Several Problems in Dimension Reduction

降维在大数据、复杂数据的现代几乎是一种不可避免的做法，也是本次提案的主要议题。得益于我们在这方面的成功，我们的研究项目将进一步拓宽降维研究领域的范围，解决该领域的几个难点、微妙和未被研究的问题。这些结果将引起广泛的兴趣，并促进在不同领域的应用，其中测量可能包含误差和/或数据可能不是完全随机的，例如在医学研究中。首席调查员(P.I.)将研究降维问题中的五个主题，开发新的方法，并分析它们的性质和性能。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，了解为什么当前这些条件的实施是有害的，规定最佳利用这些条件的程序，并显示由此产生的效率收益。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。P.I.将同时处理中心方差空间和中心均值空间，并说明为什么需要这样做。她将同时考虑两个空间何时不同以及何时重合，并建立两种情况下的估计和推理方法。第三个话题涉及多变量反应。P.I.将使用一种新的建模方法，将中心空间概念扩展到总体框架，作为当前两种方法之间的中间地带。她将展示新模型的优势，并提出估计和推理方法。P.I.还将建立多元响应中心空间估计的半参数效率界，并说明为什么和如何进行估计以获得局部有效性。第四个主题涉及协变量包含测量误差时的降维。P.I.将研究一个一般的半参数降维模型，当有误差地测量感兴趣的协变量并对其进行参数建模时。将设计一种不需要对不可观测变量分布进行建模的偏差校正程序。估计量将被证明是一致的，渐近正态的。第五个主题涉及在二次分析中分析病例对照数据时所产生的降维。P.I.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建模，但数据的病例对照性质需要使用多变量协变量的各种非参数估计，从而导致降维的需要。原始模型的内在联系进一步导致了具有特殊结构的降维，对于这种降维，P.I.将设计一致的估计器，并针对回归误差分布的错误指定建立其局部有效性和稳健性。

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.