Topics in Dimensionality Reduction in Nonparametric Statistical Modelling

调查员计划进行研究，以确定结构的多变量数据，而不强加严格的结构假设和结构适应技术的发展，在非参数回归和密度估计的降维。 在非参数和半参数估计的最新进展的基础上，提出了开发迭代算法，该算法交替使用平均导数泛函的估计来识别低维结构，并通过使用当前的结构信息来改进模型估计。 本文将讨论部分线性回归模型和独立成分分析模型中线性和非线性成分的识别和估计。随着大型复杂数据库和计算机能力的急剧增加，越来越需要和可能开发非参数模型、概念、和程序，可用于研究变量之间的关系，并建立模型，而不依赖于严格的参数假设的结构上的平均响应和误差分布。本研究中开发的算法将提供新的统计学习工具，用于降低多变量数据的维数，以识别和可视化其结构。 在对模拟数据进行研究并对其参数进行微调后，它们将被应用于生物信息学，风险管理和其他领域的统计学习问题。

The investigator plans to conduct research on identifying structure in multivariate data without imposing rigid structural assumptions and on the development of structural adaptation techniques for dimensionality reduction in nonparametric regression and density estimation. Building on recent advances in nonparametric and semiparametric estimation, it is proposed to develop iterative algorithms which alternate between identification of the lower dimensional structure, using estimates of average derivative functionals, and model estimation improved by using the current structural information. Identification and estimation of linear and nonlinear components in partially linear regression models and of independent component analysis model with unspecified component densities will be considered.With the dramatic increase in large, complex data bases and in computer power, it has become increasingly more desirable and possible to develop nonparametric models, concepts, and procedures that can be used to study relationships between variables and to construct models without relying on rigid parametric assumptions on the structure of mean responses and error distributions. Algorithms developed in this research will provide new statistical learning tools for reducing dimensionality of multivariate data in order to identify and visualize its structure. After the algorithms are investigated on simulated data and their parameters are fine-tuned, they will be applied to statistical learning problems from bioinformatics, risk management, and other areas.