Saddlepoint and Bootstrap Methods in Systems Theory and Survival Analysis

系统理论和生存分析中的鞍点和引导方法

• 批准号: 0202284
• 负责人: Ronald Butler
• 金额: $ 12.6万
• 依托单位: Colorado State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-08-01 至 2006-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0202284&HistoricalAwards=false
• 关键词: Saddlepoint; Bootstrap; Methods; Systems; Theory

AbstractPI: R. W. ButlerDMS-0202284Title: Saddlepoint and Bootstrap Methods in System Theory and Survival AnalysisStochastic systems underlie the models applied in most areas of modern science. These concepts have evolved over a long period of time but were heavily influenced more recently by the cybernetics movement spanning the late 40s until the late 70s. Those involved on the electrical engineering side of this movement used flow graphs to represent the systems. Interesting time-dependent system characteristics were naturally characterized in terms of the Laplace transforms that could be associated with the flow graph. Their efforts lead to a general theory of finite state semi-Markov systems that generalized the very restrictive Markov systems that are commonly applied even today. Unfortunately, the inversion of the transforms in this approach proved to be a challenge that ultimately limited the impact of the flow graphs. This proposal addresses these inversions along with other missing tools so that a general systems theory may be fully developed into a complete mathematical discipline. These inversion tools, using saddlepoint methods, allow for the determination of the transient behavior of complex systems. In addition, the bootstrap is introduced for nonparametric statistical inference about the true system based on system observation. The range of applications in the engineering and biomedical sciences include communication and computer networks, queueing theory, multi-state survival models, and right/left and interval censoring in the context of the competing risks associated with such models. More specifically, this proposal addresses the following issues: (1) New techniques are required for inverting the Laplace transforms that characterize the complex behavior of systems; the author proposes several new saddlepoint methods for achieving this by using the method of steepest descents. (2) Transforms describing system characteristics need to be specified in ways that make the saddlepoint methods easy to use. Often Mason's rule is used but, because of its form, it is much too complicated to apply to large systems. The author has proposed several alternative co-factor rules that greatly simplify computations for the saddlepoint inversions. (3) Nonparametric statistical inference is to be developed for such semi-Markov systems using the bootstrap and its relationship to empirical transforms. The single and double bootstrap lead to a practical means for computing confidence bands of survival and hazard functions related to the semi-Markov process. The proposed double bootstrap is implemented through saddlepoint inversions.This proposal studies the statistical and probabilistic traits of dynamic stochastic systems with feedback. The models for such systems are commonly used to make extrapolations and predictions in most areas of modern science. In engineering, for example, a communication or computer system changes from state to state and the dynamics of these state transitions determine the evolution of the system. This proposal considers the computation and estimation of reliability or performance evaluation for such an evolving dynamic system. The proposed methods have important applications in reliability analysis, electrical engineering, biomedical sciences and manufacturing.

摘要：R·W·巴特勒DMS-0202284标题：系统论和生存分析中的鞍点和自举方法随机系统是现代科学大多数领域应用的模型的基础。这些概念经过了很长一段时间的演变，但最近受到跨越40年代末至70年代末的控制论运动的严重影响。参与这场运动的电气工程方面的人使用流程图来表示系统。有趣的依赖于时间的系统特征可以自然地用可以与流图相关联的拉普拉斯变换来表征。他们的努力导致了有限状态半马尔可夫系统的一般理论，该理论推广了即使在今天仍然普遍应用的非常严格的马尔可夫系统。不幸的是，这种方法中转换的反转被证明是一个挑战，最终限制了流程图的影响。这一建议解决了这些倒置以及其他缺失的工具，从而使一般系统理论可以完全发展成为一门完整的数学学科。这些反演工具，使用鞍点方法，允许确定复杂系统的瞬变行为。此外，还将Bootstrap引入到基于系统观测的真实系统的非参数统计推断中。在工程和生物医学科学中的应用范围包括通信和计算机网络、排队理论、多状态生存模型，以及在与这些模型相关的竞争风险的背景下的右/左和区间审查。更具体地说，这个建议解决了以下问题：(1)需要新的技术来求逆表征系统复杂行为的拉普拉斯变换；作者提出了几种新的鞍点方法来实现这一点，使用最陡下降法。(2)描述系统特征的转换需要以使鞍点方法易于使用的方式来指定。梅森规则经常被使用，但由于其形式，它太复杂了，不适用于大型系统。作者提出了几种可供选择的余因式规则，它们大大简化了鞍点求逆的计算。(3)利用Bootstrap及其与经验变换的关系，对这类半马尔可夫系统进行非参数统计推断。单自举和双自举为计算与半马尔可夫过程有关的生存函数和风险函数的置信度提供了一种实用的方法。提出的双Bootstrap方法是通过鞍点求逆来实现的，研究了具有反馈的动态随机系统的统计和概率特性。在现代科学的大多数领域，这类系统的模型通常被用来进行推断和预测。例如，在工程中，通信或计算机系统从一个状态到另一个状态变化，这些状态转换的动态决定了系统的演化。该方案考虑了这样一个动态演化系统的可靠性或性能评估的计算和估计。所提出的方法在可靠性分析、电气工程、生物医学科学和制造业中具有重要的应用。