标题： 基于条件互信息的马尔可夫链阶数估计 显示英文标题

标题： Markov Chain Order estimation with Conditional Mutual Information

摘要： 我们引入条件互信息（CMI）来估计马尔可夫链的阶数。对于一个由$K$个符号组成的马尔可夫链，我们定义阶数为$m$和$I_c(m)$的 CMI 是链中两个变量相隔$m$步时的互信息，并且以链中的中间变量为条件。我们基于 CMI 的估计偏差找到近似的解析显著性界限，并开发了一种基于$I_c(m)$随机化显著性检验的方法，其中随机化符号序列通过对原始符号序列的成分随机排列形成。显著性检验应用于递增的$m$值，马尔可夫链的阶数通过最后一个拒绝零假设的阶数来估计。 我们展示了蒙特卡洛模拟中 CMI 检验的适用性，并将其与赤池信息量准则（AIC）、贝叶斯信息量准则（BIC）、最大波动法（Peres-Shields 估计器）以及使用$\phi$-散度的递增阶数似然比检验进行了比较。CMI 检验的阶数标准对于大于一的阶数表现更优，但其对大阶数的有效性取决于数据的可用性。根据模拟结果，我们解释了 CMI 检验与其他准则估计的 DNA 链基因和基因间区域的阶数。 显示英文摘要

摘要： 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.