An Adaptive Markov Chain Monte Carlo Approach to Time Series Clustering of Processes with Regime Transition Behavior

A numerical framework for clustering of time series via a Markov chain Monte Carlo (MCMC) method is presented. It combines concepts from recently introduced variational time series analysis and regularized clustering functional minimization [I. Horenko, SIAM J. Sci. Comput., 32 (2010), pp. 62--83] with MCMC. A conceptual advantage of the presented combined framework is that it allows us to address the two main problems of the existent clustering methods, e.g., the nonconvexity and the ill-posedness of the respective functionals, in a unified way. Clustering of the time series and minimization of the regularized clustering functional are based on the generation of samples from an appropriately chosen Boltzmann distribution in the space of cluster affiliation paths using simulated annealing and the Metropolis algorithm. The presented method is applied to sets of generic ill-posed clustering problems, and the results are compared to those obtained by the standard methods. As demonstrated in numerical examples, the presented MCMC formulation of the regularized clustering problem allows us to avoid the locality of the obtained minimizers, provides good clustering results even for very ill-posed problems with overlapping clusters, and is the computationally superior method for long time series. [PUBLICATION ABSTRACT]