On 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 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 SISC vol. 32(1):62-83 ) with Bayesian approach and MCMC. Conceptual advantage of the presented combined framework is that it allows to address the two main problems of the existent clustering methods, e.g. the non-convexity and the ill-posedness of the respective functionals, in a unified way. Clustering of the time series and minimization of the regularized clustering functional is based on 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 the ones obtained by the standard methods. As demonstrated in numerical examples, presented Bayesian formulation of the regularized clustering problem allows to avoid the locality of the obtained minimizers and to calculate the a priori confidence intervals for the clustering results, even in the case of very ill-posed problems with overlapping clusters.