Self-normalized processes, U-statistics, changepoint analysis, long-memory processes, anisotropic random walks

自归一化过程、U 统计、变点分析、长记忆过程、各向异性随机游走

• 批准号: 8182-2011
• 负责人: Csorgo, Miklos
• 金额: $ 1.53万
• 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: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=568566
• 关键词: Self; normalized; processes; statistics; changepoint

Title of Proposal: Self-normalized processes, U-statistics, Changepoint Analysis, Long-memory processes, Anisotropic random walks 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, including those of anisotropic random walks that are also studied in the physics literature.

提案题目：自规范化过程，U统计，变点分析，长记忆过程，各向异性随机游动 我们从伯特兰·罗素的一句话开始：“尽管这看起来似乎是一个悖论，但所有科学都被近似的想法所支配。“不变性原理关注的是理论上可识别的各种可观察（基于数据）过程的近似率。 因此，前者在原则上和实践上都可以由后者来研究。在这样做的同时，人们还必须意识到潜在过程中可能发生的变化。的确，突变问题出现在各种实验和数学科学中。例如，在流行病学中，人们可能感兴趣的是检验一种疾病的发病率是否随着时间的推移而保持不变，如果不是，则估计变化的时间，以便提出可能的原因。 另一个例子是心电图中的节律分析，其中变化检测方法的使用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称之为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的主要信息和灵感来源之一。 当用另一个过程来近似一个过程时，也应该考虑到在这样做时可能出现的错误。 这反过来又导致研究独立和长距离相关序列的误差过程的波动规律。 所有这一切提供了一个美丽的相互作用的方法的各个领域的理论和应用的统计，概率和随机过程。 特别是，需要研究高斯和相关的随机过程的路径特性进入并加强了我们的建议中提到的所有领域的研究，包括那些也在物理学文献中研究的各向异性随机游动。