基于规则化轨道算法和加速梯度算法的高维协方差矩阵估计的研究
发布时间：2018-12-26 10:10
【摘要】：在高维数据分析中,协方差结构扮演着十分重要的角色,而协方差矩阵的正定性是许多多元统计程序有效性的核心要求。本文主要讨论了高维情形下协方差矩阵估计的两种算法程序。在第一部分,我们主要介绍了正定的高维协方差矩阵ADMM算法的规则化轨道。在过去的十年里,高维协方差矩阵估计问题已经变得越来越流行,然而,关于规则化轨道的计算,或者通过全部规则化参数范围解决最优问题来获得稀疏协方差模型序列却很少受到关注。在这部分,我们对正定的大维协方差矩阵运用ADMM算法规则化轨道去快速逼近稀疏协方差模型序列,从而实现统计模型选择的目的。模拟结果表明我们的方法不仅计算快、易操作,而且可以高效的探索稀疏协方差模型空间。在第二部分,我们提出了高维正定协方差估计的一种有效算法。为了同时获得正定的、稀疏的高维协方差估计量,我们考虑用一个正定约束下l1-惩罚最小值问题去估计高维协方差矩阵。我们运用加速梯度算法去解决这个最优化问题,并建立了它的收敛速率为O(1/(k2)),其中k表示迭代次数。模拟结果表明,我们的方法在计算时间、FPR、FNR及F-范数和谱范数下的收敛速率等指标上更具有竞争优势。
[Abstract]:Covariance structure plays a very important role in high-dimensional data analysis, and the positive definiteness of covariance matrix is the core requirement of the validity of many multivariate statistical procedures. In this paper, two algorithms for covariance matrix estimation in high dimensional cases are discussed. In the first part, we mainly introduce the regular orbit of the ADMM algorithm of positive definite high dimensional covariance matrix. In the past decade, the problem of estimating high-dimensional covariance matrices has become more and more popular. Or the sparse covariance model sequence can be obtained by solving the optimal problem in the range of all regularized parameters, but little attention has been paid to it. In this part, we use the ADMM regularized orbit to quickly approximate the sparse covariance model sequence for the positive definite large dimensional covariance matrix, so as to achieve the purpose of statistical model selection. The simulation results show that our method is not only fast and easy to operate, but also can efficiently explore the sparse covariance model space. In the second part, we propose an efficient algorithm for estimating high dimensional positive definite covariance. In order to obtain simultaneously positive definite and sparse high dimensional covariance estimators, we consider estimating the high dimensional covariance matrix by using a positive definite constraint l 1-penalty minimum problem. We use the accelerated gradient algorithm to solve the optimization problem and establish the convergence rate of O (1 / (k2),) where k denotes the number of iterations. The simulation results show that our method is more competitive in computing time, convergence rate of FPR,FNR and F-norm and spectral norm.
【学位授予单位】：安徽师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O212.4
[Abstract]:Covariance structure plays a very important role in high-dimensional data analysis, and the positive definiteness of covariance matrix is the core requirement of the validity of many multivariate statistical procedures. In this paper, two algorithms for covariance matrix estimation in high dimensional cases are discussed. In the first part, we mainly introduce the regular orbit of the ADMM algorithm of positive definite high dimensional covariance matrix. In the past decade, the problem of estimating high-dimensional covariance matrices has become more and more popular. Or the sparse covariance model sequence can be obtained by solving the optimal problem in the range of all regularized parameters, but little attention has been paid to it. In this part, we use the ADMM regularized orbit to quickly approximate the sparse covariance model sequence for the positive definite large dimensional covariance matrix, so as to achieve the purpose of statistical model selection. The simulation results show that our method is not only fast and easy to operate, but also can efficiently explore the sparse covariance model space. In the second part, we propose an efficient algorithm for estimating high dimensional positive definite covariance. In order to obtain simultaneously positive definite and sparse high dimensional covariance estimators, we consider estimating the high dimensional covariance matrix by using a positive definite constraint l 1-penalty minimum problem. We use the accelerated gradient algorithm to solve the optimization problem and establish the convergence rate of O (1 / (k2),) where k denotes the number of iterations. The simulation results show that our method is more competitive in computing time, convergence rate of FPR,FNR and F-norm and spectral norm.
【学位授予单位】：安徽师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O212.4
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