高维数据建模与分析的若干问题

As modern science and technology develops rapidly, a huge number of complex high-dimensional data arise in the life sciences, information science, material science, industrial engineering, economy, and finance. Besides high dimensionality, these data have the feature of strong nonlinearity with strong noises and correlations. It is an important and challenging issue how to model and analyze such data, and this provides an opportunity to develop statistics. This project will study several pressing problems in modeling and analysis of high-dimensional data, including that, for high-dimensional regression models, we will study the algorithms of non-convex penalized estimation, and present their convergence properties and statistical properties; Based on the previous results, we will also study the model averaging methods, hypotheses testing and interval estimation; we will study robust estimation with high efficiency for high-dimensional linear models; we will study the modeling methods for high-dimensional data from computer experiments, and provide the inferential and predictive methods under the models.

随着现代科技的迅猛发展，在生命科学、信息科学、材料科学、工业工程以及经济金融等许多领域都产生了大量的复杂高维数据。这些数据不仅维数很高，而且具有强噪声、强相依性和强非线性等特点。如何对这些复杂高维数据进行统计建模和分析是非常重要而且又极具挑战性的一类问题，也是发展统计学的极好机遇。本项目拟研究高维数据的建模与分析中急待解决的若干问题，包括：针对高维回归模型，研究非凸惩罚估计的算法，给出算法的收敛性质与统计性质，并在此基础上研究高维模型平均方法、参数的假设检验和区间估计等；研究高维线性模型参数高效率的稳健估计方法；研究高维计算机实验数据的建模方法，并给出模型的推断和预报方法。

本项目研究了高维数据的建模与分析中急待解决的若干问题，包括：针对高维回归模型，给出了一般惩罚估计方法的新算法，并给出算法的收敛性质与在非凸惩罚下的统计性质，并在此基础上研究了高维模型平均方法、参数的假设检验和区间估计等；给出了高维线性模型参数高效率的稳健估计与变量选择方法；给出了高维计算机实验数据的校准方法、序贯建模方法和变量选择方法等。这些研究成果很多发表在Annals of Statistics, Biometrika等统计学一流期刊上。本项目的研究基本实现了拟定的研究目标。