Simple, fast and accurate evaluation of the action of the exponential of a rate matrix on a probability vector
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Given a time-homogeneous, finite-statespace Markov chain with a rate matrix, or infinitesimal generator of Q, an initial distribution of ν and a time interval of t, the distribution across states at time t, ν^ (t):=ν^ (Q t) is typically required for one or many time intervals, t, either as a contribution to the likelihood to enable statistical inference, or to understand the evolution of the process itself. When, as in many statistical applications in, for example, epidemics, systems biology and ecology, Q arises from a reaction network, or when Q corresponds to a negative graph Laplacian, it is usually sparse. Building on a relatively recent development for the action of the exponential of a general sparse matrix on a vector, we use the special properties of rate matrices to create the Single Positive Series method, which accurately and quickly evaluates the distribution across states at a single time point in O((r+1)ρ d) operations, where d is the dimension of the statespace, ρ=_i=1,...,d|Q_ii| and r is the average number of positive entries in the rows of Q. We also present the Multiple-Use Single Positive Series algorithm which simultaneously evaluates the distribution vector at k different time points in O((k+r)ρ d) operations. We demonstrate across a range of examples that the Single Positive Series method is both more accurate and more than an order of magnitude more efficient than the nearest generic and generally available competitor, and that the Multiple-Use Single Positive Series algorithm improves upon this still further.
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