Modeling nonlinear time series

It is argued that the ubiquity of strange attractors in nature suggests that using nonlinear modeling techniques might improve performance in some signal processing applications. A synthetic data set generated by numerically integrating a simple nonlinear differential equation is described, and the case with which crude nonlinear methods outperform linear methods is illustrated. The synthetic data are fit by linear autoregressive moving average (ARMA) models and three nonlinear methods: piecewise linear, hidden Markov models (HMM) with discrete outputs, and HMMs with continuous autoregressive outputs (ARHMM). Criteria for assessing model performance are discussed, and connections between these criteria and fundamental invariants developed in ergodic theory are noted