Volume 29, Issue 2 pp. 141-156
Special Issue Paper

Improvement of expectation–maximization algorithm for phase-type distributions with grouped and truncated data

Hiroyuki Okamura

Corresponding Author

Hiroyuki Okamura

Department of Information Engineering, Graduate School of Engineering, Hiroshima University, 1-4-1 Kagamiyama, Higashi-Hiroshima, 739-8527 Japan

Correspondence to: Hiroyuki Okamura, Department of Information Engineering, Graduate School of Engineering, Hiroshima University, 1-4-1 Kagamiyama, Higashi-Hiroshima, 739-8527 Japan.

E-mail: [email protected]

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Tadashi Dohi

Tadashi Dohi

Department of Information Engineering, Graduate School of Engineering, Hiroshima University, 1-4-1 Kagamiyama, Higashi-Hiroshima, 739-8527 Japan

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Kishor S. Trivedi

Kishor S. Trivedi

Department of Electrical and Computer Engineering, Duke University, Hudson Hall, Durham, NC, 27707 USA

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First published: 24 April 2012
Citations: 19

Abstract

This paper proposes an improved expectation–maximization (EM) algorithm for phase-type (PH) distributions with grouped and truncated data. Olsson (1996) derived an EM algorithm for PH distributions under censored data, and the similar technique can be utilized to the PH fitting even under grouped and truncated data. However, it should be noted that Olsson's algorithm has a drawback in terms of computation speed. Because the time complexity of the algorithm is a cube of number of phases, it does not work well in the case where the number of phases is large. This paper proposes an improvement of the EM algorithm under grouped and truncated observations. By applying a uniformization-based technique for continuous-time Markov chains, it is shown that the time complexity of our algorithm can be reduced to the square of number of phases. In particular, when we consider the PH fitting using a canonical form of PH distributions, the time complexity is linear in the number of phases. Copyright © 2012 John Wiley & Sons, Ltd.

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