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领域（如时间）中结构化的数据。双向结构化功能数据由两个功能域（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，分析结果会更加准确。首席PI最近在双向正则化方面的工作提供了一些初步结果，并为研究双向功能数据的降维、特征提取、回归和分类的新方法提供了一个良好的起点。本研究小组计划在几个重要方向上进一步发展双向功能数据的方法：(a)发展双向正则奇异值分解（SVD）的再现核希尔伯特空间理论；(b)为按流形等一般领域编制索引的数据制订新的降维方法；(c)发展有效分析离散双向功能数据的降维方法；(d)发展解决脑磁图（MEG）逆问题的双向正则化方法；(e)开发新的分类器，利用动态脑磁图作为预测指标来诊断精神障碍；(f)发展能抵抗异常值的可靠方法。该研究的成功将为功能数据分析增加一个新的维度，并显著丰富该领域。双向结构化功能数据出现在各个学科中，包括医学、社会科学、地球科学、经济学和商业。但是，很少有现有的方法充分考虑到这类功能数据的双向结构。本研究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、社会和经济现象更好的理解，并做出更准确的预测。特别是，双向正则SVD为分析高通量生物信息学数据和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提供了一种新的方差分析方法。新的MEG逆求解器将促进非侵入性脑功能区域的术前绘图。新的分类器将有助于诊断和评估精神疾病使用动态脑磁图图像。拟议的活动包括培训参与拟议项目的博士生，并指导心理学系的女性初级统计学家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。将建立网页，使人们能够迅速获得新方法的用户友好和易于使用的软件实施，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.