Collaborative Research: Segmentation of Time Series via Self-Normalization

该项目旨在为时间序列数据的变点分析开发新的统计方法和理论。变点模型在许多科学领域都有广泛的应用，包括模拟美国金融市场的每日波动，以及冠状病毒等传染病的每周增长率等。与现有方法相比，本研究将为灵活的变点模型提供推理，该模型在真实数据集表现出的复杂依赖关系下仍然有效。该项目所采用的方法将通过出版物、会议和研讨会介绍以及开放源码软件的开发向相关科学界传播。首席调查员(PI)将联合指导一名博士生和本科生参与研究，并提供高级主题课程，介绍最先进的时间序列分析技术。时间序列分割，也称为变点估计，是统计学中的基本问题之一，即时间序列被分割成分段的同质分段，使得每一部分具有相同的行为。有大量文献致力于独立观测中的变点估计；然而，能够适应时间相关性的稳健的方法和严谨的理论仍然很少。自归一化(SN)方法最近取得了成功，这是由一个PI开发的，用于时间序列中的结构突变测试和其他推理问题，PI将把自归一化技术推广到时间序列分割。具体地说，PI将开发一种系统和统一的基于SN的变点估计方法和相关理论，用于(I)将分段平稳时间序列分割成均匀片段，以便在每一片段中有限维参数是恒定的；(Ii)将具有平稳和弱相依误差的线性趋势模型分割为具有恒定斜率的周期。即将开发的分割算法广泛适用于固定维度的时间序列数据，通过适当修改自我归一化测试统计数据，可以进一步扩展到涵盖高维和局部平稳的时间序列。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。

This project aims to develop new statistical methodology and theory for change-point analysis of time series data. Change-point models have wide applications in many scientific areas, including modeling the daily volatility of the U.S. financial market, and the weekly growth rate of an infectious disease such as coronavirus, among others. Compared with existing methodologies, this research will provide inference for a flexible range of change point models, which will remain valid under complex dependence relationships exhibited by real datasets. The methodologies ensuing from the project will be disseminated to the relevant scientific communities via publications, conference and seminar presentations, and the development of open-source software. The Principal Investigators (PIs) will jointly mentor a Ph.D. student and involve undergraduate students in the research, and offer advanced topic courses to introduce the state-of-the-art techniques in time series analysis.Time series segmentation, also known as change-point estimation, is one of the fundamental problems in statistics, where a time series is partitioned into piecewise homogeneous segments such that each piece shares the same behavior. There is a vast body of literature devoted to change-point estimation in independent observations; however, robust methodology and rigorous theory that can accommodate temporal dependence is still scarce. Motivated by the recent success of the self-normalization (SN) method, which was developed by one of the PIs for structural break testing and other inference problems in time series, the PIs will advance the self-normalization technique to time series segmentation. Specifically, the PIs will develop a systematic and unified SN-based change-point estimation methodology and the associated theory for (i) segmenting a piecewise stationary time series into homogeneous pieces so within each piece a finite dimensional parameter is constant; (ii) segmenting a linear trend model with stationary and weakly dependent errors into periods with constant slope. The segmentation algorithms to be developed are broadly applicable to fixed-dimensional time series data and can be further extended to cover high-dimensional and locally stationary time series with proper modification of the self-normalized test statistics.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.