Various functions to perform statistical inference of DTMC

These functions verify the Markov property, assess the order and stationarity of the Markov chain.

This function tests whether an empirical transition matrix is statistically compatible with a theoretical one. It is a chi-square based test. In case a cell in the empirical transition matrix is >0 that is 0 in the theoretical transition matrix the null hypothesis is rejected.

Verifies that the s elements in the input list belongs to the same DTMC

verifyMarkovProperty(sequence, verbose = TRUE)

assessOrder(sequence, verbose = TRUE)

assessStationarity(sequence, nblocks, verbose = TRUE)

verifyEmpiricalToTheoretical(data, object, verbose = TRUE)

verifyHomogeneity(inputList, verbose = TRUE)

Arguments

sequence

An empirical sequence.

verbose

Should test results be printed out?

nblocks

Number of blocks.

data

matrix, character or list to be converted in a raw transition matrix

object

a markovchain object

inputList

A list of items that can coerced to transition matrices

Value

Verification result

a list with following slots: statistic (the chi - square statistic), dof (degrees of freedom), and corresponding p-value. In case a cell in the empirical transition matrix is >0 that is 0 in the theoretical transition matrix the null hypothesis is rejected. In that case a p-value of 0 and statistic and dof of NA are returned.

a list of transition matrices?

References

Anderson and Goodman.

See also

markovchain

Author

Tae Seung Kang, Giorgio Alfredo Spedicato

Examples

sequence <- c("a", "b", "a", "a", "a", "a", "b", "a", "b",
              "a", "b", "a", "a", "b", "b", "b", "a")
mcFit <- markovchainFit(data = sequence, byrow = FALSE)
verifyMarkovProperty(sequence)
#> Testing markovianity property on given data sequence
#> Chi - square statistic is: 0.28 
#> Degrees of freedom are: 5 
#> And corresponding p-value is: 0.9980024 
assessOrder(sequence)
#> Warning: The accuracy of the statistical inference functions has been questioned. It will be thoroughly investigated in future versions of the package.
#> Warning: Chi-squared approximation may be incorrect
#> Warning: Chi-squared approximation may be incorrect
#> The assessOrder test statistic is:  1.55307e-31 
#> The Chi-Square d.f. are:  2 
#> The p-value is:  1 
assessStationarity(sequence, 1)
#> Warning: The accuracy of the statistical inference functions has been questioned. It will be thoroughly investigated in future versions of the package.
#> Warning: Chi-squared approximation may be incorrect
#> Warning: Chi-squared approximation may be incorrect
#> The assessStationarity test statistic is:  0.1960191 
#> The Chi-Square d.f. are:  0 
#> The p-value is:  0 



#Example taken from Kullback Kupperman Tests for Contingency Tables and Markov Chains

sequence<-c(0,1,2,2,1,0,0,0,0,0,0,1,2,2,2,1,0,0,1,0,0,0,0,0,0,1,1,
2,0,0,2,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0,2,1,0,
0,2,1,0,0,0,0,0,0,1,1,1,2,2,0,0,2,1,1,1,1,2,1,1,1,1,1,1,1,1,1,0,2,
0,1,1,0,0,0,1,2,2,0,0,0,0,0,0,2,2,2,1,1,1,1,0,1,1,1,1,0,0,2,1,1,
0,0,0,0,0,2,2,1,1,1,1,1,2,1,2,0,0,0,1,2,2,2,0,0,0,1,1)

mc=matrix(c(5/8,1/4,1/8,1/4,1/2,1/4,1/4,3/8,3/8),byrow=TRUE, nrow=3)
rownames(mc)<-colnames(mc)<-0:2; theoreticalMc<-as(mc, "markovchain")

verifyEmpiricalToTheoretical(data=sequence,object=theoreticalMc)
#> Testing whether the
#>    0  1  2
#> 0 51 11  8
#> 1 12 31  9
#> 2  6 11 10
#> transition matrix is compatible with
#>       0     1     2
#> 0 0.625 0.250 0.125
#> 1 0.250 0.500 0.250
#> 2 0.250 0.375 0.375
#> [1] "theoretical transition matrix"
#> ChiSq statistic is 6.551795 d.o.f are 6 corresponding p-value is 0.3642899 
#> $statistic
#>        0 
#> 6.551795 
#> 
#> $dof
#> [1] 6
#> 
#> $pvalue
#>         0 
#> 0.3642899 
#> 


data(kullback)
verifyHomogeneity(inputList=kullback,verbose=TRUE)
#> Warning: The accuracy of the statistical inference functions has been questioned. It will be thoroughly investigated in future versions of the package.
#> Testing homogeneity of DTMC underlying input list 
#> ChiSq statistic is 275.9963 d.o.f are 35 corresponding p-value is 0 
#> $statistic
#> [1] 275.9963
#> 
#> $dof
#> [1] 35
#> 
#> $pvalue
#> [1] 0
#>