Check exact lumpability of a Markov chain

Checks whether a Markov chain is (strongly) lumpable with respect to a partition of its states: for every pair of blocks \(A, B\) of the partition, all the states of \(A\) have the same probability of moving into \(B\) (Kemeny and Snell 1960, Theorem 6.3.2). The aggregated process is then a Markov chain, for every initial distribution.

Usage

is.lumpable(object, partition, tol = 1e-10)

# S4 method for class 'markovchain'
is.lumpable(object, partition, tol = 1e-10)

Arguments

object

A markovchain object.

partition

A named list of character vectors, the blocks of the partition (macro-states); every state must belong to exactly one block.

tol

Non-negative tolerance for the equality of the probabilities (default 1e-10).

Value

TRUE if the chain is lumpable with respect to partition, FALSE otherwise.

References

Kemeny, J. G. and Snell, J. L. (1960). Finite Markov Chains. Van Nostrand.

See also

Other state aggregation: autoLump(), lump()

Examples

statesNames <- c("a", "b", "c", "d")
P <- matrix(c(0.1, 0.2, 0.3, 0.4,
              0.2, 0.1, 0.4, 0.3,
              0.3, 0.3, 0.2, 0.2,
              0.2, 0.4, 0.1, 0.3), nrow = 4, byrow = TRUE,
            dimnames = list(statesNames, statesNames))
mc <- new("markovchain", states = statesNames, transitionMatrix = P)
partition <- list(ab = c("a", "b"), cd = c("c", "d"))
is.lumpable(mc, partition) # rows of a and b give 0.3/0.7 to ab/cd
#> [1] TRUE
is.lumpable(mc, list(ac = c("a", "c"), bd = c("b", "d")))
#> [1] FALSE