平稳过程的隐马尔可夫模型逼近
Approximation of stationary processes by Hidden Markov Models
摘要：
我们的目标是在建设一个隐马尔可夫模型（HMM）的分配的复杂性（底层马尔可夫链的状态数），最好的近似，在Kullback-Leibler发散率，一个给定的平稳过程。在较弱的条件下，我们证明了平稳过程与HMM之间的发散率的存在性。由于在一般情况下，没有可用的解析表达式，这个发散率，我们近似它与适当定义的，易于计算，汉克尔矩阵之间的分歧，我们使用我们的近似标准。我们提出了一个三步算法，非负矩阵分解技术的基础上，实现了HMM最佳定义的近似标准。在马尔可夫近似的特殊情况下，给出了该算法的完整的理论分析。
英文摘要：
We aim at the construction of a Hidden Markov Model (HMM) of assigned complexity (number of states of the underlying Markov chain) which best approximates, in Kullback-Leibler divergence rate, a given stationary process. We establish, under mild conditions, the existence of the divergence rate between a stationary process and an HMM. Since in general there is no analytic expression available for this divergence rate, we approximate it with a properly defined, and easily computable, divergence between Hankel matrices, which we use as our approximation criterion. We propose a three-step algorithm, based on the Nonnegative Matrix Factorization technique, which realizes an HMM optimal with respect to the defined approximation criterion. A full theoretical analysis of the algorithm is given in the special case of Markov approximation.
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