Some Problems in Spectral Methods and Discrete Probability

谱方法和离散概率中的一些问题

• 批准号: RGPIN-2019-06751
• 负责人: Takahara, Glen
• 金额: $ 1.31万
• 依托单位: Queen's University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759143
• 关键词: Some; Problems; Spectral; Methods; Discrete

In statistics, a fundamental goal of time series analysis is to estimate temporal structure in data. With the proliferation of such data as part of the big data revolution the need for more accurate, realistic, and computationally efficient methods is increasing. Periodic structure is key in processes of the natural world and in many man-made environments, and spectral analysis is the proper approach to estimate periodic, or near-periodic, structure in time series. Two areas in which important practical issues remain are time series regression and long range dependent time series. Standard regression models in statistics do not efficiently account for temporal structure while estimation of long range dependence can be unreliable in practical scenarios. In probability theory, a basic problem is to compute the probability of a finite union of events, which requires knowing the probability of the intersection of every subset of the events. Often, only single event and pairwise intersection event probabilities are known or can be computed efficiently. Therefore, tight and low complexity bounds using limited information are desirable. Considerable interest in this problem has persisted for over 50 years as applications of such bounds in system design and statistics has expanded. The proposed research will focus on designing and analyzing novel statistical procedures that meet contemporary challenges posed by time series regression and estimation of long range dependence, and on novel methodology for bounding a union probability. New and substantial research challenges arise when trying to estimate relevant temporal structure yet maintain interpretability of fitted parameters in many statistical regression contexts, when trying to estimate long range dependence accurately in the face of structural or extra-variation contamination in the time series, and when constructing bounds under complexity constraints. The research objectives are divided into three main themes: (1) The creation of tools to incorporate modern spectral methods into standard regression models, the improvement of robustness and flexibility of current frequency domain methods, and the statistical analysis of the new procedures; (2) The development of robust techniques to estimate long range dependence and the statistical analysis of these techniques; (3) The investigation of optimality of bounds and the construction and performance of low complexity suboptimal bounds under information constraints. The training component of the proposed research will provide on average 2 M.Sc. and 3 Ph.D students each year with stimulating research challenges and immerse them in important current topics in statistics and probability. The research is expected to provide practical tools to increase the usefulness and practical application of time series regression models, to increase the applicability of long range dependent models, and to advance knowledge in an important problem in probability.

在统计数据中，时间序列分析的基本目标是估计数据中的时间结构。随着大数据革命的一部分，此类数据的扩散，需要更准确，现实和计算有效的方法的需求正在增加。周期性结构是自然世界和许多人造环境的过程中的关键，而光谱分析是估计时间序列中周期性或近乎周期结构的适当方法。仍然存在重要实际问题的两个领域是时间序列回归和远程依赖时间序列。统计中的标准回归模型不能有效地说明时间结构，而在实际情况下，远程依赖性的估计可能是不可靠的。在概率理论中，一个基本问题是计算事件有限结合的概率，这需要知道事件每个子集的相交的概率。通常，仅知道或可以有效地计算单个事件和成对交叉事件概率。因此，需要使用有限信息的紧密和低复杂性边界。由于系统设计和统计数据中这种界限的应用已扩大，因此对这个问题的兴趣持续了50多年。拟议的研究将着重于设计和分析应对时间序列回归和远距依赖性估计以及界限联合概率的新方法所带来的当代挑战的新型统计程序。当试图估计相关的时间结构时，出现了新的和实质性的研究挑战，但在许多统计回归环境中保持拟合参数的可解释性，当试图在时间序列中的结构性或外部变化污染时，以及在复杂性约束下构建界限时，试图准确地估算远距离依赖性。研究目标分为三个主要主题：（1）创建将现代光谱方法纳入标准回归模型的工具，改善当前频域方法的鲁棒性和灵活性以及对新程序的统计分析； （2）发展强大的技术以估计远程依赖性和这些技术的统计分析； （3）在信息约束下，界限的最优性以及低复杂性次优界限的构建和性能的研究。拟议研究的培训部分将平均提供2 M.Sc。每年都有3个博士学位学生刺激研究挑战，并将其沉浸在统计和概率上的重要主题中。预计该研究将提供实用的工具，以增加时间序列回归模型的有用性和实际应用，以提高远程依赖模型的适用性，并在概率上提高知识。