Various function to perform structural analysis of DTMC

These functions return absorbing and transient states of the markovchain objects.

period(object)

communicatingClasses(object)

recurrentClasses(object)

transientClasses(object)

transientStates(object)

recurrentStates(object)

absorbingStates(object)

canonicForm(object)

Arguments

object

A markovchain object.

Value

period

returns a integer number corresponding to the periodicity of the Markov chain (if it is irreducible)

absorbingStates

returns a character vector with the names of the absorbing states in the Markov chain

communicatingClasses

returns a list in which each slot contains the names of the states that are in that communicating class

recurrentClasses

analogously to communicatingClasses, but with recurrent classes

transientClasses

analogously to communicatingClasses, but with transient classes

transientStates

returns a character vector with all the transient states for the Markov chain

recurrentStates

returns a character vector with all the recurrent states for the Markov chain

canonicForm

returns the Markov chain reordered by a permutation of states so that we have blocks submatrices for each of the recurrent classes and a collection of rows in the end for the transient states

References

Feres, Matlab listing for markov chain.

See also

markovchain

Author

Giorgio Alfredo Spedicato, Ignacio Cordón

Examples

statesNames <- c("a", "b", "c")
mc <- new("markovchain", states = statesNames, transitionMatrix =
          matrix(c(0.2, 0.5, 0.3,
                   0,   1,   0,
                   0.1, 0.8, 0.1), nrow = 3, byrow = TRUE,
                 dimnames = list(statesNames, statesNames))
         )

communicatingClasses(mc)
#> [[1]]
#> [1] "a" "c"
#> 
#> [[2]]
#> [1] "b"
#> 
recurrentClasses(mc)
#> [[1]]
#> [1] "b"
#> 
recurrentClasses(mc)
#> [[1]]
#> [1] "b"
#> 
absorbingStates(mc)
#> [1] "b"
transientStates(mc)
#> [1] "a" "c"
recurrentStates(mc)
#> [1] "b"
canonicForm(mc)
#> Unnamed Markov chain 
#>  A  3 - dimensional discrete Markov Chain defined by the following states: 
#>  b, a, c 
#>  The transition matrix  (by rows)  is defined as follows: 
#>     b   a   c
#> b 1.0 0.0 0.0
#> a 0.5 0.2 0.3
#> c 0.8 0.1 0.1
#> 

# periodicity analysis
A <- matrix(c(0, 1, 0, 0, 0.5, 0, 0.5, 0, 0, 0.5, 0, 0.5, 0, 0, 1, 0), 
            nrow = 4, ncol = 4, byrow = TRUE)
mcA <- new("markovchain", states = c("a", "b", "c", "d"), 
          transitionMatrix = A,
          name = "A")

is.irreducible(mcA) #true
#> [1] TRUE
period(mcA) #2
#> [1] 2

# periodicity analysis
B <- matrix(c(0, 0, 1/2, 1/4, 1/4, 0, 0,
                   0, 0, 1/3, 0, 2/3, 0, 0,
                   0, 0, 0, 0, 0, 1/3, 2/3,
                   0, 0, 0, 0, 0, 1/2, 1/2,
                   0, 0, 0, 0, 0, 3/4, 1/4,
                   1/2, 1/2, 0, 0, 0, 0, 0,
                   1/4, 3/4, 0, 0, 0, 0, 0), byrow = TRUE, ncol = 7)
mcB <- new("markovchain", transitionMatrix = B)
period(mcB)
#> [1] 3