Markov chain order estimation with conditional mutual information

Abstract

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 φ-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.

DNA viewed as an out-of-equilibrium structure

“…The order of the process is then estimated to be the smallest value of r which produces a nonsignificant test statistics [18,19]. Tables II and III for the first row of Table II and subsequently for the second raw to a value 0.392362 × 10 1 , smaller than the χ 2 value of 5.99 at 5% level.…”

Investigating long range correlation in DNA sequences using significance tests of conditional mutual information

“…see Miller (1955) and Roulston (1999). In Papapetrou and Kugiumtzis (2013), the significance of I C (m) is tested using randomization to form the null distribution ofÎ C (m). The Markov chain order estimation from the scheme of randomization CMI testing for increasing orders m was found to perform consistently well for different scenarios of correlation structure and order of Markov chains, and it was favorably compared to other known order estimation methods, such as the AIC and BIC criteria (Tong, 1975;Katz, 1981;Guttorp, 1995;Csiszár and Shields, 2000), the Peres-Shields estimator (Peres and Shields, 2005), and the likelihood ratio test using a -divergence measure (Menéndez et al, 2001(Menéndez et al, , 2011.…”

“…Apparently, for sequences with LRC the estimation fails as there is no characteristic order, and one expects that increasing the length of the symbol sequence the order estimate increases as well. However, this idea is met with practical limitations, mainly due to the inefficiency of the current methods in estimating high orders (Dalevi et al, 2006;Lu et al, 2012;Papapetrou and Kugiumtzis, 2013). We deal with this problem here and propose a method that can assess whether a given symbol sequence has a characteristic order (the Markov chain order) found by saturation of the estimated order with the increase of the subsequence length (up to the actual sequence length), or alternatively the symbol sequence has LRC structure or a high Markov chain order that cannot be estimated on the basis of the given symbol sequence length.…”

“…However, the focus is mostly on Markov chains of relatively small order. Recently, we introduced a scheme for estimating the Markov chain order by sequentially applying a significance test of the conditional mutual information (CMI), I C (m), at increasing orders m (Papapetrou and Kugiumtzis, 2013). The comparison to other methods showed that the CMI testing can effectively estimate high orders when the length of the symbol sequence is accordingly large.…”

“…The comparison to other methods showed that the CMI testing can effectively estimate high orders when the length of the symbol sequence is accordingly large. In Papapetrou and Kugiumtzis (2013), the computationally intensive randomization test for the significance of CMI was used and we have later found that parametric significance test relying on Gamma null distribution matches well the performance of the randomization test (Papapetrou and Kugiumtzis, 2014). This result is particularly relevant for the current study dealing with high orders and long symbol sequences, and thus we use the parametric significance test of CMI to estimate the hypothesized Markov chain http://dx.doi.org/10.1016/j.compbiolchem.2014.08.007 1476-9271/© 2014 Elsevier Ltd. All rights reserved.…”