  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.
