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

近年来，快速而廉价的计算机的广泛使用使我们能够积累大量的数据，需要有效地使用数据库来创造创新和价值，并解决重要的科学和工程问题。在生命科学中的基因组数据库、来自地球观测卫星的遥感数据、营销和经济数据中的POS数据等领域，出现了大量数据集带来的统计挑战。通过本课题的研究，我们对各种统计非线性建模策略的构建问题进行了研究，取得了以下结果：(1)提出了一种非线性建模技术：确定一组基函数，通过正则化估计未知参数，然后对所构建的模型进行评估，在竞争模型中选择一个合适的模型。我们描述了基于函数方法的建模，并引入了一种广义信息准则。对贝叶斯信息准则BIC进行了扩展，使其可以应用于正则化估计模型的评价。(2)利用高斯径向基函数和正则化技术，提出了函数回归建模和函数判别分析。将该方法应用于酵母细胞周期基因表达数据的分析。(3)嵌入维度和延迟时间的近似选择一直是混沌动力系统的中心问题。引入了延迟时间，并考虑了嵌入维度和延迟时间的估计。(4)从信息论的角度给出了随机过程的模型选择准则。我们得到了与小扩散相关的统计量分布的渐近展开式，并将其应用于经济学中的期权定价。

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.