Spiked模型中特征值和特征向量的理论分析与推断

The high dimensional data analysis is one of the most important research areas in modern statistics. The high dimensional data observed in wireless communication, economic and speech recognition often contains a lot of noises and a small number of principal components. Due to the influence of the dimension of the observations, the classical statistical methods always lead to inaccurate inference in dealing with such high dimensional data. During the explotation of finding appropriate methods to analyze this kind of data, the spiked population model emerged. In a spiked population model, the population covariance matrix has a few eigenvalues well separated from others and both the dimension of observations and the sample size tend to infinity. It is of great importance to investigate the properties of eigenvalues and eigenvectors of the spiked population model for high dimensional data analysis because one can estimate the number of signals in wireless communications and potential factors in economic dataset and classify genetic data from these information. Thus in order to more efficiently analyze the high dimensional data, in this project we are going to study properties of the sample eigenvalues and eigenvectors and make inference about the population eigenvalues and eigenvectors for the spiked population model.

高维数据处理是现代统计学中重要的研究方向之一.无线电通讯、经济和语音识别等领域观测到的高维数据常包含大量噪声和少数主成分.由于维数的影响,经典的统计方法在处理这类高维数据时常得出不准确的结论.在探索这类高维数处理方法的过程中,spiked模型应运而生,模型假设样本维数与样本个数都趋于无穷,总体协方差阵有少数几个离群特征值.分析spiked模型中特征值与特征向量的性质对高维数据处理至关重要,据此可以估计和检验无线电通讯中的信号个数,经济数据中潜在的因子个数以及对基因数据进行分组等.本项目拟对spiked模型样本特征值和特征向量的性质进行研究进而对总体特征值和特征向量做出推断,以求更有效地分析和处理高维数据.

高维数据处理是现代统计学中重要的研究方向之一.由于维数的影响,经典的统计方法在处理这类高维数据时常得出不准确的结论.因此在高维条件下研究各种统计模型的性质对高维数据分析有非常重要的意义，本项目针对spiked模型及相关的分块随机矩阵的极限性质进行研究，以求能为处理高维数据提供可靠依据. 通过项目组成员的共同努力, 在三年内共发表SCI论文4篇, 投稿1篇, 圆满完成了预期研究目标。