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中的时间结构。作为大数据革命的一部分，随着这类数据的激增，对更准确、更现实、计算效率更高的方法的需求越来越大。周期结构在自然界和许多人造环境的过程中是关键的，而谱分析是估计时间序列中周期或近周期结构的合适方法。两个重要的实际问题仍然存在，即时间序列回归和长期相关时间序列。统计学中的标准回归模型不能有效地解释时间结构，而长期相关性的估计在实际情况下可能是不可靠的。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相交的概率。通常，只有单个事件和成对相交事件的概率是已知的，或者可以有效地计算。因此，使用有限信息的紧凑和低复杂性界限是可取的。随着这种界限在系统设计和统计中的应用范围扩大，人们对这个问题的极大兴趣已经持续了50多年。拟议的研究将集中于设计和分析新的统计程序，以应对时间序列回归和估计长期相关性所带来的当代挑战，以及确定联合概率的新方法。在许多统计回归背景下，当试图估计相关的时间结构并保持拟合参数的可解释性时，当面对时间序列中的结构性或额外变化污染时，当试图准确估计长期相关性时，当在复杂性约束下构造边界时，新的和实质性的研究挑战出现了。研究目标分为三个主要主题：(1)将现代谱方法融入标准回归模型的工具的创建，当前频域方法的稳健性和灵活性的改进，以及新方法的统计分析；(2)估计长期相关性的稳健技术的发展和这些技术的统计分析；(3)在信息约束下的界的最优性和低复杂性次优界的构造和性能的研究。拟议研究的培训部分将平均提供2名理科硕士。以及每年3名博士生，他们面临着激动人心的研究挑战，让他们沉浸在统计学和概率学的重要当前主题中。本研究可望为提高时间序列回归模型的实用性和实际应用，增加长程相关模型的适用性，增进对概率论中一个重要问题的认识提供实用工具。

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.