Computationally Tractable Estimation Methods for Markov Processes

NSF DMS-9704732 马尔可夫过程的可计算估计方法 Peter W. Glynn 斯坦福大学 马尔可夫过程理论提供了丰富的分析和概率结构，从建模的角度来看，这是本质上自然的。 大部分关于连续时间马尔可夫过程的推理的文献都假设该过程在某个时间间隔内被连续观察。然而，在实践中，许多可用的数据集只涉及观察离散的 过程的“快照”。 现有的理论参数估计在这种情况下，往往会导致计算禁止的方法。本研究针对离散观测连续时间马氏过程的参数估计问题，提出一种计算上易于处理的参数估计方法。许多结果利用蒙特卡罗模拟来实现易处理性。 的 本研究中探讨的第二个基本问题是， 误差通过随机系统传播的理论当随机系统的输出性能指标不能以封闭形式评估时，输出指标由建模系统模拟。为了解决这种情况下，调查人员开发的随机系统的泛函估计和他们的抽样性能。 统计模型一直被证明是建模，理解和预测复杂系统的强大工具。 这项研究推进了相关观测统计模型的前沿。 例如，本研究的统计方法适用于模拟空气、土壤中的污染物和污染物水平的不同领域， 和水随时间和/或空间演变的变化;预测股票市场的变化;以及预测或评估对开发最佳通信网络的需求。 在过去，为了模拟复杂系统的目的而使用统计模型是有限的，这是由于 计算能力;近年来有了明显改善的状态。 在这项研究中，研究人员开发了一个框架，用于实际实施先进的统计方法，充分利用当今可用的高性能计算。该研究解决了部分观测信息下的建模问题。 例如，电气或计算机工程师可以使用这样的模型来评估网络状态 当只能获得关于系统状况的部分信息时;在预测某一区域的空气质量时，在该区域不规则的地点和经常在不规则的时间点进行污染物水平的观测。必要的理论结构，以实现在更常见的情况下，只有部分信息是可用的模型。 此外，本研究涉及的错误评估的 再次利用高性能计算的可用性来预测或测量所估计的复杂系统的输出性能。

NSF DMS-9704732 Computationally Tractable Estimation Methods for Markov Processes Peter W. Glynn Stanford University Markov process theory provides a rich analytic and probablistic structure which is intrinsically natural from a modelling perspective. Much of the literature on inference for continuous- time Markov processes assumes that the process has been observed continuously over some time interval. However, in practice, many of the data sets available involve observing only discrete ``snapshots'' of the process. Existing theory on parameter estimation in this setting often results in computationally prohibitive methodologies. This research addresses the issue of computationally tractable parameter estimation for discretely observed continuous-time Markov processes. Many of the results utilize Monte Carlo simulation to achieve tractability. The second fundamental question examined in this research is that of error propagation through a stochastic system. When the output performance measure of the stochastic system cannot be evaluated in closed form the output measure is simulated from the modelled system. To address this situation, the investigators develop estimators and their sampling properties for functionals of the stochastic system. Statistical models have always proven a powerful tool for purposes of modelling, understanding and predicting complex systems. This research advances the frontier of statistical models for dependent observations. For example, the statistical methods of this research are applicable to the diverse areas of modelling levels of pollutants and contaminants in air, soil and water which evolve over time and/or space; forecasting changes in the stock market; and predicting or assessing demand for the development of an optimal communications network. The use of statistical models for purposes of modelling complex systems has been limited in the past due to the state of computing power; a state which has certainl y improved in recent years. In this research, the investigators develop a framework for practical implementation of advanced statistical methodologies which capitalizes fully on the high performance computing available today. The research addresses the issue of modelling under partially observed information. For example, the electrical or computer engineer may use such models to assess network status when only partial information is available on the state of the system; in predicting air quality for a given region, observations of pollutant levels are made at sites irregularly located over the region and often at irregular points in time. The theoretical constructs necessary to implement the models in the more common scenario when only partial information is available are presented. Additionally, this research involves error assessment of the predictions or output performance measure of an estimated complex system again capitalizing on the availability of high powered computing.