【摘要】：在大數(shù)據(jù)時(shí)代,高維數(shù)據(jù)呈現(xiàn)在基因組和健康科學(xué)、經(jīng)濟(jì)與金融、天文學(xué)與物理學(xué)、信號(hào)處理與成像等學(xué)科領(lǐng)域.其中一個(gè)共同特征是預(yù)測(cè)變量具有稀疏性.選擇最相關(guān)的預(yù)測(cè)變量是高維數(shù)據(jù)回歸分析的一個(gè)主要研究?jī)?nèi)容,具有十分重要的應(yīng)用價(jià)值.為此,針對(duì)線性回歸假設(shè)的許多統(tǒng)計(jì)方法被提出和廣泛研究.然而,在壓縮感知、信號(hào)處理與亞波長(zhǎng)光學(xué)成像等實(shí)際問(wèn)題中,響應(yīng)變量和回歸參數(shù)是二次關(guān)系.所以,本文引入二次度量回歸(QMR)模型,研究了其高維情形下的變量選擇問(wèn)題,并建立了相應(yīng)的優(yōu)化理論與算法.第二章引入一致正則性概念,給出相應(yīng)的判定條件,并且用于高維QMR模型的可辨識(shí)性研究.第三章針對(duì)高維QMR模型的lq(0 q 1)正則最小二乘問(wèn)題,給出了相應(yīng)估計(jì)的中偏差和弱Oracle性質(zhì),得到了解的存在性及其不動(dòng)點(diǎn)理論.在此基礎(chǔ)上,構(gòu)造了不動(dòng)點(diǎn)迭代算法,建立了其收斂性結(jié)果.最后,通過(guò)數(shù)值模擬表明該方法的有效性.第四章針對(duì)QMR模型的l0約束最小二乘問(wèn)題,給出了解的存在性及不動(dòng)點(diǎn)理論,進(jìn)而構(gòu)造了稀疏投影梯度算法,并得到該算法的收斂性.最后,通過(guò)數(shù)值模擬表明l0約束最小二乘方法的有效性.第五章針對(duì)高維QMR模型的特殊情形—線性模型,研究了加權(quán)l(xiāng)1正則分位數(shù)回歸問(wèn)題.使用交替方向乘子法提出了一種快速、有效算法,得到了算法的收斂性.利用該算法和局部線性近似技巧,還構(gòu)造了一類(lèi)非凸懲罰的分位數(shù)回歸估計(jì)的計(jì)算方法.最后,數(shù)值實(shí)驗(yàn)表明該算法的有效性.
[Abstract]:In big data's time, high dimensional data were presented in the fields of genome and health science, economics and finance, astronomy and physics, signal processing and imaging. One of the common characteristics is the sparsity of predictive variables. Choosing the most relevant predictive variables is one of the main research contents of high dimensional data regression analysis, which has very important application value. Therefore, many statistical methods for linear regression hypothesis have been proposed and widely studied. However, in practical problems such as compression sensing, signal processing and subwavelength optical imaging, the response variables and regression parameters are quadratic. Therefore, in this paper, the quadratic metric regression (QMR) model is introduced to study the variable selection problem in the case of high dimension, and the corresponding optimization theory and algorithm are established. In chapter 2, the concept of uniform regularity is introduced, and the corresponding criteria are given, which are used to study the identifiability of high-dimensional QMR model. In chapter 3, for the lq (0Q 1) regular least squares problem of high dimensional QMR model, the intermediate deviation and weak Oracle properties of the corresponding estimates are given, and the existence of the solution and its fixed point theory are obtained. On this basis, the fixed point iterative algorithm is constructed and its convergence results are established. Finally, numerical simulation shows the effectiveness of the method. In chapter 4, the existence of solution and the fixed point theory are given for the l0-constrained least square problem of QMR model, and then the sparse projection gradient algorithm is constructed, and the convergence of the algorithm is obtained. Finally, numerical simulation shows the validity of the l 0 constrained least squares method. In chapter 5, the weighted L 1 regular quantile regression problem is studied for the special case of the high dimensional QMR model, the linear model. A fast and effective algorithm is proposed by using alternating direction multiplier method, and the convergence of the algorithm is obtained. By using this algorithm and the local linear approximation technique, a new method of quantile regression estimation for nonconvex penalty is also constructed. Finally, numerical experiments show the effectiveness of the algorithm.
【學(xué)位授予單位】：北京交通大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2017
【分類(lèi)號(hào)】：O212.1

【參考文獻(xiàn)】

相關(guān)期刊論文 前3條

1 許志強(qiáng);;壓縮感知[J];中國(guó)科學(xué):數(shù)學(xué);2012年09期

2 文再文;印臥濤;劉歆;張寅;;壓縮感知和稀疏優(yōu)化簡(jiǎn)介[J];運(yùn)籌學(xué)學(xué)報(bào);2012年03期

3 常象宇;徐宗本;張海;王建軍;梁勇;;穩(wěn)健L_q(0<q<1)正則化理論:解的漸近分布與變量選擇一致性[J];中國(guó)科學(xué):數(shù)學(xué);2010年10期

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本文編號(hào)：2342303