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Lower transition probability of an imprecise continuous time Markov chain

Source: R/ctmcProbabilistic.R
impreciseProbabilityatT.Rd

For an imprecise continuous time Markov chain, whose transition rates are only known to lie in intervals, computes the lower probability of being in state i at time s, as a function of the state at time t (Krak et al. 2017).

Usage

impreciseProbabilityatT(C, i, t = 0, s, error = 10^-3, useRCpp = TRUE)

Arguments

C

An ictmc object: its slot Q is a generator and row \(k\) of the rates may be scaled by any factor in range[k, ].

i

Index (position in the states of C) of the state whose lower probability is computed.

t

Initial time (default 0).

s

Final time, larger than t.

error

Maximum approximation error (default 0.001).

useRCpp

Logical: use the C++ implementation (default)? It requires t and s to be integer-valued.

Value

A numeric vector with one element per state \(x\): the lower probability that the chain is in state i at time s given that it is in \(x\) at time t. With degenerate intervals (range equal to 1) it is column i of \(\exp((s - t) Q)\), up to error.

Details

The lower transition operator is approximated by \(n\) steps \(g \leftarrow g + \delta \underline{Q} g\), with \(\delta = (s - t) / n\) and \(n\) chosen so that the error is below error, starting from the indicator of state i; \(\underline{Q}\) takes, in each row, the smaller of the values given by the two extremes of the interval of rates.

References

Krak, T., De Bock, J. and Siebes, A. (2017). Imprecise continuous-time Markov chains. International Journal of Approximate Reasoning 88, 452-528.

See also

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

Author

Vandit Jain

Examples

states <- c("n", "y")
Q <- matrix(c(-1, 1, 1, -1), nrow = 2, byrow = TRUE,
            dimnames = list(states, states))
range <- matrix(c(1/52, 3/52, 1/2, 2), nrow = 2, byrow = TRUE)
ictmc <- new("ictmc", states = states, Q = Q, range = range,
             name = "testictmc")
impreciseProbabilityatT(ictmc, i = 2, t = 0, s = 1, error = 1e-3)
#> [1] 0.008259774 0.140983489