Dimension Reduction for Stochastic Processes

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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.