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。主要的激励应用是神经成像分析，与BRAIN Initiative相关。（然而，开发的方法和理论适用于涉及时空建模的更广泛的领域。该研究计划具有强大的多学科合作组成部分，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对临床研究产生直接影响，并在艾滋病毒/艾滋病、主要神经精神和神经变性疾病、正常大脑发育和癌症等医学研究中有更广泛的应用。所讨论的问题也引起一般社会的广泛兴趣，因为它们涉及到诸如保健政策和社会保障规划等紧迫问题。随着现代影像学技术的发展，许多大规模的研究已经或正在广泛进行，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的重要特征：（i）极高维，（ii）分段光滑，（iii）时间，（iv）空间相关。由于缺乏有效的统计工具和理论，对这些数据的分析以及将其与临床数据相结合的工作受到阻碍，这突出表明迫切需要从统计学的角度发展方法和理论。 该项目从时域和频域的三个更广泛的角度应对挑战。第一个视角开发了用于自适应函数估计的时空模型。该模型可以有效地从噪声功能数据中提取信息标记。第二个角度关注的功能数据组的降秩模型与时空相关性。此外，第三个角度开发先进的功能混合效应模型，用于模拟重复功能反应和一组相关协变量之间的不同关联函数，同时考虑复杂的时空相关性。

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.