Maximum Pseudolikelihood Estimation for a Model-Based Clustering of time series Data

Mixture of autoregressions (MoAR) models provide a model-based approach to the clustering of time series data. The maximum likelihood (ML) estimation of MoAR models requires the evaluation of products of large numbers of densities of normal random variables. In practical scenarios, these products converge to zero as the length of the time series increases, and thus the ML estimation of MoAR models becomes unfeasible in such scenarios. We propose a maximum pseudolikelihood (MPL) estimation approach to overcome the product problem. The MPL estimator is proved to be consistent and can be computed via an MM (minorization--maximization) algorithm. The MM algorithm is proved to monotonically increase the pseudolikelihood (PL) function and to be globally convergent to a stationary point of the log-PL function. Simulations are used to assess the performance of the MPL estimator against that of the ML estimator in cases where the latter was able to be calculated. An application to the clustering of time series arising from a resting-state fMRI experiment is presented as a demonstration of the methodology.