Mathematical modeling for high-dimensional nonlinear data and its application to the analysis of complex phenomena

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In recent years the wide availability of fast and inexpensive computers enables us to accumulate a huge amount of data, and the effective use of databases are required in order to create innovation and values and to solve important science and engineering problems. Statistical challenges posed by large data sets arise in such areas as genome databases in life science, remote-sensing data from earth observing satellites, POS data in marketing and economic data. Through this research project we have investigated the problem of constructing various types of statistical nonlinear modeling strategies and obtained the results in the following :(1)We proposed nonlinear modeling techniques ; determining a set of basis functions, estimating the unknown parameters by regularization and then evaluating the constructed model to select a suitable one among competing models. We describe modeling based on functional approach and introduce a generalized information criterion. Bayesian information criterion BIC is also extended in such a way that it can be applied to the evaluation of models estimated by the method of regularization.(2)We proposed functional regression modeling and functional discriminant analysis, using Gaussian radial basis functions along with the technique of regularization. The proposed method was applied to the analysis of yeast cell cycle gene expression data.(3)Approximate selection of embedding dimension and delay time have been a central issue of chaotic dynamical systems. We introduced the delay time and consider the estimation of embedding dimension and delay time.(4)Model selection criteria were presented for stochastic process from an information-theoretic approach. We derived asymptotic expansions for the distributions of statistics related to small diffusions and applied it to option pricing in economics.