Computationally Tractable Estimation Methods for Markov Processes

NSF DMS-9704648凯瑟琳·班尼特·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的推理的文献都假定该过程在某一时间间隔内是连续观察的。然而，在实践中，许多现有的数据集只涉及观察这一过程的离散“快照”。在这种情况下，现有的关于参数估计的理论往往导致计算上令人望而却步的方法。本文研究离散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处理性。本研究考察的第二个基本问题是误差在随机系统中的传播问题。当随机系统的输出性能指标不能以封闭形式估计时，从建模系统中模拟输出指标。为了解决这种情况，研究人员发展了随机系统泛函的估计量及其抽样性质。统计模型一直被证明是建模、理解和预测复杂系统的有力工具。这项研究推进了相依观测统计模型的前沿。例如，这项研究的统计方法适用于以下各个领域：对空气、土壤和水中污染物和污染物随时间和/或空间演变的水平进行建模；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，出于对复杂系统建模的目的，统计模型的使用一直受到限制，因为计算能力的状态；这种状态在最近几年肯定有所改善。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。这项研究解决了在部分观测信息下的建模问题。例如，当只有关于系统状态的部分信息可用时，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；在预测给定地区的空气质量时，污染物水平的观测是在该地区不规则的地点进行的，并且通常是在不规则的时间点上进行的。在更常见的情况下，当只有部分信息可用时，提出了实现模型所需的理论构造。此外，这项研究还涉及对估计的复杂系统的预测或输出性能测量的误差评估，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.