本文選題：最小二乘問題 + 不定最小二乘問題　； 參考：《重慶大學(xué)》2016年博士論文

【摘要】：不定最小二乘問題不僅可以作為一般最小二乘問題的一種推廣形式,同時(shí)H?估計(jì)和控制,風(fēng)險(xiǎn)敏感估計(jì)和控制(risk-sensitive estiamtion and control),有限記憶自適應(yīng)濾波(finite memory adaptive filtering)等控制論問題都可以統(tǒng)一的轉(zhuǎn)化成不定最小二乘問題來研究.因此關(guān)于不定最小二乘問題展開研究是有價(jià)值和意義的.另外,我們還對因子模型作了一些討論.由于嚴(yán)格因子模型不能很好的刻畫變量之間的聯(lián)系,而且在高維的情況下,現(xiàn)有的樣本量不足以給出一個(gè)好的估計(jì),這些因素促使關(guān)于近似因子模型的研究變得日益活躍起來.第二章是預(yù)備知識(shí)部分,我們首先給出一些基本定義,再在此基礎(chǔ)上給出條件數(shù)的統(tǒng)一定義形式.另外,一些必要的引理和概念也在這里給出,同時(shí)我們還證明了一些文章中要使用的基本定理.在第三章和第四章中,我們討論了不定最小二乘問題及其線性約束情況的條件數(shù).基于數(shù)據(jù)空間上定義的乘積范數(shù),我們首先給出其條件數(shù)的統(tǒng)一定義形式,然后在此新的框架下建立不定最小二乘問題條件數(shù)的精確表達(dá)式.作為其特殊情況,我們還給出了范數(shù)型,混合型和分量型條件數(shù)的具體表達(dá)式.在2-范數(shù)的情況下,我們給出了幾種具有存儲(chǔ)和計(jì)算優(yōu)勢的各類最小二乘問題條件數(shù)的等價(jià)形式.在第三章中,根據(jù)不定最小二乘問題與整體最小二乘問題之間的聯(lián)系,我們從不定最小二乘問題的角度重新推導(dǎo)了整體最小二乘問題的條件數(shù)表達(dá)式.在第四章,我們給出了推導(dǎo)各類線性最小二乘問題及其約束情況的條件數(shù)表達(dá)式所使用的統(tǒng)一技巧.我們的方法不再借助于奇異值分解,不需要各種復(fù)雜的構(gòu)造技巧,具有一定的普適性.同時(shí),我們還研究了系數(shù)矩陣具有特殊結(jié)構(gòu)的各類最小二乘問題,建立了相應(yīng)的結(jié)構(gòu)型條件數(shù)的表達(dá)式.針對大型矩陣精確計(jì)算條件數(shù)過于耗時(shí)的問題,我們提出了基于概率統(tǒng)計(jì)方法的幾種估計(jì)條件數(shù)的算法.最后,我們通過隨機(jī)數(shù)值實(shí)驗(yàn)來檢驗(yàn)這幾種算法估計(jì)條件數(shù)的效率,另外還對不同條件數(shù)之間的差異做了比較.第五章,我們研究了近似因子模型的估計(jì)問題.由于目前關(guān)于近似因子模型的懲罰似然估計(jì)方法不能保證誤差協(xié)方差矩陣的正定性,而且現(xiàn)有文獻(xiàn)中基于極大似然方法估計(jì)協(xié)方差矩陣通常是使用ADMM方法來保證協(xié)方差陣的正定性.但是ADMM方法需要選取懲罰參數(shù),而懲罰參數(shù)又對實(shí)際計(jì)算中算法的收斂速度有著不可忽視的影響.為了克服這些困難,我們提出了一種新的算法,該算法能在保證誤差協(xié)方差矩陣正定性的同時(shí),還有著較快的收斂速度.通過數(shù)值模擬,我們發(fā)現(xiàn)保證誤差協(xié)方差矩陣的正定性可以提高近似因子模型的估計(jì)效率和預(yù)測精度,而且根據(jù)協(xié)方差矩陣的分解公式,我們的算法還可以給出很好的協(xié)方差矩陣的估計(jì).
[Abstract]:The indeterminate least squares problem can not only be used as a generalized form of the general least squares problem, but also can be used as a generalized form of the general least squares problem. Estimation and control, risk sensitive estimation and control, risk-sensitive estiamtion and control, finite memory adaptive filtering, finite memory adaptive filtering and so on, can be uniformly transformed into indefinite least squares problems to be studied. Therefore, it is valuable and meaningful to study the indefinite least squares problem. In addition, we also discuss the factor model. Because the strict factor model can not depict the relation between variables well, and in the case of high dimension, the existing sample size is not enough to give a good estimate, these factors make the research on approximate factor model become more and more active. In the second chapter, we give some basic definitions, and then give the unified definition form of conditional number. In addition, some necessary lemmas and concepts are also given here, and we also prove some basic theorems to be used in this paper. In chapter 3 and chapter 4, we discuss the indefinite least squares problem and the number of conditions for its linear constraints. Based on the product norm defined on the data space, we first give the unified definition form of the condition number, and then establish the exact expression of the condition number of the indefinite least squares problem under this new frame. As a special case, we also give the concrete expressions of the condition numbers of norm type, mixed type and component type. In the case of 2-norm, we give some equivalent forms of condition numbers for various least squares problems with the advantages of storage and computation. In the third chapter, according to the relation between the indeterminate least squares problem and the global least squares problem, we rederive the conditional number expression of the global least squares problem from the point of view of the indefinite least squares problem. In chapter 4, we give a unified technique to derive the conditional number expressions of all kinds of linear least squares problems and their constraints. Our method no longer depends on singular value decomposition, and does not require any complicated construction techniques, so it is universal. At the same time, we study all kinds of least squares problems with special structure of coefficient matrix, and establish the expression of the corresponding structural condition number. In order to solve the problem that the accurate calculation of condition number of large matrix is too time-consuming, we propose several algorithms for estimating condition number based on probability and statistics method. Finally, we use random numerical experiments to test the efficiency of these algorithms to estimate the number of conditions. In addition, we compare the differences between different conditional numbers. In chapter 5, we study the estimation of approximate factor model. Because the present penalty likelihood estimation method for approximate factor model can not guarantee the positive definiteness of error covariance matrix, Moreover, the ADMM method is usually used in the estimation of covariance matrix based on maximum likelihood method in the existing literature to ensure the positive definiteness of covariance matrix. However, the ADMM method needs to select the penalty parameters, and the penalty parameters have an important effect on the convergence rate of the algorithm in the actual calculation. In order to overcome these difficulties, we propose a new algorithm, which can guarantee the positive definiteness of the error covariance matrix and converge fast. Through numerical simulation, we find that ensuring the positive definiteness of error covariance matrix can improve the estimation efficiency and prediction accuracy of approximate factor model, and according to the decomposition formula of covariance matrix, Our algorithm can also give a good estimate of the covariance matrix.
【學(xué)位授予單位】：重慶大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類號(hào)】：O241.5

【參考文獻(xiàn)】

相關(guān)期刊論文 前1條

1 徐洪國;A Backward Stable Hyperbolic QR Factorization Method for Solving Indefinite Least Squares Problem[J];Journal of Shanghai University;2004年04期

，

本文編號(hào)：1832712