Graph-based algorithms for phase-type distributions

• Published in: Statistics and computing Vol. 32; no. 6
• Main Authors: Røikjer, Tobias, Hobolth, Asger, Munch, Kasper
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2022; Springer Nature B.V
• Subjects: Algorithms; Artificial Intelligence; Computer Science; Distribution functions; Graphical representations; Markov analysis; Markov chains; Mathematical models; Multivariate analysis; OriginalPaper; Probability and Statistics in Computer Science; Programming languages; Run time (computers); Statistical Theory and Methods; Statistics and Computing/Statistics Programs; Phase-type distributions; Moments; Distribution; Computational statistics; Graph-based algorithms
• ISSN: 0960-3174; 1573-1375
• DOI: 10.1007/s11222-022-10174-3

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 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 graph-based algorithms for analyzing 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 open source software package written in C named ptdalgorithms. This library exposes all methods in the programming languages C and R. We compare the running time of ptdalgorithms to the fastest tools using a traditional matrix-based formulation. This comparison includes the computation of the probability distribution, which is usually computed by exponentiation of the sub-intensity or sub-transition matrix. We also compare time spent calculating the moments of (multivariate) phase-type distributions usually defined by inversion of the same matrices. The numerical results of our graph-based and traditional matrix-based methods are identical, and our graph-based algorithms are often orders of magnitudes faster. Finally, we demonstrate with a classic problem from population genetics how ptdalgorithms serves as a much faster, simpler, and completely general modeling alternative.