Traditional multivariate statistics focuses on the analysis of data that are vectors of finite length. However, modern data collection methods are now frequently returning observations that should be viewed as the result of digitized recording or sampling from stochastic processes. These types of data occur in the context of functional data analysis (where the observation process has a one dimensional index set), image analysis and spatial statistics, for example, and arise from every aspect of the modern world. Our ability to process and to make use of such data directly affects the way in which scientific inquiries are conducted and practical problems are solved. Goals for analyzing this type of high dimensional data include the detection of structure and dimensionality reduction and these are the issues that will be addressed in the proposed project. Specifically, a number of topics will be investigated concerning dimension reduction for data from stochastic processes including i) canonical correlations analysis for two or more processes, ii) mixed models methods for analysis of data from multiple processes, iii) inverse regression and iv) varying coefficient models. The unifying theme in all this work is the use of reproducing kernel Hilbert space methods to formulate both the problems and their proposed solutions.
With fast progressing modern technology in fields such as medicine, environmental science, and homeland security, the data collected in those fields today are frequently curved or spatial data, and may be even more complicated in terms of scope and structure. Traditional statistical methods were created to primarily deal with low-dimensional data, and are not suitable for the high-dimensional or ``functional'' nature of the data described above. This research is aimed at addressing a number of fundamental issues in the emerging branch of modern statistical data analysis that deals with high-dimensional data. The results to be obtained will not only potentially impact high-dimensional data analytic methodology across a myriad of disciplines, but will also provide a theoretical foundation and directions for future statistical research.
