An Aggregation Algorithm for Markov Chains
Chapter 12 uses a Markov chain model (Nix and Vose 1992) of a complete finite-population EA with selection, mutation, and recombination. Each state of the Markov model is a particular population of the EA. If there are N states, then the Markov chain model is defined by an N × N matrix Q called the “one-step probability transition matrix,” where Q(i,j) is the probability of going from state i to state j in one step. The n-step (transient) behavior of the system is described by the nth power of Q, Q n . For EAs, the number of states grows enormously as the population size (or string length) increases (e.g., see Table 12.1), which can make the models computationally intensive. Our motivation for examining the Markov chain models was to explore the differences between mutation and recombination in EAs. However, due to the large size of the models, this chapter makes an excursion and introduces a novel technique for simplifying Markov models, in order to automatically reduce the number of states in the model.
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