Topics in Dimensionality Reduction in Nonparametric Statistical Modelling

研究者计划在不强加刚性结构假设的情况下识别多变量数据中的结构，并在非参数回归和密度估计中开发用于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，提出了一种迭代算法，该算法可以在识别低维结构（使用平均导数函数估计）和使用当前结构信息改进的模型估计之间交替进行。将考虑部分线性回归模型和未指定成分密度的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，开发非参数模型、概念和程序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，并在不依赖于平均响应和误差分布结构的严格参数假设的情况下构建模型。本研究开发的算法将为多元数据的降维提供新的统计学习工具，以识别和可视化其结构。在模拟数据上研究算法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.