高阶多元Markov链及其非负张量模型的理论与数值分析

项目名称： 高阶多元Markov链及其非负张量模型的理论与数值分析

项目编号： No.11271144

项目类型： 面上项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 黎稳

作者单位： 华南师范大学

项目金额： 60万元

中文摘要： 高维Markov链是研究排队网络、制造系统和无条件数据序列等现实世界中流行的模型。概率分布对许多实际问题的分析有重要的作用。人们常常应用高阶、多元与高阶多元Markov链去建立许多实际问题的有效的数学模型。多元与高阶多元Markov链有s条链，状态的数目根据链和阶的个数呈指数级的增长，一些近似一阶的方法被应用于处理高维问题。然而，它们不能获得数据序列的长效的性态。本项目研究高阶、多元与高阶多元Markov链及其应用，并且建立相关的高阶张量模型。同时也建立一些简化模型，这些简化模型也可以获得数据序列间的负相关性。然后，我们将对这些高阶多元Markov链和非负张量推广Perron-Frobenius理论。我们将讨论各种模型的（联合）平稳概率分布的扰动分析，对应于高阶Markov链的非负张量理论以及解各种模型的参数和分布向量的有效算法和数学分析。我们也会应用所得到的模型到各类实际问题中。

中文关键词： 高阶多元Markov链；非负张量；数值分析；特征值；线性互补问题

英文摘要： High dimensional Markov chains are popular models for studying many real world systems such as queueing networks, manufacturing systems and also categorical data sequences. Suppose we are given an irreducible Markov chain, it is well-known that the Perron-Frobenius's theorem guarantees the existence and the uniqueness of the stationary probability distribution. This probability distribution is important for the performance analysis of many practical systems. However, in many real applications, one has to employ the multivariate, higher-order or higher-order multivariate Markov chain in constructing an effective mathematical model. In a conventional model of multivariate or higher-order multivariate Markov chains having s chains, the total number of states grows exponentially with respect to number of chains s or the order of the model. Some approximate first-order models have been proposed to handle this high dimensionality problem. However, they cannot capture the long-range dependence of the data sequences. In this project, we will study the models for multivariate, higher-order or high-order multivariate Markov chains and their applications, and apply the nonnegative tensor to model the higher-order or higher-order multivariate Markov chains. We will also propose simplified models in case the high-order model

英文关键词： Higher-order multivariate Markov chain；Nonnegative tensor；Numerical analysis；eigenvalue；linear complementarity problem