Transformation of Phase-Type Distributions via Rewards

Transform a variable following a phase-type distribution according to a non-negative reward vector.

reward_phase_type(phase_type, reward)

Arguments

phase_type

an object of class cont_phase_type or disc_phase_type.

reward

a vector of the same length as the number of states. The vector should contain non-negative values. Rewards for the discrete phase-type distribution can only be integers.

Value

An object of class disc_phase_type or cont_phase_type.

Details

For the reward transformation for continuous phase-type distribution, the transformation will be performed as presented in the book of Bladt and Nielsen (2017).

For the discrete phase_type distribution is based on the PhD of Navarro (2018) and Hobolth, Bladt and Andersen (2021).

References

Bladt, M., & Nielsen, B. F. (2017). *Matrix-exponential distributions in applied probability* (Vol. 81). New York: Springer.

Campillo Navarro, A. (2018). *Order statistics and multivariate discrete phase-type distributions*. DTU Compute. DTU Compute PHD-2018, Vol.. 492

Hobolth, A., Bladt, M. & Andersen, L.A. (2021). *Multivariate phase-type theory for the site frequency spectrum*. ArXiv.

See also

PH, DPH

Examples

##===========================##
## For continuous phase-type ##
##===========================##

subint_mat <- matrix(c(-3, 1, 1,
                      2, -3, 0,
                      1, 1, -3), ncol = 3)
init_probs <- c(0.9, 0.1, 0)
ph <- PH(subint_mat, init_probs)
reward <- c(0.5, 0, 4)

reward_phase_type(ph, reward)
#> $subint_mat
#>           [,1]      [,2]
#> [1,] -4.666667  3.333333
#> [2,]  0.250000 -0.750000
#> 
#> $init_probs
#>           [,1]       [,2]
#> [1,] 0.9333333 0.03333333
#> 
#> $defect
#> [1] 0.03333333
#> 
#> attr(,"class")
#> [1] "cont_phase_type"

##=========================##
## For discrete phase-type ##
##=========================##

subint_mat <- matrix(c(0.4, 0, 0,
                      0.24, 0.4, 0,
                      0.12, 0.2, 0.5), ncol = 3)
init_probs <- c(0.9, 0.1, 0)
ph <- DPH(subint_mat, init_probs)

reward <- c(1, 0, 4)

reward_phase_type(ph, reward)
#> $subint_mat
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,]  0.4  0.2    0    0    0
#> [2,]  0.0  0.0    1    0    0
#> [3,]  0.0  0.0    0    1    0
#> [4,]  0.0  0.0    0    0    1
#> [5,]  0.0  0.5    0    0    0
#> 
#> $init_probs
#>      [,1]       [,2] [,3] [,4] [,5]
#> [1,]  0.9 0.03333333    0    0    0
#> 
#> $defect
#> [1] 0.06666667
#> 
#> attr(,"class")
#> [1] "disc_phase_type"