phsum: Sum of phase-type distributions

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

Description

The sum of two independent discrete or continuous phase-type distributed variables with initial distributions initDist1 and initDist2 as well as sub-transition/sub-intensity matrices equal to P_Mat1/T_Mat1 and P_Mat2/T_Mat2.

Usage

phsum(object1, object2)

Arguments

object1, object2

two objects of class discphasetype or contphasetype for which the sum should be computed.

Details

In the discrete case, the sum of two phase-type distributed variables tau1 ~ DPH_p(α,S) and tau2 ~ DPH_q(β,T) is again discrete phase-type distributed in the following way

tau1 + tau2 ~ DPH_{p+q}((α,0),cbind((S, s β),(0,T)) ).

In the continuous case, the sum of two phase-type distributed variables X ~ PH_p(α,S) and Y ~ PH_q(β,T) is again continuous and phase-type distributed in the following way

X + Y ~ PH_{p+q}((α,0),cbind((S, s β),(0,T)) ).

Value

The function phsum returns an object of type discphasetype or contphasetype (depending on the input) holding the phase-type representation of the sum of the input objects.

Source

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

See Also

sum.

Examples

## A simple example
phsum(T_MRCA$n5,T_Total$n5)

## For n=4, the total length of branches giving rise to
## singletons is phase-type distributed with initial distribution
initDist1 <- c(1,0,0)
## and sub-intensity rate matrix
T_Mat1 <- matrix(c(-1.5, 1.5, 0,
                   0, -1.5, 1,
                   0, 0, -1), nrow = 3, byrow = TRUE)
## The total length of branches giving rise to
## double-tons is phase-type distributed with initial distribution
initDist2 <- c(1,0)
## and sub-intensity rate matrix
T_Mat2 <- matrix(c(-3, 1,
                   0, -0.5), nrow = 2, byrow = TRUE)
## Defining two objects of type "contphasetype"
T1 <- contphasetype(initDist1, T_Mat1)
T2 <- contphasetype(initDist2, T_Mat2)
## Hence, the total length of branches giving rise to
## singletons and doubletons is phase-type distributed
## in the following way
phsum(T1,T2)
## (Please compare this distribution with the distribution
## obtained directly from the reward transformation)

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