Collaborative Research: New Developments for Analysis of Two-Way Structured Functional Data

大多数现有的功能数据分析方法仅限于在一个域中结构化的数据，例如时间。双向结构化函数数据通过空间和时间两个功能域来索引，每个域都有自己的正则性概念，如平滑或稀疏性。充分考虑数据的双向结构将导致更准确的分析结果。最近PI领导的双向正则化工作为研究双向函数数据降维、特征提取、回归和分类的新方法提供了一些初步结果和良好的起点。该研究小组计划在几个重要方向上进一步发展双向函数数据的方法学：(A)发展双向正则化奇异值分解(SVD)的再生核Hilbert空间理论；(B)发展新的降维方法，以流形等一般领域为索引的数据；(C)发展有效分析离散双向函数数据的降维方法；(D)发展解决脑磁图(MEG)逆问题的双向正则化方法；(E)开发以动态脑磁图图像为预测器的用于精神疾病诊断的新分类器；(F)制定能够抵抗异常值的稳健方法。这项研究的成功将为功能数据分析增加一个新的维度，并极大地丰富该领域。双向结构化功能数据出现在不同的学科，包括医学、社会科学、地球科学、经济学和商学。但很少有现有的方法完全考虑到这种类型的函数数据的双向结构。这项研究开发的新的统计方法将为有效利用这些数据提供宝贵的工具。它们将提供对科学、社会和经济现象的更好理解，并做出更准确的预测。特别是，双向正则化的SVD提供了一种新的方差分析方法，用于分析高通量生物信息学数据，并发现与疾病表型相关的生物标记物和临床变量之间的相互作用。新的脑磁图逆解算器将有助于对大脑功能区进行非侵入性的术前标测。新的分类器将有助于使用动态脑磁图对精神疾病进行诊断和评估。拟议的活动包括培训参与拟议项目的博士生，以及指导心理学系的初级统计员女性P.I.。研究成果将通过合作工作、学术报告和期刊出版物进行传播。将创建网页，以便快速访问新方法的用户友好和可访问的软件实施以及技术报告和相关参考资料。

Most existing methodologies of functional data analysis are limited to data structured in one domain, such as time. Two-way structured functional data are indexed by two functional domains, such as space and time, and each domain has its own notion of regularity, such as smoothness or sparsity. Fully considering the two-way structure of the data will lead to more accurate analysis results. Recent work by the lead PI on two-way regularization has provided some preliminary results and a good starting point for investigating new methodology of dimension reduction, feature extraction, regression and classification for two-way functional data. This research team plans to further develop the methodology for two-way functional data in several important directions:(a) Develop a Reproducing Kernel Hilbert Space theory for two-way regularized singular value decomposition (SVD); (b) Develop novel dimension reduction methods for data that are indexed by general domains such as manifolds; (c) Develop dimension reduction methods that effectively analyze discrete two-way functional data; (d) Develop two-way regularization methods for solving the magnetoencephalography (MEG) inverse problem; (e) Develop new classifiers for diagnosis of mental disorders using dynamic MEG images as predictors; (f) Develop robust methods that are resistant to outliers. The success of the research will add a new dimension to functional data analysis and significantly enrich the field.Two-way structured functional data arise in various disciplines, including medicine, social sciences, earth sciences, economics, and business. But few existing methodologies fully take into account the two-way structure of this type of functional data. The novel statistical methods developed in this research will provide valuable tools for efficient use of such data. They will provide better understanding of scientific, social and economic phenomena and make more accurate predictions. In particular, the two-way reguarlized SVD provides a new analysis of variance method for analyzing high throughput bioinformatics data and for discovering interactions among biomarkers and clinical variables that are associated with a disease phenotype. The new MEG inverse solvers will facilitate noninvasive presurgical mapping of functional areas of the brain. The new classifiers will help diagonosis and assessment of mental diseases using dynamic MEG images. The proposed activities involve training of Ph.D. students who participate in the proposed projects and mentoring of the female, junior statistician P.I. in a psychology department. Research results will be disseminated through collaborative work, academic presentations, and journal publications. Web pages will be created to enable quick access to user-friendly and accessible software implementations of new methods as well as technical reports and relevant references.