Dimension Reduction for Stochastic Processes

传统的多元统计侧重于分析有限长度的向量数据。然而，现代数据收集方法现在经常返回应该被视为数字化记录或随机过程抽样的结果的观察结果。这些类型的数据出现在功能数据分析（其中观察过程具有一维索引集），图像分析和空间统计的背景下，例如，并且出现在现代世界的各个方面。我们处理和利用这些数据的能力直接影响着科学研究和解决实际问题的方式。分析这类高维数据的目标包括结构检测和降维，这些都是将在拟议项目中解决的问题。具体而言，将研究一些关于随机过程数据降维的主题，包括i)两个或多个过程的标准相关分析，ii)用于分析多过程数据的混合模型方法，iii)逆回归和iv)变系数模型。所有这些工作的统一主题是使用再现核希尔伯特空间方法来制定问题及其提出的解决方案。随着现代技术在医学、环境科学、国土安全等领域的快速发展，这些领域收集的数据往往是曲线数据或空间数据，并且在范围和结构上可能更加复杂。传统的统计方法主要是为了处理低维数据而创建的，不适合上述数据的高维或“功能”性质。本研究旨在解决现代统计数据分析新兴分支中处理高维数据的一些基本问题。所获得的结果不仅将潜在地影响跨无数学科的多维数据分析方法，而且还将为未来的统计研究提供理论基础和方向。

Traditional multivariate statistics focuses on the analysis of data thatare vectors of finite length. However, modern data collection methods arenow frequently returning observations that should be viewed as the resultof digitized recording or sampling from stochastic processes. These types ofdata occur in the context of functional data analysis (where theobservation process has a one dimensional index set), image analysis andspatial statistics, for example, and arise from every aspect of the modernworld. Our ability to process and to make use of such data directlyaffects the way in which scientific inquiries are conducted and practicalproblems are solved. Goals for analyzing this type of high dimensionaldata include the detection of structure and dimensionality reduction andthese are the issues that will be addressed in the proposed project.Specifically, a number of topics will be investigated concerning dimensionreduction for data from stochastic processes including i) canonicalcorrelations analysis for two or more processes, ii) mixed models methodsfor analysis of data from multiple processes, iii) inverse regression andiv) varying coefficient models. The unifying theme in all this work is theuse of reproducing kernel Hilbert space methods to formulate both theproblems and their proposed solutions.With fast progressing modern technology in fields such as medicine,environmental science, and homeland security, the data collected in thosefields today are frequently curved or spatial data, and may be even morecomplicated in terms of scope and structure. Traditional statisticalmethods were created to primarily deal withlow-dimensional data, and are not suitable for the high-dimensional or``functional'' nature of the data described above. This research is aimedat addressing a number of fundamental issues in the emerging branch ofmodern statistical data analysis that deals with high-dimensional data.The results to be obtained will not only potentially impacthigh-dimensional data analytic methodology across a myriad ofdisciplines, but will also provide a theoretical foundation anddirections for future statistical research.