Advanced Statistical Tools for Ultra-High Dimensional Functional Data with Spatial-Temporal Correlation

该项目涉及开发创新的先进统计工具，用于分析具有时空相关性的超高维功能数据。主要的激励应用是与大脑倡议相关的神经成像分析。(然而，发展的方法和理论适用于更广泛的领域，包括时空建模。)该研究项目有一个强大的多学科协作部分，主要团队成员来自生物统计学/统计学、计算机科学、精神病学、放射学和心理学。正在开发的工具和软件可以在临床研究中产生立竿见影的效果，并在艾滋病毒/艾滋病、重大神经精神和神经退行性疾病、正常大脑发育和癌症等医学研究中有更广泛的应用。所解决的问题也是广大社会广泛关注的问题，因为它们涉及诸如保健政策和社会保障规划等紧迫问题。随着现代成像技术的发展，许多大规模的研究已经或正在被广泛地进行，以收集丰富的功能数据和临床数据。函数数据共有四个共同和重要的特征：(I)极高的维度，(Ii)分段平滑，(Iii)时间上的，和(Iv)空间上的。由于缺乏有效的统计工具和理论，这类数据的分析及其与临床数据的结合一直受到阻碍，这凸显了从统计学角度发展方法和理论的迫切需要。该项目从三个更广泛的角度解决了时间和频率领域的挑战。第一种观点发展了自适应函数估计的时空模型。该模型能有效地从含有噪声的函数数据中提取信息标记。第二种观点涉及具有时空相关性的功能数据组的降阶模型。此外，第三个视角发展了先进的功能混合效应模型，用于模拟重复功能反应和一组感兴趣的协变量之间的不同关联函数，同时考虑到复杂的时空相关性。

This project concerns developing innovative advanced statistical tools for the analysis of ultra-high dimensional functional data with spatial-temporal correlation. The primary motivating application is neuroimaging analysis, relevant to the BRAIN Initiative. (However, the developed methods and theory are applicable to a much broader range of fields involving spatial-temporal modeling.) The research program has a strong multidisciplinary collaborative component, with key team members drawn from biostatistics/statistics, computer science, psychiatry, radiology, and psychology. The tools and software under development can have immediate impacts in clinical research, and have wider applications in medical studies of HIV/AIDS, major neuropsychiatric and neurodegenerative disorders, normal brain development, and cancer, among many others. The problems addressed are also of broad interest to general society, since they relate to pressing issues such as health care policies and social security planning. With modern imaging techniques, many large-scale studies have been or are being widely conducted to collect a wealthy set of functional data and clinical data. Functional data share four common and important features: (i) extremely high dimensional, (ii) piecewise smooth, (iii) temporally, and (iv) spatially dependent. The analysis of such data and integration of them with clinical data have been hindered by lack of effective statistical tools and theory, underscoring the great need for methodological and theoretical development from a statistical perspective. The project addresses challenges from three broader perspectives in both time and frequency domains. The first perspective develops spatial-temporal models for adaptive function estimation. The models can effectively extract informative markers from noisy functional data. The second perspective concerns reduced rank models for groups of functional data with spatial-temporal correlation. In addition, the third perspective develops advanced functional mixed effects models for modeling varying association function between repeated functional responses and a set of covariates of interest, while accounting for complex spatial-temporal correlation.