A simple noise reduction method based on nonlinear forecasting

Abstract

Non-parametric detrending or noise reduction methods are often employed to separate trends from noisy time series when no satisfactory models exist to fit the data. However, conventional detrending methods depend on subjective choices of detrending parameters. Here, we present a simple multivariate detrending method based on available nonlinear forecasting techniques. These are in turn based on state space reconstruction for which a strong theoretical justification exists for their use in non-parametric forecasting. The detrending method presented here is conceptually similar to Schreiber's noise reduction method using state space reconstruction. However, we show that Schreiber's method contains a minor flaw that can be overcome with forecasting. Furthermore, our detrending method contains a simple but nontrivial extension to multivariate time series. We apply the detrending method to multivariate time series generated from the Van der Pol oscillator, the Lorenz equations, the Hindmarsh-Rose model of neuronal spiking activity, and a univariate real-life measles data set. It is demonstrated that detrending heuristics can be objectively optimized with in-sample forecasting errors that correlate well with actual detrending errors.