Computationally Tractable Estimation Methods for Markov Processes

恩索尔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。许多关于连续时间马尔可夫过程推理的文献假设该过程在一段时间间隔内连续被观察到。然而，在实践中，许多可用的数据集只涉及观察过程的离散“快照”。在这种情况下，现有的参数估计理论往往导致计算上的限制方法。本文研究了离散观测连续时间马尔可夫过程的参数估计问题。许多结果利用蒙特卡罗模拟来实现可追溯性。本研究考察的第二个基本问题是随机系统中的误差传播问题。当随机系统的输出性能测度不能以封闭形式进行评价时，从被建模系统中模拟输出测度。为了解决这种情况，研究人员开发了随机系统泛函的估计器及其抽样性质。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。本研究开拓了依赖观测统计模型的前沿。例如，这项研究的统计方法适用于模拟污染物水平的不同领域，以及空气、土壤和水中随时间和/或空间演变的污染物水平；预测股票市场的变化；预测或评估发展最佳通信网络的需求。过去，由于计算能力的限制，统计模型用于复杂系统建模的使用受到了限制；这个国家近年来确实有所改善。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，当只有部分系统状态信息可用时，电气或计算机工程师可以使用这种模型来评估网络状态；在预测某一地区的空气质量时，对污染物水平的观测是在该地区的不规则地点进行的，而且往往是在不规则的时间点进行的。在只有部分信息可用的更常见场景中，给出了实现模型所需的理论构造。此外，本研究还涉及对预测的误差评估，或对估计的复杂系统的输出性能度量，再次利用高性能计算的可用性。

NSF DMS-9704648 Katherine Bennett Ensor 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 certainly 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.