Markov Chain class

The S4 class that describes markovchain objects.

Arguments

states: Name of the states. Must be the same of colnames and rownames of the transition matrix
byrow: TRUE or FALSE indicating whether the supplied matrix is either stochastic by rows or by columns
transitionMatrix: Square transition matrix
name: Optional character name of the Markov chain

Note

markovchain object are backed by S4 Classes.
Validation method is used to assess whether either columns or rows totals to one. Rounding is used up to .Machine$double.eps * 100. If state names are not properly defined for a probability matrix, coercing to markovchain object leads to overriding states name with artificial "s1", "s2", ... sequence. In addition, operator overloading has been applied for $+,*,^,==,!=$ operators.

Creation of objects

Objects can be created by calls of the form new("markovchain", states, byrow, transitionMatrix, ...).

Methods

*: signature(e1 = "markovchain", e2 = "markovchain"): multiply two markovchain objects
*: signature(e1 = "markovchain", e2 = "matrix"): markovchain by matrix multiplication
*: signature(e1 = "markovchain", e2 = "numeric"): markovchain by numeric vector multiplication
*: signature(e1 = "matrix", e2 = "markovchain"): matrix by markov chain
*: signature(e1 = "numeric", e2 = "markovchain"): numeric vector by markovchain multiplication
[: signature(x = "markovchain", i = "ANY", j = "ANY", drop = "ANY"): ...
^: signature(e1 = "markovchain", e2 = "numeric"): power of a markovchain object
==: signature(e1 = "markovchain", e2 = "markovchain"): equality of two markovchain object
!=: signature(e1 = "markovchain", e2 = "markovchain"): non-equality of two markovchain object
absorbingStates: signature(object = "markovchain"): method to get absorbing states
canonicForm: signature(object = "markovchain"): return a markovchain object into canonic form
coerce: signature(from = "markovchain", to = "data.frame"): coerce method from markovchain to data.frame
conditionalDistribution: signature(object = "markovchain"): returns the conditional probability of subsequent states given a state
coerce: signature(from = "data.frame", to = "markovchain"): coerce method from data.frame to markovchain
coerce: signature(from = "table", to = "markovchain"): coerce method from table to markovchain
coerce: signature(from = "msm", to = "markovchain"): coerce method from msm to markovchain
coerce: signature(from = "msm.est", to = "markovchain"): coerce method from msm.est (but only from a Probability Matrix) to markovchain
coerce: signature(from = "etm", to = "markovchain"): coerce method from etm to markovchain
coerce: signature(from = "sparseMatrix", to = "markovchain"): coerce method from sparseMatrix to markovchain
coerce: signature(from = "markovchain", to = "igraph"): coercing to igraph objects
coerce: signature(from = "markovchain", to = "matrix"): coercing to matrix objects
coerce: signature(from = "markovchain", to = "sparseMatrix"): coercing to sparseMatrix objects
coerce: signature(from = "matrix", to = "markovchain"): coercing to markovchain objects from matrix one
dim: signature(x = "markovchain"): method to get the size
names: signature(x = "markovchain"): method to get the names of states
names<-: signature(x = "markovchain", value = "character"): method to set the names of states
initialize: signature(.Object = "markovchain"): initialize method
plot: signature(x = "markovchain", y = "missing"): plot method for markovchain objects
predict: signature(object = "markovchain"): predict method
print: signature(x = "markovchain"): print method.
show: signature(object = "markovchain"): show method.
sort: signature(x = "markovchain", decreasing=FALSE): sorting the transition matrix.
states: signature(object = "markovchain"): returns the names of states (as names.
steadyStates: signature(object = "markovchain"): method to get the steady vector.
summary: signature(object = "markovchain"): method to summarize structure of the markov chain
transientStates: signature(object = "markovchain"): method to get the transient states.
t: signature(x = "markovchain"): transpose matrix
transitionProbability: signature(object = "markovchain"): transition probability

References

A First Course in Probability (8th Edition), Sheldon Ross, Prentice Hall 2010

Author

Giorgio Spedicato

Examples

#show markovchain definition
showClass("markovchain")
#> Class "markovchain" [package "markovchain"]
#> 
#> Slots:
#>                                                                           
#> Name:            states            byrow transitionMatrix             name
#> Class:        character          logical           matrix        character
#create a simple Markov chain
transMatr<-matrix(c(0.4,0.6,.3,.7),nrow=2,byrow=TRUE)
simpleMc<-new("markovchain", states=c("a","b"),
              transitionMatrix=transMatr, 
              name="simpleMc")
#power
simpleMc^4
#> simpleMc^4 
#>  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 0.3334 0.6666
#> b 0.3333 0.6667
#> 
#some methods
steadyStates(simpleMc)
#>              a         b
#> [1,] 0.3333333 0.6666667
absorbingStates(simpleMc)
#> character(0)
simpleMc[2,1]
#> [1] 0.3
t(simpleMc)
#> Unnamed Markov chain 
#>  A  2 - dimensional discrete Markov Chain defined by the following states: 
#>  a, b 
#>  The transition matrix  (by cols)  is defined as follows: 
#>     a   b
#> a 0.4 0.3
#> b 0.6 0.7
#> 
is.irreducible(simpleMc)
#> [1] TRUE
#conditional distributions
conditionalDistribution(simpleMc, "b")
#>   a   b 
#> 0.3 0.7 
#example for predict method
sequence<-c("a", "b", "a", "a", "a", "a", "b", "a", "b", "a", "b", "a", "a", "b", "b", "b", "a")
mcFit<-markovchainFit(data=sequence)
predict(mcFit$estimate, newdata="b",n.ahead=3)
#> [1] "a" "b" "a"
#direct conversion
myMc<-as(transMatr, "markovchain")

#example of summary
summary(simpleMc)
#> simpleMc  Markov chain that is composed by: 
#> Closed classes: 
#> a b 
#> Recurrent classes: 
#> {a,b}
#> Transient classes: 
#> NONE 
#> The Markov chain is irreducible 
#> The absorbing states are: NONE
if (FALSE) plot(simpleMc) # \dontrun{}