Generator matrix from a transition matrix

Computes the generator \(Q\) of a continuous time Markov chain whose transition matrix over a period of length t is P, as the matrix logarithm \(Q = \log(P) / t\), so that \(\exp(t Q) = P\).

Usage

transition2Generator(P, t = 1, method = "logarithm")

Arguments

P

A transition matrix (rows summing to one) over a period of length t.

t

Length of the period, a positive number.

method

Method used: only "logarithm", the principal matrix logarithm computed by expm::logm(), is available.

Value

A square matrix with the dimensions of P, the generator \(\log(P) / t\).

Details

Not every transition matrix is the transition matrix of a continuous time chain (the embedding problem): the matrix returned is a valid generator only if its off-diagonal entries are non-negative, which is not checked. freq2Generator() gives regularized estimates that are always valid generators.

See also

Other continuous time Markov chains: ExpectedTime(), ctmcFit(), freq2Generator(), generatorToTransitionMatrix(), impreciseProbabilityatT(), is.CTMCirreducible(), is.TimeReversible(), probabilityatT(), rctmc()

Examples

P <- matrix(c(0.4, 0.6, 0.1, 0.9), nrow = 2, byrow = TRUE)
Q <- transition2Generator(P)
Q
#>            [,1]       [,2]
#> [1,] -1.0319767  1.0319767
#> [2,]  0.1719961 -0.1719961
expm::expm(Q) # gives back P
#>      [,1] [,2]
#> [1,]  0.4  0.6
#> [2,]  0.1  0.9