多维因变量充分降维与多总体共同充分降维方法研究

With the rapid developments of bioinformatics and life sciences, research into high dimensional or ultra-high dimensional statistical analysis has gained considerable momentum. This project is devoted to research problems regarding high dimensional data with complex structures among multivariate responses and multi-population. Classical theory and methodology of sufficient dimension reduction will be generalized to the complex situations of multivariate responses and multi-population. Based on our proposed methodology for sufficient dimension reduction with multivariate responses or multi-population, we further develop a general statistical theory for marginal dimension test and marginal coordinate test. Combining our proposed method with forward stepwise regression and sparse precision matrix, we will develop a new class of feature screening or selection methods and establish the screening or selection consistency. For multi-population, we will further introduce common sufficient dimension reduction theory and the corresponding methodology, which is certainly an important extension of the classical common principal component analysis. We will also dedicate to applying the new theory and methodology we propose to bioinformatics or life sciences to make positive contributions.

随着生物信息学和生命科学的发展，高维甚至是超高维数据分析得到了统计学界的广泛关注。本课题将针对高维数据中复杂的多因变量结构以及多总体结构展开深入研究。我们将经典的充分降维理论推广至多因变量以及多总体情形。基于所提的多因变量或多总体的充分降维方法，我们将发展一套完整的结构维数以及变量显著性检验理论。我们还将向前逐步回归以及稀疏协方差逆矩阵与所提新方法结合，发展一类新型的超高维变量筛选或选择方法，并建立变量筛选或选择的相合性。对于多总体情形，我们还将引入共同充分降维理论和方法，这将是经典的共同主成分分析理论的重要推广。本课题还将致力于将所发展的新理论新方法应用于生物信息学或生命科学，争取对这些领域做出有益的贡献。

Our research related to this project are three fold. Firstly, we conduction research on sufficient dimension reduction based on sparse estimation, minimax estimation, test theory and post inference. Secondly, we generalize the result of sufficient dimension reduction with single population to multi-population. Finally, we develop new sufficient dimension reduction method for multivariate response data based on conditional characteristic function. With this specific NSF support, our team published 9 papers, 3 in Annals of Statistics, 1 in Journal of the American Statistical Association, 2 in Statistical Sinica, 1 in Journal of Multivariate Analysis, 1 in Journal of Statistical Computation and Simulation, 1 in Journal of Nonaprametic Statistics.

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