Time delay estimation with hidden Markov models

Most traditional methods for extracting the relationships between two time series are based on cross-correlation. In a nonlinear non-stationary environment, these techniques are not sufficient. We show how to use hidden Markov models (HMMs) to identify the lag (or delay) between different variables for such data. Adopting an information-theoretic approach, we develop a procedure for training HMMs to maximise the mutual information (MMI) between delayed time series. The method is used to model the oil drilling process. We show that cross-correlation gives no information and that the MMI approach outperforms the maximum likelihood approach.