Aggregate a Markov chain over a partition

Builds the chain on the blocks (macro-states) of a partition of the states. By default the chain must be exactly lumpable with respect to the partition (see is.lumpable()); with force = TRUE an approximate aggregation is returned for any partition.

Usage

lump(object, partition, force = FALSE)

# S4 method for class 'markovchain'
lump(object, partition, force = FALSE)

Arguments

object: A markovchain object.
partition: A named list of character vectors, the blocks of the partition; the names become the states of the result.
force: Logical: if FALSE (default) stop unless the chain is exactly lumpable; if TRUE aggregate anyway.

Value

A row-stochastic markovchain object on the blocks.

Details

The probability of moving from block \(A\) to block \(B\) is the average over the states of \(A\) of their probability of moving into \(B\), weighted by the stationary distribution (the average of the stationary distributions if there are several; equal weights within a block with zero stationary mass). For a lumpable chain all the states of a block give the same value, so the weights do not matter; otherwise the result is the usual stationary-weighted approximation, which preserves the stationary distribution of the blocks.

References

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

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"))
lump(mc, partition)
#> Unnamed Markov chain (Lumped) 
#>  A  2 - dimensional discrete Markov Chain defined by the following states: 
#>  ab, cd 
#>  The transition matrix  (by rows)  is defined as follows: 
#>     ab  cd
#> ab 0.3 0.7
#> cd 0.6 0.4
#> 
# an approximate aggregation over a partition that is not lumpable
lump(mc, list(ac = c("a", "c"), bd = c("b", "d")), force = TRUE)
#> Unnamed Markov chain (Lumped) 
#>  A  2 - dimensional discrete Markov Chain defined by the following states: 
#>  ac, bd 
#>  The transition matrix  (by rows)  is defined as follows: 
#>           ac        bd
#> ac 0.4543103 0.5456897
#> bd 0.4395833 0.5604167
#>