Mean First Passage Time for irreducible Markov chains — meanFirstPassageTime • markovchain

Given an irreducible (ergodic) markovchain object, this function calculates the expected number of steps to reach other states

meanFirstPassageTime(object, destination)

Arguments

object: the markovchain object
destination: a character vector representing the states respect to which we want to compute the mean first passage time. Empty by default

Value

a Matrix of the same size with the average first passage times if destination is empty, a vector if destination is not

Details

For an ergodic Markov chain it computes:

If destination is empty, the average first time (in steps) that takes the Markov chain to go from initial state i to j. (i, j) represents that value in case the Markov chain is given row-wise, (j, i) in case it is given col-wise.
If destination is not empty, the average time it takes us from the remaining states to reach the states in destination

References

C. M. Grinstead and J. L. Snell. Introduction to Probability. American Mathematical Soc., 2012.

Author

Toni Giorgino, Ignacio Cordón

Examples

m <- matrix(1 / 10 * c(6,3,1,
                       2,3,5,
                       4,1,5), ncol = 3, byrow = TRUE)
mc <- new("markovchain", states = c("s","c","r"), transitionMatrix = m)
meanFirstPassageTime(mc, "r")
#>        s        c 
#> 4.545455 2.727273 


# Grinstead and Snell's "Oz weather" worked out example
mOz <- matrix(c(2,1,1,
                2,0,2,
                1,1,2)/4, ncol = 3, byrow = TRUE)

mcOz <- new("markovchain", states = c("s", "c", "r"), transitionMatrix = mOz)
meanFirstPassageTime(mcOz)
#>          s c        r
#> s 0.000000 4 3.333333
#> c 2.666667 0 2.666667
#> r 3.333333 4 0.000000