mathit{t}}$: M\ensuremath{\rightarrow}M existing on some low m-dimensional manifold M. The surface M' in ${\mathit{openR}}^{\mathit{d}}$ is the assumed embedded image of M. We give results of systematic studies of linear (tangent plane) projection schemes. We describe in detail the basic algorithm for implementing these schemes. We apply the algorithm iteratively to known map and flow time series to which white noise has been added. In controlled studies, we measure the signal-to-noise ratio improvements, iterating ${\mathit{n}}_{\mathit{M}}$ times until a stable maximum ${\mathrm{\ensuremath{\delta}}}_{\mathit{M}}$ is achieved. We present extensive results for ${\mathrm{\ensuremath{\delta}}}_{\mathit{M}}$ and ${\mathit{n}}_{\mathit{M}}$ for a wide range of values of embedding trial dimension d, projection dimension k, number of nearest-neighbor points for local approximation \ensuremath{\nu}, embedding delay \ensuremath{\Delta}, sampling interval \ensuremath{\Delta}T, initial noise amplitude scrN, and trajectory length N.We give results for very low and very high noise amplitudes 0%\ensuremath{\le}scrN\ensuremath{\le}100%. We develop an empirical method for estimating the initial noise level for a given experimental time series, and for the optimal choice of algorithm parameters to achieve peak reduction. We present interesting results of application of the noise-reduction algorithm to a chaotic time series produced from a periodically driven magnetoelastic ribbon experiment on the control of chaos. Two noteworthy elements of the noise-reduction method we describe result in certain stabilizing and efficiency features. The first is our use of a physical replacement time series, which is a unique scalar time series with the property that its corresponding delay coordinate construction data state vector time series in ${\mathit{openR}}^{\mathit{d}}$ is optimally close to the noise-reduced replacement vector time series generated by the projection. The second is the introduction of a ``measure-ordered'' cover, which produces notable improvement in reliability, control, and computational efficiency of the whole algorithm.