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.
phsum(object1, object2)
| object1, object2 | two objects of class |
|---|
Mogens Bladt and Bo Friis Nielsen (2017): Matrix-Exponential Distributions in Applied Probability. Probability Theory and Stochastic Modelling (Springer), Volume 81.
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.
In the discrete case, the sum of two phase-type distributed variables \(tau1 ~ DPH_p(\alpha,S)\) and \(tau2 ~ DPH_q(\beta,T)\) is again discrete phase-type distributed in the following way $$tau1 + tau2 ~ DPH_{p+q}((\alpha,0),cbind((S, s \beta),(0,T)) ).$$ In the continuous case, the sum of two phase-type distributed variables \(X ~ PH_p(\alpha,S)\) and \(Y ~ PH_q(\beta,T)\) is again continuous and phase-type distributed in the following way $$X + Y ~ PH_{p+q}((\alpha,0),cbind((S, s \beta),(0,T)) ).$$
sum.
## A simple example phsum(T_MRCA$n5,T_Total$n5)#> $initDist #> [1] 1 0 0 0 0 0 0 0 #> #> $T_Mat #> [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] #> [1,] -10 10 0 0 0 0.0 0.0 0.0 #> [2,] 0 -6 6 0 0 0.0 0.0 0.0 #> [3,] 0 0 -3 3 0 0.0 0.0 0.0 #> [4,] 0 0 0 -1 1 0.0 0.0 0.0 #> [5,] 0 0 0 0 -2 2.0 0.0 0.0 #> [6,] 0 0 0 0 0 -1.5 1.5 0.0 #> [7,] 0 0 0 0 0 0.0 -1.0 1.0 #> [8,] 0 0 0 0 0 0.0 0.0 -0.5 #> #> attr(,"class") #> [1] "contphasetype"## 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)#> $initDist #> [1] 1 0 0 0 0 #> #> $T_Mat #> [,1] [,2] [,3] [,4] [,5] #> [1,] -1.5 1.5 0 0.0 0.0 #> [2,] 0.0 -1.5 1 0.5 0.0 #> [3,] 0.0 0.0 -1 1.0 0.0 #> [4,] 0.0 0.0 0 -3.0 1.0 #> [5,] 0.0 0.0 0 0.0 -0.5 #> #> attr(,"class") #> [1] "contphasetype"## (Please compare this distribution with the distribution ## obtained directly from the reward transformation)