Dynamical mixture modeling with fast, automatic determination of Markov chains

Markov state modeling has gained popularity in various scientific fields due to its ability to reduce complex time series data into transitions between a few states. Yet, current frameworks are limited by assuming a single Markov chain describes the data, and they suffer an inability to discern heterogeneities. As a solution, this paper proposes a variational expectation-maximization algorithm that identifies a mixture of Markov chains in a time-series data set. The method is agnostic to the definition of the Markov states, whether data-driven (e.g. by spectral clustering) or based on domain knowledge. Variational EM efficiently and organically identifies the number of Markov chains and dynamics of each chain without expensive model comparisons or posterior sampling. The approach is supported by a theoretical analysis and numerical experiments, including simulated and observational data sets based on ${\tt Last.fm}$ music listening, ultramarathon running, and gene expression. The results show the new algorithm is competitive with contemporary mixture modeling approaches and powerful in identifying meaningful heterogeneities in time series data.