Some Problems in Spectral Methods and Discrete Probability

在统计学中，时间序列分析的基本目标是估计数据的时间结构。随着大数据革命中此类数据的激增，对更准确、更现实和计算效率更高的方法的需求也在不断增加。周期性结构是自然世界和许多人造环境过程的关键，而谱分析是估计时间序列中周期性或近周期性结构的正确方法。仍然存在重要实际问题的两个领域是时间序列回归和长范围相关时间序列。统计学中的标准回归模型不能有效地解释时间结构，而长程依赖性的估计在实际场景中可能不可靠。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的概率是已知的或可以有效地计算。因此，使用有限信息的严格且低复杂度的界限是可取的。随着这种界限在系统设计和统计中的应用不断扩大，人们对这个问题的兴趣已经持续了 50 多年。拟议的研究将侧重于设计和分析新颖的统计程序，以应对时间序列回归和长期依赖性估计所带来的当代挑战，以及限制联合概率的新颖方法。当试图估计相关时间结构但在许多统计回归环境中保持拟合参数的可解释性时，当试图在时间序列中的结构或外变污染的情况下准确估计长期依赖性时，以及在复杂性约束下构建边界时，会出现新的实质性研究挑战。 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，提高当前频域方法的鲁棒性和灵活性，以及​​新程序的统计分析； (2) 开发稳健的技术来估计长期依赖性以及这些技术的统计分析； (3)信息约束下边界最优性的研究以及低复杂度次优边界的构造和性能。 拟议研究的培训部分将平均提供 2 个硕士学位。每年有 3 名博士生接受刺激的研究挑战，让他们沉浸在统计和概率领域的当前重要主题中。该研究有望提供实用工具来提高时间序列回归模型的实用性和实际应用，提高长程相关模型的适用性，并增进对概率重要问题的认识。

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.