Mean first passage times

The mean first passage time from class \(i\) to class \(j\) is the expected number of periods for a unit currently in \(i\) to first reach \(j\). Together with the transition matrix these times summarise how quickly a distribution mixes. The diagonal holds the mean recurrence time \(1/\pi_j\) (the expected return time to a class), where \(\pi\) is the ergodic distribution.

Computed with the Kemeny-Snell fundamental matrix \(Z = (I - P + A)^{-1}\), where \(A\) has every row equal to \(\pi\); then \(M_{ij} = (\delta_{ij} - Z_{ij} + Z_{jj}) / \pi_j\).

Usage

mfpt(x, ...)

# Default S3 method
mfpt(x, ...)

# S3 method for class 'sddr_markov'
mfpt(x, ...)

Arguments

x

A row-stochastic matrix, or an object of class sddr_markov.

...

Currently unused.

Value

A numeric matrix of mean first passage times.

References

Kemeny, J. G. and Snell, J. L. (1976). Finite Markov Chains. Springer.

See also

steady_state(), sojourn_time()

Examples

P <- matrix(c(0.8, 0.2,
              0.3, 0.7), nrow = 2, byrow = TRUE)
mfpt(P)
#>          [,1] [,2]
#> [1,] 1.666667  5.0
#> [2,] 3.333333  2.5