Prediction of Chaotic Time Series in the Presence of Measurement Error: the Importance of Initial Conditions

In this paper we argue that even if a dynamic relationship can be well described by a deterministic system, retrieving this relationship from an empirical time series has to take into account some, although possibly very small measurement error in the observations. Therefore, measuring the initial conditions for prediction may become much more difficult since one now has a combination of deterministic and stochastic elements. We introduce a partial smoothing estimator for estimating the unobserved initial conditions. We will show that this estimator allows to reduce the effects of measurement error for predictions although the reduction may be small in the presence of strong chaotic dynamics. This will be illustrated using the logistic map.