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，随着此类数据的激增，对更准确、更真实、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为环境过程中的关键，而谱分析是估计时间序列中的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，对长期相关性的估计可能不可靠。概率论中的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，只有单个事件和两两相交事件的概率是已知的或可以有效地计算。因此，使用有限信息的紧凑和低复杂性边界是可取的。随着这种界限在系统设计和统计中的应用的扩大，对这个问题的相当大的兴趣已经持续了50多年。提出的研究将侧重于设计和分析新的统计程序，以满足时间序列回归和长期依赖估计所带来的当代挑战，以及限制联合概率的新方法。当试图估计相关的时间结构时，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面对结构或额外变化污染时准确估计长期依赖性时，以及在复杂性约束下构建边界时，会出现新的和实质性的研究挑战。

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.