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，如时间。双向结构化函数数据由两个函数域（如空间和时间）索引，每个域都有自己的规则性概念，如平滑性或稀疏性。充分考虑数据的双向结构将导致更准确的分析结果。PI领导的双向正则化最近的工作提供了一些初步的结果，并为研究双向函数数据的降维，特征提取，回归和分类的新方法提供了一个很好的起点。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：（a）发展用于双向正则化奇异值分解（SVD）的再生核希尔伯特空间理论;（B）开发用于一般域（如流形）索引的数据的新的降维方法;（c）开发有效分析离散双向函数数据的降维方法;（d）开发用于离散双向函数数据的降维方法。（d）开发双向正则化方法，解决脑磁图逆问题;（e）开发新的分类器，利用动态脑磁图图像作为预测指标诊断精神障碍;（f）开发抗异常值的稳健方法。该研究的成功将为函数数据分析增加一个新的维度，并大大丰富了该领域。双向结构化函数数据出现在各个学科，包括医学，社会科学，地球科学，经济学和商业。但是，现有的方法很少充分考虑到这种类型的功能数据的双向结构。本研究中开发的新的统计方法将为有效利用这些数据提供有价值的工具。它们将提供对科学、社会和经济现象的更好理解，并作出更准确的预测。特别是，双向正则化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，并用于发现与疾病表型相关的生物标志物和临床变量之间的相互作用。新的脑磁图逆解算器将促进脑功能区的非侵入性术前映射。新的分类器将有助于使用动态MEG图像诊断和评估精神疾病。拟议的活动包括培训博士。学生谁参加拟议的项目和指导的女性，初级统计学家P. I。在心理学系。研究成果将通过合作工作、学术报告和期刊出版物传播。将建立网页，以便能够迅速查阅方便用户和便于查阅的新方法的软件执行情况以及技术报告和有关参考资料。

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.