Markov Chain Order estimation with Conditional Mutual Information

We introduce the Conditional Mutual Information (CMI) for the estimation of the Markov chain order. For a Markov chain of $K$ symbols, we define CMI of order $m$, $I_c(m)$, as the mutual information of two variables in the chain being $m$ time steps apart, conditioning on the intermediate variables of the chain. We find approximate analytic significance limits based on the estimation bias of CMI and develop a randomization significance test of $I_c(m)$, where the randomized symbol sequences are formed by random permutation of the components of the original symbol sequence. The significance test is applied for increasing $m$ and the Markov chain order is estimated by the last order for which the null hypothesis is rejected. We present the appropriateness of CMI-testing on Monte Carlo simulations and compare it to the Akaike and Bayesian information criteria, the maximal fluctuation method (Peres-Shields estimator) and a likelihood ratio test for increasing orders using $\phi$-divergence. The order criterion of CMI-testing turns out to be superior for orders larger than one, but its effectiveness for large orders depends on data availability. In view of the results from the simulations, we interpret the estimated orders by the CMI-testing and the other criteria on genes and intergenic regions of DNA chains.