Analyzing Markov Chains using Kronecker Products by Tuǧrul Dayar

Author:Tuǧrul Dayar
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

We see in the following section and next chapter that given the state-space size, , of each subsystem for h = 1, …, H, the number of possibilities available in choosing the number of partitions, N, is large enough to accommodate a representation of the state space without unreachable states. Once N is determined, the CTMC is represented as an (N ×N) block matrix, with block (p, w) being defined as a sum of Kronecker products capturing the transitions from states in to states in for p, w = 1, …, N. This is essentially the idea behind HMMs [19]. All iterative methods discussed in Chap. 3 are implemented in the APNN toolbox [7] for an (N ×N) block matrix whose blocks are sums of Kronecker products. The off-diagonal blocks of this block matrix need not be square, hence the vector–Kronecker product multiplication algorithm in Sect. 2.2 involving rectangular factors.

We now turn to a class of problems that can be treated with approximative decompositional iterative methods using the understanding developed in this section.

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