Several Problems in Dimension Reduction

在数据大而复杂的现代时代，降维几乎是一种不可避免的实践，也是本提案的主要主题。得益于我们在这一领域的成功，我们的研究项目将进一步拓宽降维研究领域的范围，并解决该领域的一些困难、微妙和尚未研究的问题。这些结果将引起广泛的兴趣，并迅速应用于测量可能包含错误和/或数据可能不完全随机的不同领域，如医学研究。首席研究员（P.I.）将研究降维问题中的五个主题，开发新的方法并分析其性质和性能。第一个主题涉及降维中的两个常见条件：线性条件和常方差条件。P.I.将揭示这些条件的真实影响，了解为什么目前这些条件的实施是有害的，规定一个程序来优化利用这些条件，并显示由此产生的效率增益。这项研究将改变目前实施这些条件的做法。第二个主题涉及中心方差空间估计。pi将同时处理中心方差空间和中心均值空间，并说明为什么有必要这样做。她将考虑两个空间不同和重合的情况，并建立两种情况下的估计和推理方法。第三个主题涉及多元反应。P.I.将使用一种新的建模方法将中心空间概念扩展到一般框架，该方法作为两种当前方法之间的中间地带。她将展示新模型的优点，并提出估计和推理方法。本文还将建立多元响应中心空间估计的半参数效率界，并说明为什么以及如何进行估计以实现局部效率。第四个主题涉及协变量包含测量误差时的降维。当对感兴趣的协变量进行误差测量并参数化建模时，pi将研究一般的半参数降维模型。将设计一种偏差校正程序，它不需要对不可观测变量分布进行建模。估计量将被证明是一致的，渐近正态的。第五个主题涉及二次分析中病例对照数据分析时出现的降维问题。pi将说明，当多个协变量可用时，尽管对回归均值函数进行了完全参数化建模，但数据的病例控制性质需要使用多变量协变量进行各种非参数估计，因此需要降维。原始模型所赋予的内在关系进一步导致了具有特殊结构的降维，为此，pi将设计一致估计量，并针对回归误差分布的不规范建立其局部效率和鲁棒性。

Dimension reduction is almost an unavoidable practice in the modern era with big and complex data, and is the main topic in this proposal. Benefit from our success in this area, our research projects will further broaden the scope of the dimension reduction research domain and resolve several difficult, subtle and under-studied issues in this field. The results will generate wide interest and prompt applications in different fields where measurements may contain errors and/or where the data may not be completely random such as in medical studies. The principal investigator (P.I.) will study five topics in dimension reduction problems, develop new methodologies and analyze their properties and performances. The first topic concerns two common conditions in dimension reduction: the linearity condition and the constant variance condition. The P.I. will reveal the true effect of these conditions, understand why the current implementation of these conditions are detrimental, prescribe a procedure to optimally utilize these conditions and show the resulting efficiency gain. This research will change the current practice of implementing these conditions. The second topic concerns central variance space estimation. The P.I. will treat the central variance space simultaneously with the central mean space and demonstrate why it is necessary to do so. She will consider both when the two spaces differ and when they coincide, and establish methods for estimation and inference in both cases. The third topic concerns multivariate responses. The P.I. will extend the central space concept to the general framework using a new way of modeling that serves as a middle ground between two current approaches. She will demonstrate the advantages of the new model and propose estimation and inference methods. The P.I. will also establish the semiparametric efficiency bound for the multivariate response central space estimation,and illustrate why and how to conduct estimation to achieve local efficiency. The fourth topic concerns dimension reduction when covariates contain measurement errors. The P.I. will study a general semiparametric dimension reduction model when the covariate of interest is measured with error and modeled parametrically. A bias-correction procedure will be devised which does not require modeling the unobservable variable distribution. The estimators will be shown to be consistent, asymptotically normal. The fifth topic concerns dimension reduction arising when analyzing case-control data in secondary analysis. The P.I. will illustrate that when multiple covariates are available, despite of a completely parametric modeling of the regression mean function, the case-control nature of the data requires various nonparametric estimation with multivariate covariates, hence leading to the need of reducing dimension. The inherent relation imposed by the original model further leads to dimension reduction with special structure, for which the P.I. will devise consistent estimators and establish their local efficiency and robustness against the misspecification of the regression error distribution.