Collaborative Research: Empirical Frequency Band Analysis for Functional Time Series

长期监测季节性风暴是了解长期大气趋势和作出可靠的季节性预报的重要组成部分。一种行之有效的测量中纬度风暴位置和强度的方法是利用某一频带内风速的时间变化。有两个变化很大的区域，分别从亚洲东海岸和北美东海岸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在变化。然而，风暴路径的测量结构和强度对频段的选择很敏感，目前还没有数据驱动的技术来确定适当表征风速变化趋势的频段。了解高频率（气旋生长和传播）与低频率（气旋闭塞和衰减）的相对强度与位置和季节的关系，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的空间和时间趋势。该项目旨在开发数据驱动程序，通过建立表征不同频率之间风速趋势差异的最佳总结措施来加强这一努力。研究团队将开发、验证并公开共享用于自适应频带估计的软件和分析工具，以充分总结时变动力学，并解释大气条件下时空依赖性之间的复杂相互作用。此外，该项目还包括通过大气科学、计算和理论统计的跨学科方案进行专业培训。这些活动包括指导博士论文和本科生毕业项目，以及为对统计和大气科学交叉研究感兴趣的当地高中生举办讲座。非平稳泛函时间序列的频域特性往往包含有价值的信息。这些特性通过它们随时间变化的功率谱来表征。寻求功率谱的低维总结测量的从业者通常将频率划分为频带，并在这些频带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能够最好地总结频谱信息的频段，这对于非平稳函数时间序列来说，由于其高维性，尤其具有挑战性。该项目旨在建立一个新的数据驱动框架，用于非平稳函数时间序列的自适应频带估计，该框架充分总结了序列的时变动态，同时解释了功能和时间依赖结构之间的复杂相互作用。与这一努力有关的三个具体目标是：(1)开发非平稳泛函时间序列的局部频带估计方法，以最好地保留泛函域内的非平稳频谱信息；(2)开发多元非平稳泛函时间序列的局部频带估计新方法，以最好地保留泛函域内多个泛函时间序列之间的联合非平稳频谱信息；(3)开发非平稳函数时间序列的多变量频带估计新方法，以最好地保留定位在多维频域内的非平稳频谱信息。将建立这些程序的理论有效性，并设计计算效率高的估计程序以确保可扩展性。将进行广泛的模拟研究，以探索新方法的经验和计算特性，这些新方法有望增强对风暴动力学背后隐藏机制的理解，从而有助于增强对不利气候事件的抵御能力。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.