discretization: Discretizing a continuous phase-type distribution

In aumath-advancedr2019/PhaseTypeGenetics: The Phase-Type Distribution with Applications in Population Genetics

Description

Discretizes a continuous phase-type distribution with initial distribution initDist and sub-intensity rate matrix T_Mat.

Usage

discretization(object, a = NULL, lambda = NULL)

Arguments

object	a continuous phase-type distributed object of class contphasetype.
a	a constant that is larger than the maximum of all diagonal entries of the sub-intensity rate matrix.
lambda	the positive mutation rate at the locus.

Details

The relation between continuous and discrete phase-type distributions is given in the following way. If T is the sub-intensity rate matrix of a continuous phase-type distribution with representation PH(initDist,T), then there exists a constant a>0 such that P := I + 1/a * T is a sub-transition probability matrix and DPH(initDist, P) is a representation for a discrete phase-type distribution. This holds for any a larger than the maximum of all diagonal entries in T, as all entries in a sub-transition probability matrix have to be between zero and one. It even holds that for a genealogical model where the total branch length τ ~ PH(initDist, T) and the mutation rate at the locus is λ = θ/2, that the number of segregating sites S_{Total} plus one is discrete phase-type distributed with initial distribution initDist and sub-transition probability matrix P = (I-λ^{-1} * T)^{-1}, i.e.

S + 1 ~ DPH(initDist, P).

Value

Depending on the input, the function returns the discretized phase-type distribution with sub-transition probability matrix equal to either

P := I + 1/a * T

(if a is provided) or

P = (I-lambda^{-1} * T)^{-1}

(if λ is provided). If both a and λ are provided, the function returns both distributions in a list. In all three cases, the returned objects are of type discphasetype.

Source

Mogens Bladt and Bo Friis Nielsen (2017): Matrix-Exponential Distributions in Applied Probability. Probability Theory and Stochastic Modelling (Springer), Volume 81.

Examples

## For n=4, the total branch length is phase-type
## distributed with initial distribution
initDist <- c(1,0,0,0)
## and sub-intensity rate matrix
T_Mat <- matrix(c(-1.5, 1.5, 0, 0,
                  0, -1, 2/3, 1/3,
                  0, 0, -0.5, 0,
                  0, 0, 0, -0.5), nrow = 4, byrow = TRUE)

TTotal <- contphasetype(initDist,T_Mat)

## Hence, for theta=2, the number of segregating sites plus one is
## discrete phase-type distributed with the same initial
## distribution and sub-transition probability matrix
discretization(TTotal, lambda=1)$P_Mat

aumath-advancedr2019/PhaseTypeGenetics documentation built on Dec. 3, 2019, 7:16 a.m.