Collaborative Research: Empirical Frequency Band Analysis for Functional Time Series

随着时间的推移监测季节性风暴是了解长期大气趋势和产生可靠的季节预报的重要组成部分。一种公认的中纬度风暴位置和强度的测量方法是使用某一频带内风速的时间变化。有两个变化很大的地区，分别从亚洲东海岸和北美延伸到太平洋和大西洋，这些“风暴路径”的位置和强度每年都有所不同。然而，测量到的风暴路径的形状和强度对频带的选择很敏感，目前还没有数据驱动的技术来确定适当描述风速变化趋势的频带。了解高频(气旋增长和传播)与低频(气旋遮挡和衰减)作为位置和季节的函数的相对强度，以及风暴路径位置和特征的长期趋势，有助于气候研究人员评估大气条件下的空间和时间趋势。该项目旨在开发以数据为导向的程序，通过为确定不同频率之间风速趋势的差异确定最佳汇总措施来加强这一努力。研究小组将开发、验证和公开共享用于自适应频带估计的软件和分析工具，这些软件和分析工具充分总结了时变动力学，并考虑了大气条件下空间和时间相关性之间的复杂相互作用。此外，该项目还包括通过大气科学以及计算和理论统计方面的跨学科方案进行的专业培训。这些活动包括指导博士论文和本科顶峰项目，以及为当地高中生举办讲座，他们对统计学和大气科学的交叉领域的研究感兴趣。非平稳函数时间序列的频域特性通常包含有价值的信息。这些特性通过它们的时变功率谱来表征。寻求功率谱低维汇总测量的从业者通常将频率划分为频段，并在这些频段内创建崩溃的功率测量。然而，标准频带可能不能提供足够的功率谱汇总测量。需要一种标准化的、定量的方法来客观地识别最能概括光谱信息的频段，这对于非平稳函数时间序列来说尤其具有挑战性，因为它的高维。该项目旨在为非平稳函数时间序列建立一个新的数据驱动的自适应频带估计框架，该框架充分总结了序列的时变动力学，同时考虑了函数依赖结构和时间依赖结构之间的复杂相互作用。这项工作的三个具体目标是：(1)发展最好地保持函数域内非平稳谱信息的非平稳函数时间序列的局部频带估计方法；(2)发展最好地保持函数域内多个函数时间序列之间的联合非平稳谱信息的多变量非平稳函数时间序列的局部频带估计新方法；(3)发展最好地保持多维频域内非平稳谱信息的非平稳函数时间序列的多变量频带估计的新方法。将建立这些程序的理论有效性，并将设计计算效率高的估计程序，以确保可伸缩性。将进行广泛的模拟研究，以探索新方法的经验和计算特性，预计将增强对风暴动力学背后隐藏的机制的理解，从而有助于增强对不利气候事件的适应能力。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Monitoring seasonal storms over time is an essential component of understanding long-term atmospheric trends and producing reliable seasonal forecasts. A well-established measure of the location and intensity of mid-latitude storms uses the temporal variance of wind velocity within a certain frequency band. There are two areas of large variance, extending from the East Coasts of Asia and North America out into the Pacific and Atlantic Oceans respectively, and the location of these "storm tracks" and their intensities vary from year to year. However, the measured configurations and strength of the storm tracks are sensitive to the choice of frequency band, and there are currently no data-driven techniques for identifying frequency bands that appropriately characterize trends in wind velocity variability. Understanding the relative strength of higher frequencies (cyclone growth and propagation) vs. lower frequencies (cyclone occlusion and decay) as a function of location and season, as well as long-term trends in the locations and characteristics of storm tracks, aids climate researchers in assessing spatial and time trends in atmospheric conditions. This project aims to develop data-driven procedures to enhance this effort by establishing optimal summary measures for characterizing differences in wind velocity trends among frequencies. The research team will develop, validate, and openly share software and analytical tools for adaptive frequency band estimation that adequately summarizes time-varying dynamics and accounts for the complex interaction between the spatial and temporal dependence in atmospheric conditions. In addition, the project includes professional training through a transdisciplinary program in atmospheric sciences and computational and theoretical statistics. The activities include supervision of doctoral theses and undergraduate capstone projects, as well as talks for local high school students interested in research at the intersection of statistics and atmospheric sciences.The frequency-domain properties of nonstationary functional time series often contain valuable information. These properties are characterized through their time-varying power spectra. Practitioners seeking low-dimensional summary measures of the power spectrum often partition frequencies into bands and create collapsed measures of power within these bands. However, standard frequency bands may not provide adequate summary measures of the power spectrum. There is need for a standardized, quantitative approach to objectively identify frequency bands that can best summarize spectral information, which for nonstationary functional time series is especially challenging due to the high dimensionality. This project seeks to establish a new data-driven framework for adaptive frequency band estimation for nonstationary functional time series that adequately summarizes the time varying dynamics of the series and simultaneously accounts for the complex interaction between the functional and temporal dependence structures. The three specific aims associated with this effort are: (1) to develop methodology for local frequency band estimation of a nonstationary functional time series that best preserves nonstationary spectral information localized within the functional domain, (2) to develop new methodology for local frequency band estimation of a multivariate nonstationary functional time series that best preserves the joint nonstationary spectral information among multiple functional time series localized within the functional domain, and (3) to develop new approaches for multivariate frequency band estimation of a nonstationary functional time series that best preserves nonstationary spectral information localized within a multidimensional frequency domain. Theoretical validity of these procedures will be established, and computationally efficient estimation procedures will be designed to ensure scalability. Extensive simulation studies will be conducted to explore the empirical and computational properties of the new methods, which are expected to enhance understanding of hidden mechanisms behind storm dynamics and thereby contribute to enhanced resilience to adverse climatic events.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.