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 ChiSquareMarkovChainStatistics

Source Notebook

Calculate a chi-square statistic reflecting the homogeneity of a Markov chain's transition matrix over several time periods

Contributed by: Konstantin Nosov

ResourceFunction["ChiSquareMarkovChainStatistics"][poolmat,permats,classcts]

tests the homogeneity of a Markov chain's transition matrix poolmat across all time periods, where permats is the list of transition matrices for each period and classcts is the number of observations.

Details

The entire sample (a sequence of states of a Markov chain) is assumed to be divided into T periods, where T is Length[permats].
If the Markov chain has N states, the pooled matrix poolmat is an N×N matrix.
If the number of periods is T, permats is a T×N×N list of period matrices, where permats[[t]] is the transition matrix for the tth period.
The number of observations in class, classcts, is a T×N list of lists where classcts[[t,i]] denotes the number of observations initially falling into the ith class (state) within the tth sub-period.
ResourceFunction["ChiSquareMarkovChainStatistics"] returns an Association with the following keys:
"States"number of states
"Periods"number of periods
"DegreesOfFreedom"degrees of freedom
"ChiSquareStatistics"χ2 statistics
"PValue"p-value

Examples

Basic Examples (1) 

Return χ2 statistic and associated values for a Markov chain with 5 states over 3 time periods:

In[1]:=
ResourceFunction[ "ChiSquareMarkovChainStatistics"][pooledMat, periodMats, obsInClass]
Out[1]=

Publisher

Konstantin

Version History

  • 1.0.0 – 18 May 2021

Source Metadata

  • Citation:
    • Billingsley, P., "Statistical Methods in Markov Chains." Annals of Mathematical Statistics, vol. 32, no. 1, 12–40, 1961.
    • Cochran, W.G., "The X² Test of Goodness of Fit." Annals of Mathematical Statistics, vol. 23, no. 3, 315–345, 1952.
    • Kullback, S., Kupperman M., Ku, H.H., "Tests for Contingency Tables and Markov Chains." Technometrics, vol. 4, no. 4, 573–608, 1962.
    • Tan, B., Yilmaz, K., "Markov chain test for time dependence and homogeneity: An analytical and empirical evaluation." European Journal of Operational Research, vol. 137, no. 3, 524–543, 2002. https://ideas.repec.org/a/eee/ejores/v137y2002i3p524-543.html

Related Resources

Author Notes

The function has no limitation

License Information