# PhaseType R package :package:

This is a package for working with Phase-type (PHT) distributions in the R programming language. The entire of the MCMC portion of the code has been written in optimised C for higher performance and very low memory use, whilst being easy to call from wrapper R functions.

## Definition of a Phase-type Distribution

Consider a continuous-time Markov chain (CTMC) on a finite discrete state space of size $$n+1$$, where one of the states is absorbing. Without loss of generality the generator of the chain can be written in the form:

$\mathbf{T} = \left( \begin{array}{cc} \mathbf{S} & \mathbf{s} \\ \mathbf{0}^\mathrm{T} & 0 \end{array} \right)$

where $$\mathbf{S}$$ is the $$n \times n$$ matrix of transition rates between non-absorbing states; $$\mathbf{s}$$ is an $$n$$ dimensional vector of absorption rates; and $$\mathbf{0}$$ is an $$n$$ dimensional vector of zeros. We take $$\boldsymbol{\pi}$$ as the initial state distribution: an $$n$$ dimensional vector of probabilities $$\left(\sum_i \pi_i=1\right)$$ such that $$\pi_i$$ is the probability of the chain starting in state $$i$$.

Then, we define a Phase-type distribution to be the distribution of the time to absorption of the CTMC with generator $$\mathbf{T}$$, or equivalently as the first passage time to state $$n+1$$. Thus, a Phase-type distribution is a positively supported univariate distribution having distribution and density functions:

$\begin{array}{rcl} F_X(x) &=& 1 - \boldsymbol{\pi}^\mathrm{T} \exp\\{x \mathbf{S}\\} \mathbf{e}\\ f_X(x) &=& \boldsymbol{\pi}^\mathrm{T} \exp\\{x \mathbf{S}\\} \mathbf{s} \end{array} \qquad \mbox{for } x \in [0,\infty)$

where $$\mathbf{e}$$ is an $$n$$ dimensional vector of $$1$$’s; $$x$$ is the time to absorption (or equivalently first-passage time to state $$n+1$$); and $$\exp\\{x \mathbf{S}\\}$$ is the matrix exponential. We denote that a random variable $$X$$ is Phase-type distributed with parameters $$\boldsymbol{\pi}$$ and $$\mathbf{T}$$ by $$X \sim \mathrm{PHT}(\boldsymbol{\pi},\mathbf{T})$$.

Note that $$\displaystyle \sum_{j=1}^n S_{ij} = -s_i \ \forall\\,i$$, so often a Phase-type is defined merely by providing $$\mathbf{S}$$, $$\mathbf{T}$$ then being implicitly known.

## Install

You can install the latest release directly from CRAN.

install.packages("PhaseType")

Installing directly from GitHub is not supported by the install.packages command. You could use the devtools package to install the development version if desired.

install.packages("remotes")
remotes::install_github("louisaslett/PhaseType")

Under releases, the tree/commit from which CRAN releases were made are recorded, so historic source can be downloaded from there.

## Citation

If you use this software, please cite the following:

Aslett, L. J. M. (2012), MCMC for Inference on Phase-type and Masked System Lifetime Models, PhD thesis, Trinity College Dublin.

@phdthesis{Aslett2012,
}