Graph-based algorithms for phase-type distributions

Abstract

Phase-type distributions model the time until absorption in continuous or discrete-time Markov chains on a finite state space. The multivariate phase-type distributions have diverse and important applications by modeling rewards accumulated at visited states. However, even moderately-sized state spaces make the traditional matrix-based equations computationally infeasible. State spaces of phase-type distributions are often very large but sparse, with only a few transitions from a state. This sparseness makes a graph-based representation of the phase-type distribution more natural and efficient than the traditional matrix-based representation. In this paper, we develop graphbased algorithms for analyses of phase-type distributions. In addition to algorithms for state-space construction, reward transformation, and moments calculation, we give algorithms for the marginal distribution functions of multivariate phase-type distributions and for the state probability vector of the underlying Markov chains of both time-homogeneous and time-inhomogeneous phase-type distributions. The algorithms are available as a numerically stable and memory-efficient opensource software package written in C named ptdalgorithms. This library exposes all methods in the programming languages C and R. We demonstrate with a classic problem from population genetics how ptdalgorithms serves as a much faster, much simpler, and completely general modeling alternative.