Symplectic Principal Component Analysis: A Noise Reduction Method for Continuous Chaotic Systems

Abstract

In real environments, data obtained from a dynamic system were more or less contaminated by noise. In addition, the dynamic system was often very complicated, non-linear, even chaotic, and sometimes unknown. These might generally lead the measured data to be very complex and seemingly stochastic. In order to obtain the more intrinsic characteristics of the dynamical system, especially chaotic systems, from the measured data, how to effectively reduce noise was still a crucial issue. The aim of this chapter is to introduce a method of noise reduction based on symplectic geometry for the continuous chaotic systems with noise called symplectic principal component analysis (SPCA). The symplectic geometry was a kind of phase space geometry that could preserve the dynamical structure of the system, especially non-linear structure. In symplectic space, the SPCA method could give the dominant principal component values of the data and the component values of the noise floor by the measure-preserving symplectic transform. In the end, this chapter investigated the performance of the SPCA method and applied it to reduce noise in the chaotic time series and experimental data.