Topics in Dimensionality Reduction in Nonparametric Statistical Modelling

研究人员计划在不强加刚性结构假设的情况下，在多变量数据中识别结构，并开发用于非参数回归和密度估计中降维的结构适应技术。基于非参数和半参数估计的最新进展，建议开发迭代算法，在使用平均导数泛函估计的低维结构识别和利用当前结构信息改进的模型估计之间交替进行。部分线性回归模型和独立分量分析模型中的线性和非线性分量的识别和估计将被考虑。随着大型、复杂的数据库和计算机能力的急剧增加，开发非参数模型、概念和程序已变得越来越可取和可能，这些模型、概念和程序可用于研究变量之间的关系，并且不依赖于基于均值响应和误差分布结构的僵硬的参数假设来构建模型。本研究开发的算法将为降维多变量数据提供新的统计学习工具，以便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，这些算法将被应用于生物信息学、风险管理等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.