Phase-type distribution

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Phase-type
Probability density function
Cumulative distribution function
Parameters S,\; m\times m subgenerator matrix
\boldsymbol{\alpha}, probability row vector
Support x \in [0; \infty)\!
Probability density function (pdf) \boldsymbol{\alpha}e^{xS}\boldsymbol{S}^{0}
See article for details
Cumulative distribution function (cdf) 1-\boldsymbol{\alpha}e^{xS}\boldsymbol{1}
Mean -1\boldsymbol{\alpha}{S}^{-1}\mathbf{1}
Median no simple closed form
Mode no simple closed form
Variance 2\boldsymbol{\alpha}{S}^{-2}\mathbf{1}
Skewness -6\boldsymbol{\alpha}{S}^{-3}\mathbf{1}/\sigma^{3}
Excess kurtosis 24\boldsymbol{\alpha}{S}^{-4}\mathbf{1}/\sigma^{4}
Entropy
Moment-generating function (mgf) \boldsymbol{\alpha}(-tI-S)^{-1}\boldsymbol{S}^{0}+\alpha_{m+1}
Characteristic function \boldsymbol{\alpha}(itI-S)^{-1}\boldsymbol{S}^{0}+\alpha_{m+1}

A phase-type distribution is a probability distribution that results from a system of one or more inter-related Poisson processes occurring in sequence, or phases. The sequence in which each of the phases occur may itself be a stochastic process. The distribution can be represented by a random variable describing the time until absorption of a Markov process with one absorbing state. Each of the states of the Markov process represents one of the phases.

It has a discrete time equivalent the discrete phase-type distribution.

The phase-type distribution is dense in the field of all positive-valued distributions, that is, it can be used to approximate any positive valued distribution.

Contents

[edit] Definition

There exists a continuous-time Markov process with m + 1 states, where m\geq1. The states 1,\dots,m are transient states and state m + 1 is an absorbing state. The process has an initial probability of starting in any of the m + 1 phases given by the probability vector (\boldsymbol{\alpha},\alpha_{m+1}).

The continuous phase-type distibution is the distribution of time from the processes starting until absorption in the absorbing state.

This process can be written in the form of a transition rate matrix,

{Q}=\left[\begin{matrix}{S}&\mathbf{S}^0\\\mathbf{0}&0\end{matrix}\right],

where S is a m\times m matrix and \mathbf{S}^0=-{S}\mathbf{1}. Here \mathbf{1} represents an m\times 1 vector with every element being 1.

[edit] Characterization

The distribution of time X until the process reaches the absorbing state is said to be phase-type distributed and is denoted \operatorname{PH}(\boldsymbol{\alpha},{S}).

The distribution function of X is given by,

F(x)=1-\boldsymbol{\alpha}\exp({S}x)\mathbf{1},

and the density function,

f(x)=\boldsymbol{\alpha}\exp({S}x)\mathbf{S^{0}},

for all x > 0, where \exp(\cdot) is the matrix exponential. It is usually assumed the probability of process starting in the absorbing state is zero. The moments of the distribution function are given by,

E[X^{n}]=(-1)^{n}n!\boldsymbol{\alpha}{S}^{-n}\mathbf{1}.

[edit] Special cases

The following probability distributions are all considered special cases of a continuous phase-type distribution:

  • Degenerate distribution, point mass at zero or the empty phase-type distribution - 0 phases.
  • Exponential distribution - 1 phase.
  • Erlang distribution - 2 or more identical phases in sequence.
  • Deterministic distribution (or constant) - The limiting case of an Erlang distribution, as the number of phases become infinite, while the time in each state becomes zero.
  • Coxian distribution - 2 or more (not necessarily identical) phases in sequence, with a probability of transitioning to the terminating/absorbing state after each phase.
  • Hyper-exponential distribution (also called a mixture of exponential) - 2 or more non-identical phases, that each have a probability of occurring in a mutually exclusive, or parallel, manner. (Note: The exponential distribution is the degenerate situation when all the parallel phases are identical.)
  • Hypoexponential distribution - 2 or more phases in sequence, can be non-identical or a mixture of identical and non-identical phases, generalises the Erlang.

As the phase-type distribution is dense in the field of all positive-valued distributions, we can represent any positive valued distribution. However, the phase-type is a light-tailed or platikurtic distribution. So the representation of heavy-tailed or leptokurtic distribution by phase type is an approximation, even if the precision of the approximation can be as good as we want.

[edit] Examples

In all the following examples it is assumed that there is no probability mass at zero, that is αm + 1 = 0.

