Delay Estimation for MultivariateTime

Most traditional methods for extracting the relationships between two time series are based on cross-correlation. In a non-linear non-stationary environment, these techniques are not suucient. We show in this paper how to use hidden Markov models (HMMs) to identify the lag (or delay) between diierent variables for such data. We rst present a method using maximum likelihood estimation and propose a simple algorithm which is capable of identifying associations between variables. We also adopt an information-theoretic approach and develop a novel procedure for training HMMs to maximise the mutual information between delayed time series. Both methods are successfully applied to real data: we show that HMMs are capable of modelling the oil drilling process and that they outperform existing methods for computing a crucial parameter, namely the lag for return.