Fit a continuous time Markov chain

Maximum likelihood estimation of a continuous time Markov chain from one observed trajectory: the sequence of states visited and the times at which they were entered.

Arguments

data

A list of two vectors of equal length: the states visited (character) and the times at which they were entered (numeric, increasing, the first equal to 0).

byrow

Logical: return the generator, and the transition matrix of the embedded jump chain in the confidence intervals, with the rates out of each state in its row (default) or in its column?

name

Optional name of the fitted chain.

confidencelevel

Confidence level of the intervals (default 0.95).

Value

A list with two elements:

estimate

the fitted ctmc object;

errors

a list with dtmcConfidenceInterval, the confidence level and the lower and upper endpoint matrices of the transition probabilities of the embedded jump chain (as in markovchainFit()), and lambdaConfidenceInterval, the lower and upper endpoints of the exit rate of each state.

Details

The exit rate of state \(i\) is estimated as the number of departures from \(i\) divided by the total time spent in \(i\), and the jump probabilities by the transition frequencies of the sequence of states; the generator is their product. The interval for each rate is the normal approximation \(\hat\lambda_i (1 \pm z / \sqrt{n_i})\), with \(n_i\) the number of departures from \(i\), truncated at 0.

References

Yalamanchi, S. B. and Spedicato, G. A. (2015). Continuous time Markov chains. Vignette of the markovchain package.

See also

Other continuous time Markov chains: ExpectedTime(), freq2Generator(), generatorToTransitionMatrix(), impreciseProbabilityatT(), is.CTMCirreducible(), is.TimeReversible(), probabilityatT(), rctmc(), transition2Generator()

Author

Sai Bhargav Yalamanchi

Examples

data <- list(c("a", "b", "c", "a", "b", "a", "c", "b", "c"),
             c(0, 0.8, 2.1, 2.4, 4, 5, 5.9, 8.2, 9))
fit <- ctmcFit(data)
fit$estimate
#> An object of class "ctmc"
#> Slot "states":
#> [1] "a" "b" "c"
#> 
#> Slot "byrow":
#> [1] TRUE
#> 
#> Slot "generator":
#>            a          b          c
#> a -0.9090909  0.6060606  0.3030303
#> b  0.3225806 -0.9677419  0.6451613
#> c  0.3846154  0.3846154 -0.7692308
#> 
#> Slot "name":
#> [1] ""
#> 
fit$errors$lambdaConfidenceInterval
#> $lowerEndpointVector
#> [1] 0 0 0
#> 
#> $upperEndpointVector
#> [1] 1.937805 2.062825 1.835311
#>