【摘要】：在高維數(shù)據(jù)分析中,協(xié)方差結(jié)構(gòu)扮演著十分重要的角色,而協(xié)方差矩陣的正定性是許多多元統(tǒng)計(jì)程序有效性的核心要求。本文主要討論了高維情形下協(xié)方差矩陣估計(jì)的兩種算法程序。在第一部分,我們主要介紹了正定的高維協(xié)方差矩陣ADMM算法的規(guī)則化軌道。在過去的十年里,高維協(xié)方差矩陣估計(jì)問題已經(jīng)變得越來越流行,然而,關(guān)于規(guī)則化軌道的計(jì)算,或者通過全部規(guī)則化參數(shù)范圍解決最優(yōu)問題來獲得稀疏協(xié)方差模型序列卻很少受到關(guān)注。在這部分,我們對(duì)正定的大維協(xié)方差矩陣運(yùn)用ADMM算法規(guī)則化軌道去快速逼近稀疏協(xié)方差模型序列,從而實(shí)現(xiàn)統(tǒng)計(jì)模型選擇的目的。模擬結(jié)果表明我們的方法不僅計(jì)算快、易操作,而且可以高效的探索稀疏協(xié)方差模型空間。在第二部分,我們提出了高維正定協(xié)方差估計(jì)的一種有效算法。為了同時(shí)獲得正定的、稀疏的高維協(xié)方差估計(jì)量,我們考慮用一個(gè)正定約束下l1-懲罰最小值問題去估計(jì)高維協(xié)方差矩陣。我們運(yùn)用加速梯度算法去解決這個(gè)最優(yōu)化問題,并建立了它的收斂速率為O(1/(k2)),其中k表示迭代次數(shù)。模擬結(jié)果表明,我們的方法在計(jì)算時(shí)間、FPR、FNR及F-范數(shù)和譜范數(shù)下的收斂速率等指標(biāo)上更具有競(jìng)爭(zhēng)優(yōu)勢(shì)。
[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.
【學(xué)位授予單位】：安徽師范大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O212.4

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