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Apply a boundary condition to a Markov chain's first and last state

Source: R/chainTransforms.R
toBoundedChain.Rd

Replaces the transition rows of a markovchain object's first and last state (in states(object) order) with an absorbing, reflecting, or semi-reflecting rule, leaving every other row unchanged.

Usage

toBoundedChain(object, boundaryCondition)

# S4 method for class 'markovchain'
toBoundedChain(object, boundaryCondition)

Arguments

object

A markovchain object with at least 2 states.

boundaryCondition

Either:

  • the string "absorbing": the first and last state each become absorbing (\(P_{11}=1\), \(P_{nn}=1\));

  • the string "reflecting": the first state moves to the second with certainty and the last state moves to the second-to-last with certainty (\(P_{12}=1\), \(P_{n,n-1}=1\));

  • a single number \(\beta\in[0,1]\), the semi-reflecting case: the first state stays with probability \(1-\beta\) and moves to the second state with probability \(\beta\) (\(P_{11}=1-\beta\), \(P_{12}=\beta\)), and symmetrically the last state stays with probability \(1-\beta\) and moves to the second-to-last with probability \(\beta\). \(\beta=0\) is the absorbing case and \(\beta=1\) is the reflecting case.

Value

A new markovchain object, row-stochastic, on the same states as object, identical to object except in its first and last transition rows.

Details

This function assumes – as is standard for a boundary condition – that states(object) is meaningfully ordered along a line, first state to last state, as it would be e.g. for birthDeath or any other chain built to represent a bounded random walk. It does not check this (there is no general way to check it from the transition matrix alone) and applies the same first/last-row replacement regardless of object's actual structure; only the two boundary rows are ever touched, so applying it to a chain whose states are not linearly ordered simply reinterprets whichever states happen to be listed first and last.

Unlike gamblersRuin, which is absorbing at both ends by construction and cannot be un-done, toBoundedChain() can be applied to any existing chain and with any of the three conditions, including reflecting or semi-reflecting ones that gamblersRuin() does not offer directly.

The implementation touches only 2 of the \(n\) rows and is \(O(n)\) time and memory beyond copying the transition matrix.

See also

birthDeath, gamblersRuin

Examples

bd <- birthDeath(p = c(0.3, 0.4, 0.5), q = c(0.2, 0.3, 0.1))

absorbed <- toBoundedChain(bd, "absorbing")
absorbed@transitionMatrix[1, ]
#> 1 2 3 4 
#> 1 0 0 0 
absorbed@transitionMatrix[4, ]
#> 1 2 3 4 
#> 0 0 0 1 

reflected <- toBoundedChain(bd, "reflecting")
reflected@transitionMatrix[1, ]
#> 1 2 3 4 
#> 0 1 0 0 

semiReflected <- toBoundedChain(bd, 0.25)
semiReflected@transitionMatrix[1, ]
#>    1    2    3    4 
#> 0.75 0.25 0.00 0.00