Restrict a Markov chain to a subset of states

Restricts a markovchain object to a chosen subset of its states, either as a raw (generally non-stochastic) principal submatrix, or as a properly renormalized Markov chain describing behaviour conditional on staying inside the subset.

Usage

subchain(object, states, method = c("submatrix", "renormalize"))

# S4 method for class 'markovchain'
subchain(object, states, method = c("submatrix", "renormalize"))

Arguments

object: A markovchain object.
states: A character vector of at least one state name from states(object), with no duplicates: the subset to restrict to.
method: Either "submatrix" or "renormalize" (can be abbreviated). See Details: the two options answer genuinely different questions and are not interchangeable.

Value

If method = "submatrix": a plain numeric matrix (not a markovchain object, since its rows generally do not sum to one), the principal submatrix of the transition matrix restricted to states, always returned in row-stochastic orientation regardless of object's own storage convention.

If method = "renormalize": a new markovchain object on exactly the states in states, row-stochastic, describing the chain conditional on never leaving that subset.

Details

The two methods are deliberately named after two different, standard constructions, so that the choice – and its consequences – is explicit rather than implied:

"submatrix": Simply the entries of \(P\) with both indices restricted to states, with no adjustment. Its rows generally sum to less than one, because probability mass that originally went to states outside the subset is dropped, not redistributed. This is the "\(Q\)" block used, e.g., when building the fundamental matrix of an absorbing chain (see fundamentalMatrix): a useful building block for other computations, but not itself a transition matrix of any Markov chain, which is why it is returned as a plain matrix.
"renormalize": Each retained row is divided by its own sum, so the result is row-stochastic and can be wrapped in a markovchain object. This is the chain of successive positions of object, conditioned on the event that it never leaves states (sometimes called the chain "watched on" states, or its taboo probabilities; see Norris (1997), Section 3.3). It requires every state in states to have strictly positive probability of transitioning within the subset (otherwise that conditioning event has probability zero from that state, and the row cannot be renormalized); an error is raised naming any state that fails this, rather than silently producing a row of NaN.

Neither method requires object to be irreducible: restricting to a subset of states is meaningful for any chain, and is often used precisely to study one communicating class in isolation.

The implementation performs no eigendecomposition; it is \(O(k^2)\) time and memory for a subset of size \(k\), after an \(O(n^2)\) extraction from the full \(n\)-state matrix.

References

Norris, J. R. (1998). Markov Chains. Cambridge University Press.

Examples

statesNames <- c("a", "b", "c")
mc <- new("markovchain", states = statesNames,
  transitionMatrix = matrix(c(0.5, 0.3, 0.2,
                              0.2, 0.6, 0.2,
                              0.1, 0.1, 0.8), byrow = TRUE, nrow = 3,
                            dimnames = list(statesNames, statesNames)))

# Raw submatrix: rows no longer sum to 1, mass has "leaked" to "c".
subchain(mc, c("a", "b"), method = "submatrix")
#>     a   b
#> a 0.5 0.3
#> b 0.2 0.6
rowSums(subchain(mc, c("a", "b"), method = "submatrix"))
#>   a   b 
#> 0.8 0.8 

# Renormalized: a genuine markovchain, conditional on staying in {a, b}.
watched <- subchain(mc, c("a", "b"), method = "renormalize")
watched
#> Unnamed Markov chain (subchain) 
#>  A  2 - dimensional discrete Markov Chain defined by the following states: 
#>  a, b 
#>  The transition matrix  (by rows)  is defined as follows: 
#>       a     b
#> a 0.625 0.375
#> b 0.250 0.750
#> 
rowSums(watched@transitionMatrix)
#> a b 
#> 1 1