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Model Selection, confidence and Scaling in Predicting Chaotic Time-Series

Assuming a good embedding and additive noise, the traditional approach to time-series embedding prediction has been to predict pointwise by (usually linear) regression of the k-nearest neighbors; no good mathematics has been previously developed to appropriately select the model (where to truncate Taylor's series) to balance the conflict between noise fluctuations of a small k, and large k data needs of fitting many parameters of a high ordered model. We present a systematic approach to: (1) select the statistically significant neighborhood for a fixed (usually linear) model, (2) give an unbiased estimate of predicted mean response together with a statement of quality of the prediction in terms of confidence bands.