Markov and Related Models, with Application of MCMC and Networks

在过去的几年里，马尔可夫模型的使用至少在三个不同的领域出现了爆炸性的发展：马尔可夫链平稳度量的有效计算的算法的发展，例如通过使用MCMC(马尔可夫链蒙特卡罗)方法进行统计推断和空间建模；为排队网络开发更复杂的随机模型，特别是在通信和制造系统中；以及通过非线性时间序列模型对物理现象进行建模，这需要不同于传统线性模型的方法。这项建议涉及开发和应用新的方法，这些方法是由这些不同应用领域中的问题所推动的。研究人员将在随机模型理论的五个主要方面进行研究，这些方面将有益于所有这些应用：(I)应用耦合理论和最小化理论来发展完美抽样策略的方法；(Ii)模型达到稳定状态的比率的计算评估方法；(Iii)随机有序或部分随机有序模型的性质，特别是在连续时间内，对于扩散和跳跃连续过程；(Iv)模型行为对假设中的扰动的稳健性；以及(V)在描述系统和开发算法时使用非马尔科夫过程和其他一般静态过程。总体而言，该项目有望提供新的方法，这些方法将为目前越来越广泛地用于解决制造工厂设计、大气和气候学建模以及经济理论等领域的问题的模拟和建模方法提供重要的理论基础和加速方法。在复杂网络、排队和存储系统的分析中，所设想的应用包括当前制造系统的高优先级领域，以及更一般的带反馈的排队系统。用于理解环境问题的高维空间模型也应该更快地使用一些拟议的方法进行分析。在这种背景下，非线性时间序列模型的模拟方法的发展有望为物理、社会和经济现象的模型提供新的结果。

9803682TweedieThere has been a virtual explosion in the last several years of the use of Markovian models, in at least three different areas: development of algorithms for efficient computation of the stationary measures for Markov chains, fed for example by the use of MCMC (Markov chain Monte Carlo) approaches to statistical inference and spatial modeling; development of more complex stochastic models for networks of queues, especially in communication and manufacturing systems; and modeling of physical phenomena by non-linear time series models, which needs approaches different from those of traditional linear models. This proposal concerns the development and application of new methodologies motivated by problems in these diverse applications areas. The investigator will carry out research in five main aspects of the theory of stochastic models which can benefit all of these applications: (i) Methods of applying coupling theory and minorization theory to the development of perfect sampling strategies; (ii) Methods of computational evaluation of the rates at which models achieve a stable regime; (iii) Properties of stochastically ordered or partially stochastically ordered models, especially in continuous time, for both diffusions and jump-continuous processes; (iv) Robustness of model behavior against perturbations in assumptions; and (v) Use of non-Markovian and other general stationary processes in describing systems and developing algorithms.Overall this project can be expected to deliver new methodologies which will provide a significant theoretical underpinning, and methods of speedup, to simulation and modeling methodologies currently being used more and more widely in solving problems in areas as diverse as manufacturing plant design, atmospheric and climatology modeling, and economic theory. In the analysis of complex network, queueing and storage systems, applications envisaged include the currently high-priority area of manufacturing systems, and more general queueing systems with feedback. High dimensional spatial models being used for the understanding of environmental issues should also be faster to analyze using some of the proposed methods. The development of simulation approaches for non-linear time-series models in this context can be expected to provide new results in models of physical, social and economic phenomena.