Statistical Inference for Temporally Dependent Functional Data

PI发展了一套系统的方法和相关理论来推断时间依赖的函数数据。最基本的工具是自归一化(SN)，这是最近在单变量时间序列背景下发展起来的一种新的学生化技术。PI建议在函数设置中提出新的基于SN的方法，并开发(I)一类基于SN的测试统计量来测试弱相依函数数据的均值函数和相依结构中的变点；(Ii)一类基于SN的测试统计量来测试Hilbert空间中的白噪声和函数空间中AR(1)模型的有效诊断检验工具；(Iii)两个样本设置中的新的基于SN的测试。检验可用于检验两个可能相关的函数时间序列是否具有相同的均值和/或自协方差结构。在该建议中，SN是建立时间相关函数数据的关联和系统推理方法体系的基础。该建议的动机是与大气科学家持续合作，对数值模式输出与实际观测进行比较的属性进行统计评估。为了研究气候变化，这是本世纪世界面临的最紧迫的问题之一，科学家们主要依靠数值气候模式的气候预测。目前，研究数值模式输出与实际观测结果的差异及其差异的表征是一个主要的兴趣。分析这些数据是非常具有挑战性的，因为它们是海量的、高度复杂的、具有复杂的时空相关性。在本提案中，PI开发的基于SN的推理方法解决了这些问题。在泛函主成分分析的辅助下，基于SN的方法能够处理具有相关性的海量数据集，因为该方法自动考虑未知的弱相关性，不涉及任何调整参数的选择(因此在计算上是相当有效的)，并且非常容易用渐近枢轴极限分布来实现。将基于SN的方法直接应用于气候数据，有望帮助大气科学家更好地了解数值模式输出模拟真实观测的能力。此外，拟议的方法将广泛直接应用于从精细时间尺度上的非常精确的测量获得的数据，这些数据经常出现在工程、物理科学和金融中。在教育方面，PI将开发新的高级主题课程，指导本科生和研究生，并在这个项目中让他们接触到最先进的研究。

The PI develops a systematic body of methods and related theory on inference of temporally dependent functional data. The basic tool is the self-normalization (SN), a new studentizing technique developed recently in the univariate time series setting. The PI proposes to advance new SN-based methods in functional setup and develop (i) a class of SN-based test statistics to test for a change point in the mean function and dependent structure of weakly dependent functional data; (ii) a class of SN-based test statistics to test for white noise in Hilbert space and effective diagnostic checking tools for the AR(1) model in functional space; (iii) new SN-based tests in the two sample setup. The tests can be used to check if the two possibly dependent functional time series have the same mean and/or autocovariance structure. In this proposal, the SN is the foundation on which the body of connected and systematic inference methods for temporally dependent functional data is built.The proposal is motivated by ongoing collaboration with atmospheric scientists on statistical assessment of properties of numerical model outputs as compared to real observations. To study climate change, which is one of the most urgent problems facing the world this century, scientists have relied primarily on climate projections from numerical climate models.  There is currently a major interest to study how different the numerical model outputs are from real observations and the characterization of their difference. Analyzing these data are quite challenging because they are massive and highly complex with intricate spatial-temporal dependence. The SN-based inference methods that the PI develops in this proposal address these issues. With the assistance of functional principal component analysis, the SN-based methods are able to handle massive data sets with dependence, because the methods automatically take the unknown weak dependence into account, do not involve the choice of any tuning parameters (so are quite efficient computationally),   and are very straightforward to implement with asymptotically pivotal limiting distributions. A direct application of the SN-based methods to climate data is expected to help atmospheric scientists gain a better understanding of the ability of numerical model outputs in mimicking real observations.  In addition, the proposed methods will have broad direct applications to data that are obtained from very precise measurements at fine temporal scales which frequently  arise in engineering, physical science and finance. On the educational front, the PI will develop new advanced topic courses, mentor undergraduate and graduate students and expose them to the state-of-the-art research in this project.