Collaborative Research: Integral Transform Methods for Sufficient Dimension Reduction in Regression

本计画的目的在于发展理论与方法，以在包含大量预测变数的回归分析中，达到充分的降维。研究人员提出了一种通用的方法，称为积分变换的方法，以促进降维。该方法的核心思想是利用积分变换和响应变换来改变降维区域。由于广泛的转换和积分转换的可用性，这种方法导致了一个灵活和有效的框架，用于处理和解决高维性带来的挑战。通过一系列定义明确的研究问题，研究人员研究这个框架，并为许多重要的回归应用开发特定的降维方法。该项目的成功不仅为高维数据分析提供了有效的实用工具，而且代表了半参数推断理论和方法的进步。高维数据涉及大量变量，目前在科学研究、政府、商业、众所周知，高维性给这些数据的处理和分析带来了困难。这通常被称为维数灾难。有一个迫切的需求，统计工具，能够减轻灾难的维数通过降维。该项目代表了对这一需求的回答，特别是旨在实现回归的降维。该项目的结果可以广泛应用于需要涉及大量变量的回归的领域。基因表达和蛋白质序列数据分析就是这样一个例子。因此，该项目有助于加强科学研究和发现，并有利于各种社会和经济活动。

This project is aimed to develop theory and methods for sufficient dimension reduction in regression analysis involving a large number of predictor variables. The investigators propose a general approach called the integral transform approach to facilitating dimension reduction. The key idea of this approach is to use integral transform and response transformation to change the domain where dimension reduction is performed. Due to the availability of a wide range of transformations and integral transforms, this approach leads to a flexible and effective framework for addressing and resolving challenges raised by high dimensionality. Through a series of well-defined research problems, the investigators study this framework and develop specific dimension reduction methods for many important regression applications. The success of this project not only provides effective practical tools for high-dimensional data analysis but also represents an advance in the theory and methodology of semiparametric inference.High-dimensional data that involve a large amount of variables are nowadays routinely generated and collected in areas such as scientific research, government, business, etc. It is well-known that high dimensionality causes difficulties in processing and analyzing these data. This is commonly referred to as the curse of dimensionality. There is an urgent demand of statistical tools that are able to mitigate the curse of dimensionality through dimension reduction. This project represents an answer to this demand and is particularly aimed at achieving dimension reduction in regression. The results from this project can be widely applied in areas where regression involving a large number of variables is required. Gene expression and protein sequence data analysis is one such example. Therefore, this project can help enhance scientific research and discovery and benefit a variety of social and economical activities.