Spectral Polynomial Algorithms for Computing Bi-Diagonal Representations for Phase Type Distributions and Matrix-Exponential Distributions

2006 ◽  
Vol 22 (2) ◽  
pp. 289-317 ◽  
Author(s):  
Qi-Ming He ◽  
Hanqin Zhang
Keyword(s):  
Matrix Exponential ◽  
Polynomial Algorithms ◽  
Exponential Distributions ◽  
Phase Type ◽  
Phase Type Distributions

A closure characterisation of phase-type distributions

1992 ◽  
Vol 29 (01) ◽  
pp. 92-103 ◽  
Author(s):  
Robert S. Maier ◽  
Colm Art O'Cinneide
Keyword(s):  
Analogous Result ◽  
Finite Automata ◽  
Automata Theory ◽  
Phase Type Distribution ◽  
Exponential Distributions ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Discrete Phase ◽  
Rational Sequence ◽  
Family Of Distributions

We characterise the classes of continuous and discrete phase-type distributions in the following way. They are known to be closed under convolutions, mixtures, and the unary ‘geometric mixture' operation. We show that the continuous class is the smallest family of distributions that is closed under these operations and contains all exponential distributions and the point mass at zero. An analogous result holds for the discrete class. We also show that discrete phase-type distributions can be regarded as ℝ+-rational sequences, in the sense of automata theory. This allows us to view our characterisation of them as a corollary of the Kleene–Schützenberger theorem on the behavior of finite automata. We prove moreover that any summable ℝ+-rational sequence is proportional to a discrete phase-type distribution.

scholarly journals On matrix exponential distributions

2007 ◽  
Vol 39 (1) ◽  
pp. 271-292 ◽  
Author(s):  
Qi-Ming He ◽  
Hanqin Zhang
Keyword(s):  
Distribution Function ◽  
Probability Distribution ◽  
Matrix Exponential ◽  
Exponential Distributions ◽  
Hankel Matrices ◽  
Phase Type ◽  
Types Of Information ◽  
Necessary And Sufficient ◽  
The Relationship ◽  
Derivatives Of

In this paper we introduce certain Hankel matrices that can be used to study ME (matrix exponential) distributions, in particular to compute their ME orders. The Hankel matrices for a given ME probability distribution can be constructed if one of the following five types of information about the distribution is available: (i) an ME representation, (ii) its moments, (iii) the derivatives of its distribution function, (iv) its Laplace-Stieltjes transform, or (v) its distribution function. Using the Hankel matrices, a necessary and sufficient condition for a probability distribution to be an ME distribution is found and a method of computing the ME order of the ME distribution developed. Implications for the PH (phase-type) order of PH distributions are examined. The relationship between the ME order, the PH order, and a lower bound on the PH order given by Aldous and Shepp (1987) is discussed in numerical examples.

A closure characterisation of phase-type distributions

1992 ◽  
Vol 29 (1) ◽  
pp. 92-103 ◽  
Author(s):  
Robert S. Maier ◽  
Colm Art O'Cinneide
Keyword(s):  
Analogous Result ◽  
Finite Automata ◽  
Automata Theory ◽  
Phase Type Distribution ◽  
Exponential Distributions ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Discrete Phase ◽  
Rational Sequence ◽  
Family Of Distributions

We characterise the classes of continuous and discrete phase-type distributions in the following way. They are known to be closed under convolutions, mixtures, and the unary ‘geometric mixture' operation. We show that the continuous class is the smallest family of distributions that is closed under these operations and contains all exponential distributions and the point mass at zero. An analogous result holds for the discrete class.We also show that discrete phase-type distributions can be regarded as ℝ+-rational sequences, in the sense of automata theory. This allows us to view our characterisation of them as a corollary of the Kleene–Schützenberger theorem on the behavior of finite automata. We prove moreover that any summable ℝ+-rational sequence is proportional to a discrete phase-type distribution.

Concentrated matrix exponential distributions with real eigenvalues

2021 ◽  
pp. 1-17
Author(s):  
András Mészáros ◽  
Miklós Telek
Keyword(s):  
Stochastic Models ◽  
Laplace Transformation ◽  
Matrix Exponential ◽  
Phase Type Distribution ◽  
Exponential Distributions ◽  
Inverse Laplace Transformation ◽  
Complex Eigenvalues ◽  
Phase Type ◽  
The Matrix ◽  
The Laplace Transform

Abstract Concentrated random variables are frequently used in representing deterministic delays in stochastic models. The squared coefficient of variation ( $\mathrm {SCV}$ ) of the most concentrated phase-type distribution of order $N$ is $1/N$ . To further reduce the $\mathrm {SCV}$ , concentrated matrix exponential (CME) distributions with complex eigenvalues were investigated recently. It was obtained that the $\mathrm {SCV}$ of an order $N$ CME distribution can be less than $n^{-2.1}$ for odd $N=2n+1$ orders, and the matrix exponential distribution, which exhibits such a low $\mathrm {SCV}$ has complex eigenvalues. In this paper, we consider CME distributions with real eigenvalues (CME-R). We present efficient numerical methods for identifying a CME-R distribution with smallest SCV for a given order $n$ . Our investigations show that the $\mathrm {SCV}$ of the most concentrated CME-R of order $N=2n+1$ is less than $n^{-1.85}$ . We also discuss how CME-R can be used for numerical inverse Laplace transformation, which is beneficial when the Laplace transform function is impossible to evaluate at complex points.

