Some Problems in Spectral Methods and Discrete Probability

在统计学中，时间序列分析的一个基本目标是估计数据的时间结构。随着作为大数据革命一部分的此类数据的激增，对更准确、更现实和计算效率更高的方法的需求正在增加。周期结构是自然界和许多人造环境中的关键过程，谱分析是估计时间序列中周期或近周期结构的适当方法。两个重要的实际问题仍然存在的领域是时间序列回归和长期依赖的时间序列。统计学中的标准回归模型不能有效地解释时间结构，而在实际情况下，长期依赖性的估计可能是不可靠的。在概率论中，一个基本问题是计算事件的有限并集的概率，这需要知道事件的每个子集的交集的概率。通常，只有单个事件和成对交叉事件概率是已知的或可以有效地计算。因此，需要使用有限信息的紧密且低复杂度的界限。相当大的兴趣在这个问题上已经持续了50多年的应用，这样的界限在系统设计和统计已经扩大。拟议的研究将集中在设计和分析新的统计程序，以满足当代的挑战所带来的时间序列回归和估计的长期依赖性，并在新的方法界定工会的概率。新的和实质性的研究挑战出现时，试图估计相关的时间结构，但在许多统计回归的情况下，保持拟合参数的可解释性，当试图准确地估计长范围的依赖性，在面对结构或额外的变化污染的时间序列，当构建边界的复杂性约束。 研究目标分为三个主题：（1）创建将现代谱方法纳入标准回归模型的工具，改进当前频域方法的鲁棒性和灵活性，以及对新方法的统计分析：（2）发展鲁棒技术估计长程相关性，并对这些技术进行统计分析;（3）信息约束下界的最优性研究及低复杂度次优界的构造与性能。 拟议研究的培训部分将提供平均2个硕士和3个博士生每年与刺激的研究挑战，并沉浸在统计和概率的重要当前主题。该研究有望为提高时间序列回归模型的实用性和实际应用提供实用的工具，提高长期相关模型的适用性，并在概率的一个重要问题中提高知识。

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.