Nonlinear Data Analysis for Functional Data

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In most conventional data analysis methods, we assume that data set is regarded as a set of numbers with some structures, for example a set of vectors or a set of matrices etc. Nowadays, we must often analyze more complex data. One type of the complex data is functional data structure ; data themselves are represented as functions. Ramsay and Silverman have studied function data analysis (FDA) as the analysis method to function data since the 1990's. They have published excellent books on FDA (Ramsay & Silverman, 1997, 2002).In this study, we promoted the research of the function data analysis method, the building of a theory system, an application study to the practical problems and so on.Specifically, we discussed with Prof. Ramsay and the experts of the statistical science and the information engineering. Based on those discussions, we reviewed about the expansion of the coverage of FDA. Moreover, by using the knowledge of the nonlinear data analysis, the new analysis methods were developed.Study results are classified into the following.(1)The definition of the framework of the function data :We investigated a research trend about the function data analysis. We reviewed the directionality of this research task.(2)Developments of methods for Functional Data Analysis :We developed methods for FDA including functional regression, functional MDS and functional clustering.(3)Discrete functional data analysis :We proposed a method to find the structures that the discrete functional data have by utilizing the proposed high order differences.(4)The research of the related field :We dealt with virtual parallel computer environment, dimension reduction methods and variable selection as the related topics.