Data-driven learning of the number of states in multi-state autoregressive models

In this work, we consider the class of multi-state autoregressive processes that can be used to model non-stationary time series of interest. In order to capture different autoregressive (AR) states underlying an observed time series, it is crucial to select the appropriate number of states. We propose a new and intuitive model selection technique based on the Gap statistics, which uses a null reference distribution on the stable AR filters to identify whether adding a new AR state significantly improves the performance of the model. To that end, we define a new distance measure between two AR filters based on the mean squared prediction error, and propose an efficient method to generate stable filters that are uniformly distributed in the coefficient space. Numerical results are provided to evaluate the performance of the proposed approach.