On Determining and Qualifying the Number of Superstates in Aggregation\n of Markov Chains

Many studies involving large Markov chains require determining a smaller\nrepresentative (aggregated) chains. Each {\\em superstate} in the representative\nchain represents a {\\em group of related} states in the original Markov chain.\nTypically, the choice of number of superstates in the aggregated chain is\nambiguous, and based on the limited prior know-how. In this paper we present a\nstructured methodology of determining the best candidate for the number of\nsuperstates. We achieve this by comparing aggregated chains of different sizes.\nTo facilitate this comparison we develop and quantify a notion of {\\em marginal\nreturn}. Our notion captures the decrease in the {\\em heterogeneity} within the\ngroup of the {\\em related} states (i.e., states represented by the same\nsuperstate) upon a unit increase in the number of superstates in the aggregated\nchain. We use Maximum Entropy Principle to justify the notion of marginal\nreturn, as well as our quantification of heterogeneity. Through simulations on\nsynthetic Markov chains, where the number of superstates are known apriori, we\nshow that the aggregated chain with the largest marginal return identifies this\nnumber. In case of Markov chains that model real-life scenarios we show that\nthe aggregated model with the largest marginal return identifies an inherent\nstructure unique to the scenario being modelled; thus, substantiating on the\nefficacy of our proposed methodology.\n

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