A wide range of practical time series data are not stationary or ergodic, restricting many classical analytical methods. However, a time series usually exhibits few recurring patterns in different time epochs. Examples include the daily network traffic in IP networks, electroencephalogram (EEG) signals triggered by brain modes, volatility in stock markets with economic cycles, and environmental measurements under different time scales. 

In this work, we introduce a methodology that enables reliable and efficient analysis and prediction of such type of time series. The time series is assumed to consist of different time epochs, each modeled by a (non)linear autoregressive process with unknown order (referred to as a state). The number of states is unknown and the transitions of states follow a Markov process of unknown order. The inference is carried out through a three-step strategy: 1) detect the structure changes of the time series, 2) identify each segment as a state and select the most appropriate number of states, 3) estimate the Markov source based upon the symbolic sequence obtained from the last step. Novel theories and algorithms are provided to guarantee and facilitate the above inference procedure. 

The proposed methodology was also extended to the online fashion, which is especially helpful for real-time analysis of massive streaming data. Using various synthetic and real-data experiments, we show that the proposed methodology is applicable to a wide range of time series observations with satisfying accuracy and efficiency. In addition, it is not only an alternative, but also a complement to fully parametric models such as the Markov switching model widely used in economics. 

Jie Ding, Shahin Shahrampour, Kathryn Heal, Vahid Tarokh, Analysis of Multistate Autoregressive Models. pdf