scholarly journals On matrix exponential distributions

2007 ◽  
Vol 39 (01) ◽  
pp. 271-292 ◽  
Author(s):  
Qi-Ming He ◽  
Hanqin Zhang
Keyword(s):  
Distribution Function ◽  
Probability Distribution ◽  
Matrix Exponential ◽  
Exponential Distributions ◽  
Hankel Matrices ◽  
Phase Type ◽  
Types Of Information ◽  
Necessary And Sufficient ◽  
The Relationship ◽  
Derivatives Of

In this paper we introduce certain Hankel matrices that can be used to study ME (matrix exponential) distributions, in particular to compute their ME orders. The Hankel matrices for a given ME probability distribution can be constructed if one of the following five types of information about the distribution is available: (i) an ME representation, (ii) its moments, (iii) the derivatives of its distribution function, (iv) its Laplace-Stieltjes transform, or (v) its distribution function. Using the Hankel matrices, a necessary and sufficient condition for a probability distribution to be an ME distribution is found and a method of computing the ME order of the ME distribution developed. Implications for the PH (phase-type) order of PH distributions are examined. The relationship between the ME order, the PH order, and a lower bound on the PH order given by Aldous and Shepp (1987) is discussed in numerical examples.

scholarly journals A Longitudinal Study of the Bladder Cancer Applying a State-Space Model with Non-Exponential Staying Time in States

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 363
Author(s):  
Delia Montoro-Cazorla ◽  
Rafael Pérez-Ocón ◽  
Alicia Pereira das Neves-Yedig
Keyword(s):  
Bladder Cancer ◽  
Longitudinal Study ◽  
State Space Model ◽  
Exponential Distributions ◽  
Significance Level ◽  
Phase Type Distributions ◽  
And Performance ◽  
The Times ◽  
Comparison Of The Results ◽  
Staying Time

A longitudinal study for 847 bladder cancer patients for a period of fifteen years is presented. After the first surgery, the patients undergo successive ones (recurrences). A state-model is selected for analyzing the evolution of the cancer, based on the distribution of the times between recurrences. These times do not follow exponential distributions, and are approximated by phase-type distributions. Under these conditions, a multidimensional Markov process governs the evolution of the disease. The survival probability and mean times in the different states (levels) of the disease are calculated empirically and also by applying the Markov model, the comparison of the results indicate that the model is well-fitted to the data to an acceptable significance level of 0.05. Two sub-cohorts are well-differenced: those reaching progression (the bladder is removed) and those that do not. These two cases are separately studied and performance measures calculated, and the comparison reveals details about the characteristics of the patients in these groups.

Phase Type Distributions with Finite Support

2014 ◽  
Vol 30 (4) ◽  
pp. 576-597 ◽  
Author(s):  
V. Ramaswami ◽  
N. C. Viswanath
Keyword(s):  
Finite Support ◽  
Phase Type ◽  
Phase Type Distributions

Analysis of R-out-of-N repairable systems: the case of phase-type distributions

2004 ◽  
Vol 36 (1) ◽  
pp. 116-138 ◽  
Author(s):  
Yonit Barron ◽  
Esther Frostig ◽  
Benny Levikson
Keyword(s):  
Repairable System ◽  
Repairable Systems ◽  
Regenerative Processes ◽  
Numerical Examples ◽  
Independent Components ◽  
Duality Results ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Two Systems ◽  
Repair Facility

An R-out-of-N repairable system, consisting of N independent components, is operating if at least R components are functioning. The system fails whenever the number of good components decreases from R to R-1. A failed component is sent to a repair facility. After a failed component has been repaired it is as good as new. Formulae for the availability of the system using Markov renewal and semi-regenerative processes are derived. We assume that either the repair times of the components are generally distributed and the components' lifetimes are phase-type distributed or vice versa. Some duality results between the two systems are obtained. Numerical examples are given for several distributions of lifetimes and of repair times.

scholarly journals The Algebraic Degree of Phase-Type Distributions

2010 ◽  
Vol 47 (03) ◽  
pp. 611-629
Author(s):  
Mark Fackrell ◽  
Qi-Ming He ◽  
Peter Taylor ◽  
Hanqin Zhang
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Polynomial Algorithm ◽  
Algebraic Degree ◽  
Nonnegative Matrices ◽  
Stieltjes Transform ◽  
Special Cases ◽  
Phase Type ◽  
Phase Type Distributions ◽  
The Matrix

This paper is concerned with properties of the algebraic degree of the Laplace-Stieltjes transform of phase-type (PH) distributions. The main problem of interest is: given a PH generator, how do we find the maximum and the minimum algebraic degrees of all irreducible PH representations with that PH generator? Based on the matrix exponential (ME) order of ME distributions and the spectral polynomial algorithm, a method for computing the algebraic degree of a PH distribution is developed. The maximum algebraic degree is identified explicitly. Using Perron-Frobenius theory of nonnegative matrices, a lower bound and an upper bound on the minimum algebraic degree are found, subject to some conditions. Explicit results are obtained for special cases.

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