Nonlinear Time&hyphenSeries Prediction with Missing and Noisy Data

We derive solutions for the problem of missing and noisy data in nonlinear time&hyphenseries prediction from a probabilistic point of view. We discuss different approximations to the solutions &hyphen in particular, approximations that require either stochastic simulation or the substitution of a single estimate for the missing data. We show experimentally that commonly used heuristics can lead to suboptimal solutions. We show how error bars for the predictions can be derived and how our results can be applied to K&hyphenstep prediction. We verify our solutions using two chaotic time series and the sunspot data set. In particular, we show that for K&hyphenstep prediction, stochastic simulation is superior to simply iterating the predictor.