Invariance principles, self-normalized sums, changepoint analysis, error processes

不变性原理、自归一化总和、变点分析、误差过程

• 批准号: 8182-2006
• 负责人: Csorgo, Miklos
• 金额: $ 3.28万
• 依托单位: Carleton University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2006
• 资助国家: 加拿大
• 起止时间: 2006-01-01 至 2007-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=334329
• 关键词: Invariance; principles; self; normalized; sums

Title of Proposal: Invariance Principles, Self-normalized Sums, Changepoint Analysis, Error Processes We start with a quotation from Bertrand Russel: "Although this may seem a paradox, all science is dominated by the idea of approximation." Invariance Principles are concerned with rates of approximation of various observable (data based) processes by theoretically recognizable ones.  Thus the former can, in principle and also in practice, be studied by the latter. While doing this one must also be aware of possible changes in the underlying processes. Indeed, the problem of abrupt changes arises in a variety of experimental and mathematical sciences. For instance, in epidemiology one may be interested in testing whether the incidence of a disease has remained constant over time, and if not in estimating the time(s) of change(s) in order to suggest possible causes.  Another example is rhythm analysis in electrocardiograms, where the use of change detection methods constitutes a part of pattern recognition analysis. Detecting abrupt changes in seismic signal processes is yet another example of what we, in general, call Changepoint Analysis.  The theory of invariance principles for Self-normalized (partial) Sums processes has recently become one of the main sources of information and inspiration in this regard.  When approximating a process by another one, one should also think of the possible errors made while doing this.  This, in turn, leads to studying the laws of fluctuation of the thus created Error Processes for independent, and also for long-range dependent sequences.  All this provides a beautiful interplay of the methods of various fields of the theory and applications of Statistics, Probability and Stochastic Processes.  In particular, the need for studying path properties of Gaussian and related stochastic processes enters and strengthens the research done in all the areas mentioned in our proposal.

提案的标题：不变性原则，自归一化和，变点分析，错误过程我们以伯特兰·罗素的一句话开始：“尽管这似乎是一个悖论，但所有的科学都是由近似的思想主导的。”不变性原理关注的是各种可观察到的（基于数据的）过程在理论上可识别的过程中的近似率。因此，前者在原则上和实践上都可以为后者所研究。在这样做的同时，还必须意识到底层流程中可能发生的变化。的确，突变问题出现在各种实验科学和数学科学中。例如，在流行病学中，人们可能对检验某种疾病的发病率是否随时间保持不变感兴趣，如果不感兴趣，则可能对估计变化的时间感兴趣，以便提出可能的原因。另一个例子是心电图中的节奏分析，其中使用变化检测方法构成模式识别分析的一部分。检测地震信号过程中的突变是我们通常所说的变点分析的另一个例子。自归一化（部分）和过程的不变性原理理论最近已成为这方面信息和灵感的主要来源之一。当用另一个过程近似一个过程时，还应该考虑在这样做时可能产生的错误。这反过来又导致研究由此产生的误差过程的涨落规律，对于独立的，也对于远程依赖序列。所有这些都为统计学、概率论和随机过程的理论和应用的各个领域的方法提供了一个美丽的相互作用。特别是，研究高斯和相关随机过程的路径特性的需要进入并加强了我们建议中提到的所有领域的研究。