PhaseTypeDistributions.jl

A Julia package for working with phase-type (PH) distributions and their multi-absorbing generalization (MAPH). PH distributions are implemented as subtypes of Distributions.ContinuousUnivariateDistribution, integrating with the Julia statistics ecosystem.

The package accompanies the paper Inference for Multi-Absorbing Phase Type Distributions, Algorithms, and Applications (Qiao, Surya, Asanjarani, Nazarathy; in preparation).

Background

A continuous-time PH distribution is the distribution of the absorption time of a finite-state continuous-time Markov chain with a single absorbing state. Concretely, given an initial distribution α over m transient phases and a sub-generator matrix T, the absorption time τ has

\[F_τ(x) = 1 - α^⊤ \exp(Tx) \mathbf{1}, \qquad f_τ(x) = α^⊤ \exp(Tx) t^0, \qquad t^0 = -T \mathbf{1}.\]

The MAPH generalization replaces the single absorbing state with n distinct ones; the joint distribution of (τ, κ) (absorption time and the index of the absorbing state reached) is the natural model for competing risks.

Installation

using Pkg
Pkg.add(url = "https://github.com/Julia-Matrix-Analytic-Probability/PhaseTypeDistributions.jl")

Scope

Continuous-time, finite state space, fully observed absorption times — for PH and MAPH.

Type	Description
`PHDist`	General PH from `(α, T)`
`HyperExponentialDist`	Mixture of exponentials (SCV ≥ 1)
`HypoExponentialDist`	Convolution of exponentials (SCV ≤ 1)
`ErlangPHDist`	`k` phases with equal rate `λ`
`CoxianDist`	Sequential phases with per-phase exit probs
`MAPHDist`	Multi-absorbing PH: joint distribution of (τ,κ)

Quick start

using PhaseTypeDistributions
using Distributions

ph  = PHDist([0.6, 0.4], [-3.0 1.0; 0.0 -2.0])
mean(ph), var(ph), pdf(ph, 1.0), cdf(ph, 1.0)

(0.5, 0.25, 0.27067056647322535, 0.8646647167633873)

he  = HyperExponentialDist(2.0, 3.0)        # by mean and SCV
ho  = HypoExponentialDist(2.0, 0.5)
er  = ErlangPHDist(3, 2.0)
cox = CoxianDist([3.0, 4.0, 5.0], [0.2, 0.3])

mean.([he, ho, er, cox])

4-element Vector{Float64}:
 2.0
 2.0
 1.5
 0.6453333333333333

α = [0.6, 0.4]
T = [-3.0 1.0; 0.5 -2.0]
D = [1.5 0.5; 1.0 0.5]
d = MAPHDist(α, T, D)
marginal_absorption(d), rand(d)

([0.709090909090909, 0.2909090909090909], (0.29851058127056845, 2))

PH distributions — walk through every PH type and the full API.
MAPH distributions — MAPH type, joint distribution API, bridges to PH.
API reference — index of every documented name.

Not yet supported