A Novel Approach to Study Nonlinearity and Interaction in Regression

全球许多经济部门以及各种科学学科对大型数据库的访问日益增多，带来了重大挑战，需要新的统计方法来进行数据分析。 这些数据库的一个共同特征是所研究的主题数量众多，并且描述每个主题的各种属性的大量变量。 在简约和准确性双重原则的指导下，可以开发统计模型作为数据生成过程复杂机制背后的未知物理现实的信息近似。 然而，缺乏对系统中涉及的变量之间复杂的非线性相互作用的本质和模式的理解，在很大程度上阻碍了高维数据分析领域的进展。 在该项目中，将开发新的统计理论、方法和软件。这些结果将解决关键的基础问题并开辟新的研究途径。回归是统计学中最基本的概念之一，几乎适用于所有学科。 人们对高维回归的兴趣日益浓厚，无论是从正向还是逆向角度来看，都带来了许多方面，这些方面提高了其在复杂数据建模方面的灵活性，并具有经过验证的统计效率。 在基因组学的数据密集型领域，一种新的统计概念——液体关联（LA）已经被开发出来，它使研究人员能够研究基因之间共同调节的动态模式。 该项目将研究在高维回归中采用 LA 进行变量选择的方法。 结合包括切片逆回归（SIR）在内的逆建模技术，可以揭示非线性相关变量簇之间隐藏的相互作用模式。 在许多应用中，通常需要合并一组已知在模型中很重要的背景变量。 这增加了揭示互锁的变量与变量交互模式的另一层复杂性。 还将开发边缘化背景变量造成的干扰的方法。

The increasing access to large databases across many global economic sectors, as well as various scientific disciplines, poses significant challenges that demand novel statistical methodologies for conducting data analytics. A common feature of these databases is the large number of subjects under study, and the large number of variables describing a wide variety of attributes attached to each subject. Guided by the dual principles of parsimony and accuracy, statistical models can be developed to serve as informative approximations to the unknown physical reality underlying the intricate mechanism of the data generation process. However, a lack of understanding of the nature and the pattern of complex nonlinear interactions between variables involved in the system has largely hindered the progress in the field of high-dimensional data analysis. In this project, novel statistical theory, methods, and software will be developed. These results will address key foundational issues and open up new avenues for research.Regression is one of the most fundamental concepts in statistics, with applications in almost all disciplines. The growing interest in high-dimensional regression either from the forward or from the inverse perspective has brought out many facets that increase its flexibility in modeling complex data with proven statistical efficiency. Working in the data-intensive area of genomics, a novel statistical notion, liquid association (LA), has been developed that allows researchers to study dynamic patterns of co-regulation between genes. This project will investigate methods of employing LA for variable selection in high-dimensional regression. In combination with inverse modeling techniques including sliced inverse regression (SIR), hidden patterns of interaction between clusters of nonlinearly-correlated variables can be revealed. In many applications, it is often necessary to incorporate a certain set of background variables that are known to be important in the model. This adds another layer of complexity in revealing patterns in the interlocked variable-to-variable interactions. Methods of marginalizing the interference caused by background variables will also be developed.