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.

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.

Please feel free to:

- submit suggestions and bug-reports at: https://github.com/louisaslett/PhaseType/issues
- compose an e-mail to: louis.aslett@durham.ac.uk

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")
::install_github("louisaslett/PhaseType") remotes
```

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

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,
title={MCMC for Inference on Phase-type and Masked System Lifetime Models},
author={Aslett, L. J. M.},
year={2012},
school={Trinity College Dublin}
}
```

Thank-you :smiley: