Collaborative Research: Analysis of Functional and High-Dimensional Data with Applications

该项目处理导致大量，多维和/或功能数据集的现象的统计推断。主要的例子是地球物理、生物医学和互联网相关数据。除了高维之外，这些数据通常具有自亲和性，并且需要非标准（功能）模型来进行建模和随后的统计分析。将要开发的方法将推进功能数据分析的理论和实践，这是一个非常快速发展的现代统计领域。这里提出的统计方法的共同和新颖之处在于分析数据的性质。数据集是海量的、多维的、功能性的，并且可能是自仿射的（分形或多重分形）。多尺度数据表示的最新进展为(i)开发用于估计、测试、分类和反卷积的尺度敏感分析工具，以及（ii）描述、总结和建模自相似数据提供了自然而有效的环境。贝叶斯方法将被使用时，只要可用的先验信息可以合并或任何时候，明智的自动先验是可能的。新的推理方法的发展对最近的科学倡议和新兴技术的统计支持至关重要。所提出的研究是应用驱动的，因此应用领域的特殊性影响了方法的设计和重点。该方案提出的技术解决了新医学治疗方法的有效性检验、目标检测与分类、医学图像分类、雷达卫星数据更精确恢复等问题。因此，该提案所产生的方法适用于保健和医药以及国土安全等具有战略利益的领域。除了方法上的影响外，拟议的研究还具有很强的教育成分，包括培训研究生，让本科生参与研究项目，举办部门间研讨会，提高劳动力对数学教育的认识，以及吸引少数民族和女性学生。

This project deals with statistical inference inphenomena that result in massive, multidimensional and/orfunctional data sets. Prime examples are geophysical, biomedical,and internet related data. In addition to high dimensionality, suchdata are often characterized by self-affinity and requirenon-standard (functional) models for their modeling and subsequentstatistical analysis. The methodology to be developed will advanceboth the theory and practice of functional data analysis, a veryfast-developing and modern area of statistics. The common and novelfeatures of the statistical methods proposed here lie in the natureof analyzed data. The data sets are massive, multidimensional,functional, and possibly self-affine (fractal or multifractal).Recent progress in multiscale data representations provide naturaland efficient environments for (i) developing scale-sensitiveanalyzing tools for estimation, testing, classification, anddeconvolution, and (ii) describing, summarizing, and modelingself-similar data. Bayesian methodology will be used wheneveravailable prior information can be incorporated or whenever sensibleautomatic priors are possible.Development of new inferential methodologies is critical for thestatistical support of recent scientific initiatives and newlyemerging technologies. The proposed research is application driven,so the specificities of the application fields influence the designand focus of the methodology. Techniques suggested in the proposaldeal with problems of testing of efficiency of new medicaltreatments, target detection and classification as well asclassification of medical images, or more accurate recovery of radaror satellite data. Hence, the methodologies which result from theproposal are applicable in such areas of strategic interest ashealth and medicine and homeland security. In addition tomethodological impact, the proposed research has a strongeducational component consisting of training graduate students,involving undergraduate students in research projects, conductinginter-departmental seminars, increasing awareness of mathematicseducation among the work force, and attracting minority and femalestudents.