基于极大似然的马尔可夫链初始状态估计的数学规划  

Mathematical Programming Models for Estimating the Initial States of Markov Chains by Maximum Likelihood

作　　者：楼振凯 侯福均[1] 楼旭明 Lou Zhenkai;Hou Fujun;Lou Xuming(School of Management and Economics,Beijing Institute of Technology,Beijing 100081,China;School of Economics and Management,Xi'an University of Posts and Telecommunications,Xi'an 710121,China)

机构地区：[1]北京理工大学管理与经济学院,北京100081 [2]西安邮电大学经济与管理学院,陕西西安710121

出　　处：《南京师大学报（自然科学版）》2019年第2期50-56,60,共8页Journal of Nanjing Normal University(Natural Science Edition)

基　　金：国家自然科学基金面上项目(71571019)

摘　　要：参数估计是马尔可夫模型中的常见问题.基于初始状态的重要性,本文对初始状态未知的马尔可夫链模型的初始状态进行估计,并根据状态可见与否将模型分成一般马尔可夫模型和隐马尔可夫模型.考虑观测状态或观测符号的数量,基于极大似然原理分别建立了线性规划和非线性规划模型,并证明各阶段状态的概率满足规范性.对于线性规划模型,指出其可以用单纯形法求解,并给出了解的表达.对于非线性模型,指出其最优解的存在性,并利用库恩-塔克条件(K-T条件)将模型转化成方程组的形式.算例分析中,在基于库恩-塔克条件的方程组不易求解的情形下,运用lingo得到了满足模型的解.Parameters estimationis a common issue in Markov models. Owing to the importance of the initial state,in this paper we estimate the initial state for Markov chain models which the initial state is unknown. According to whether the states are visible,we divide the models into general Markov models and hidden Markov models. We build linear programming models and non-linear programming models by maximum likelihood with considering the amount of states or observation symbols,and prove that the probabilities of state at each stage meet the normalization of probability. For linear programming models,we point out that they can be solved by the simplex method,and show the expression of solution. For non-linear programming models,we account for the existence of the optimal solution,and turn models into equations by K-T condition. In the examples,we apply lingo to obtain the optimal solution while the equations are hard to solve.

关 键 词：初始状态估计 极大似然 数学规划模型 K-T条件 

分 类 号：O221[理学—运筹学与控制论]