Distribution of the first passage times of a Markov chain
started in state: for every state \(j\) and step \(k\), the
probability that the chain enters \(j\) for the first time at step
\(k\).
Arguments
- object
A
markovchainobject.- state
Name of the initial state.
- n
Number of steps, a positive integer.
Value
A matrix with n rows (steps \(1, \dots, n\)) and one column per
state: entry \((k, j)\) is the probability that the first passage to
\(j\) happens at step \(k\). The column sums are at most one; the
remainder is the probability of a first passage after step n, or never.
Details
For \(j\) equal to state the probabilities are those of the
first return. The algorithm follows the Matlab listings of Feres.
References
Feres, R. Notes for Math 450: Matlab listings for Markov chains. Washington University in St. Louis.
See also
Other first passage and hitting:
absorptionProbabilities(),
committorAB(),
firstPassageMultiple(),
hittingProbabilities(),
meanAbsorptionTime(),
meanFirstPassageTime(),
meanNumVisits(),
meanRecurrenceTime(),
noofVisitsDist()
Examples
simpleMc <- new("markovchain", states = c("a", "b"),
transitionMatrix = matrix(c(0.4, 0.6, 0.3, 0.7),
nrow = 2, byrow = TRUE))
firstPassage(simpleMc, "b", 20)
#> a b
#> 1 0.3000000000 7.000000e-01
#> 2 0.2100000000 1.800000e-01
#> 3 0.1470000000 7.200000e-02
#> 4 0.1029000000 2.880000e-02
#> 5 0.0720300000 1.152000e-02
#> 6 0.0504210000 4.608000e-03
#> 7 0.0352947000 1.843200e-03
#> 8 0.0247062900 7.372800e-04
#> 9 0.0172944030 2.949120e-04
#> 10 0.0121060821 1.179648e-04
#> 11 0.0084742575 4.718592e-05
#> 12 0.0059319802 1.887437e-05
#> 13 0.0041523862 7.549747e-06
#> 14 0.0029066703 3.019899e-06
#> 15 0.0020346692 1.207960e-06
#> 16 0.0014242685 4.831838e-07
#> 17 0.0009969879 1.932735e-07
#> 18 0.0006978915 7.730941e-08
#> 19 0.0004885241 3.092376e-08
#> 20 0.0003419669 1.236951e-08