Nonlinear modeling and its prediction range

The linear theory of prediction is capable of performing long term forecasting when the observed time series is linear. For the chaotic signals, where linear modeling techniques are insufficient, nonlinear approximation theory can be used effectively for predicting. Long-term prediction however, is always difficult to achieve for such signals because of the amplification of errors in each prediction step. Prediction of chaotic time series dates back to the work by Farmer and Sidorowich [1] who employed the local linear polynomial approximation based on K nearest neighbors after time delay embedding of the signal in state space. Other studies such as higher order polynomial functions [1, 2], radial basis functions [2, 3], and neural networks [4, 5, 6, 7] have limited success in achieving long-term prediction of chaotic signals especially in the presence of noise. Since nonlinear signals are very sensitive to noise, it often becomes a problem to apply nonlinear techniques to estimate parameters such as correlation dimension, Lyapunov exponents and prediction error.