[edit] Exponential distribution

The simplest non-trivial example of a phase-type distribution is the exponential distribution of parameter λ. The parameter of the phase-type distribution are : \boldsymbol{S}=-\lambda and \boldsymbol{\alpha} =1

[edit] Hyper-exponential or mixture of exponential distribution

The mixture of exponential or hyper-exponential distribution with parameter (α1,α2,α3,α4,α5) (such that \sum \alpha_i =1 and ,\alpha_i > 0 \forall i) and (λ1,λ2,λ3,λ4,λ5) can be represented as a phase type distribution with

\boldsymbol{\alpha}=(\alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_5),

and

{S}=\left[\begin{matrix}-\lambda_1&0&0&0&0\\0&-\lambda_2&0&0&0\\0&0&-\lambda_3&0&0\\0&0&0&-\lambda_4&0\\0&0&0&0&-\lambda_5\\\end{matrix}\right].

The mixture of exponential can be characterized through its density

f(x)=\sum_{i=1}^5 \alpha_i \lambda_i e^{-\lambda_i x}

or its distribution function

F(x)=1-\sum_{i=1}^5 \alpha_i e^{-\lambda_i x}.

This can be generalized to a mixture of n exponential distributions.

[edit] Erlang distribution

The Erlang distribution has two parameters, the shape an integer k > 0 and the rate λ > 0. This is sometimes denoted E(k,λ). The Erlang distribution can be written in the form of a phase-type distribution by making S a k\times k matrix with diagonal elements − λ and super-diagonal elements λ, with the probability of starting in state 1 equal to 1. For example E(5,λ),

\boldsymbol{\alpha}=(1,0,0,0,0),

and

{S}=\left[\begin{matrix}-\lambda&\lambda&0&0&0\\0&-\lambda&\lambda&0&0\\0&0&-\lambda&\lambda&0\\0&0&0&-\lambda&\lambda\\0&0&0&0&-\lambda\\\end{matrix}\right].

The hypoexponential distribution is a generalisation of the Erlang distribution by having different rates for each transition (the non-homogeneous case).

[edit] Mixture of Erlang distribution

The mixture of two Erlang distribution with parameter E(3,β1), E(3,β2) and (α1,α2) (such that α1 + α2 = 1 and \forall i,\alpha_i \geq 0 ) can be represented as a phase type distribution with

\boldsymbol{\alpha}=(\alpha_1,0,0,\alpha_2,0,0),

and

{S}=\left[\begin{matrix} -\beta_1&\beta_1&0&0&0&0\\ 0&-\beta_1&\beta_1&0&0&0\\ 0&0&-\beta_1&0&0&0\\ 0&0&0&-\beta_2&\beta_2&0\\ 0&0&0&0&-\beta_2&\beta_2\\ 0&0&0&0&0&-\beta_2\\ \end{matrix}\right].

[edit] Coxian distribution

The Coxian distribution is a generalisation of the hypoexponential. Instead of only being able to enter the absorbing state from state k it can be reached from any phase. The phase-type representation is given by,

S=\left[\begin{matrix}-\lambda_{1}&p_{1}\lambda_{1}&0&\dots&0&0\\ 0&-\lambda_{2}&p_{2}\lambda_{2}&\ddots&0&0\\ \vdots&\ddots&\ddots&\ddots&\ddots&\vdots\\ 0&0&\ddots&-\lambda_{k-2}&p_{k-2}\lambda_{k-2}&0\\ 0&0&\dots&0&-\lambda_{k-1}&p_{k-1}\lambda_{k-1}\\ 0&0&\dots&0&0&-\lambda_{k} \end{matrix}\right]

and

\boldsymbol{\alpha}=(1,0,\dots,0),

where 0<p_{1},\dots,p_{k-1}\leq 1, in the case where all pi = 1 we have the hypoexponential distribution. The Coxian distribution is extremly important as any acyclic phase-type distribution has an equivalent Coxian representation.

The generalised Coxian distribution relaxes the condition that requires starting in the first phase.

[edit] See also

[edit] References

  • M. F. Neuts. Matrix-Geometric Solutions in Stochastic Models: an Algorthmic Approach, Chapter 2: Probability Distributions of Phase Type; Dover Publications Inc., 1981.
  • G. Latouche, V. Ramaswami. Introduction to Matrix Analytic Methods in Stochastic Modelling, 1st edition. Chapter 2: PH Distributions; ASA SIAM, 1999.
  • C. A. O'Cinneide (1990). Characterization of phase-type distributions. Communications in Statistics: Stocahstic Models, 6(1), 1-57.
  • C. A. O'Cinneide (1999). Phase-type distribution: open problems and a few properties, Communication in Statistic: Stochastic Models, 15(4), 731-757.
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