A new approximate likelihood

Hidden Markov models (HMM) are successfully applied in various elds of time series analysis. Colored noise, e.g. due to ltering, violates basic assumptions of the model. While it is well-known how to consider auto-regressive (AR) ltering, there is no algorithm to take into account moving-average (MA) ltering in parameter estimation exactly. We present an approximate likelihood estimator for MA-ltered HMM which is generalized to deal with auto-regressive moving-average (ARMA) ltered HMM. The approximation order of the likelihood calculation can be chosen. So, we obtain a sequence of estimators for the HMM parameters as well as for the lter coeecients. The recursion equations for an eecient algorithm are derived from exact expressions for the forward-iterations. By simulations, we show that the derived estimators are unbiased in lter situations where standard HMM are not able to recover the true dynamics. Special implementation strategies together with small approximations yield further acceleration of the algorithm. keywords hidden Markov model, Markov-switching model, linear ltered hidden Markov model, hidden Markov model with correlated noise, auto-regressive moving-average (ARMA) l-ter, innovations algorithm, maximum likelihood estimate, approximate likelihood estimate SP EDICS number: SP 3.6.1 parameter estimation in single-channel time series The authors are with the Center for Data Analysis and Model Building