Deconvolutive Clustering of Markov States

In this paper we formulate the problem of grouping the states of a discrete Markov chain of arbitrary order simultaneously with decon- volving its transition probabilities. As the name indicates, this problem is related to deconvolutive blind signal separation. However, whilst the latter has been studied in the context of continuous signal processing, e.g. as a model of a real-room mixing of sound signals, our technique tries to model computer-mediated group-discussion participation from a discrete event-log sequence. In this context, convolution occurs due to various time-delay factors, such as the network transmission bandwidth or simply the typing speed of the participants. We derive a computation- ally ecien t maximum likelihood estimation algorithm associated with our model, which exploits the sparsity of state transitions and scales linearly with the number of observed higher order transition patterns. Results obtained on a full day worth dynamic real-world Internet Relay Chat participation sequence demonstrate the advantages of our approach over state grouping alone, both in terms of penalised data likelihood and cluster clarity. Other potential applications of our model, viewed as a novel compact approximation of large Markov chains, are also discussed.