Mixture of Linear Additive Markov Processes: A Probabilistic Model for Joint Clustering and History-Dependent Transition Estimation

Abstract

In this paper, we propose a new probabilistic model called mixture of linear additive Markov processes (MoL) that can analyze sequential data such as histories of people's visit locations, songs sampled, and web-browsing tracks. The proposed method is constructed by combining linear additive Markov process (LAMP) and mixture models. By combining their benefits, the proposed model can capture long-range history-dependency of sequences as well as individual preference for each user. In addition, the steady state distribution (or equivalently, PageRank vector) of MoL can be obtained by power iteration similar to Markov Chain (MC). We also develop the MM algorithm for parameter estimation and give proof of its convergence. The algorithm enables us to use an exponential family as a component of LAMP and so, by using the distribution whose support lies in positive integers, we can also deal with infinite order LAMP and infinite order MoL. The effectiveness of MoL is confirmed by experiments on synthetic and real data.