發(fā)布時(shí)間：2018-06-15 09:09

  本文選題：EIV模型 + 估計(jì)偏差��； 參考：《武漢大學(xué)》2013年博士論文

【摘要】：經(jīng)典的Gauss-Markov模型只假定觀測向量包含隨機(jī)誤差,系數(shù)矩陣是非隨機(jī)的固定值,當(dāng)模型為線性形式時(shí),采用最小二乘估計(jì)方法(LS:least squers)可得到模型參數(shù)的最優(yōu)解。但實(shí)際應(yīng)用中,許多情況下觀測向量和系數(shù)矩陣均包含隨機(jī)誤差,這類平差模型稱為EIV (errors-in-variables)模型。 EIV模型于19世紀(jì)末就已提出,20世紀(jì)80年代前主要是統(tǒng)計(jì)領(lǐng)域開展了少量研究工作。1980年,Golub和van Loan發(fā)表了著名的奇異值分解算法(該方法實(shí)質(zhì)上與1901年P(guān)earson提出的正交回歸解法相同),之后EIV模型引起了各領(lǐng)域的廣泛關(guān)注。到目前為止,EIV模型已成為基本的數(shù)學(xué)模型之一,廣泛應(yīng)用于信號(hào)處理、通信工程、計(jì)算機(jī)視覺等眾多科學(xué)研究和工程應(yīng)用領(lǐng)域。 EIV模型最簡單的算法是忽略系數(shù)矩陣誤差,采用最小二乘方法求解,但其解為近似解,不再具有最優(yōu)統(tǒng)計(jì)特性(Xu等2012)。為了得到EIV模型的最優(yōu)解,通過對(duì)最小二乘準(zhǔn)則進(jìn)行擴(kuò)展,得到的能同時(shí)顧及觀測向量和系數(shù)矩陣誤差的整體最小二乘估計(jì)方法(TLS:total least squares)是EIV模型的嚴(yán)密估計(jì)方法。然而,TLS的計(jì)算量遠(yuǎn)大于LS方法,在觀測值和參數(shù)數(shù)量大的情況下,TLS甚至無法求解。同時(shí),有些參考文獻(xiàn)實(shí)例結(jié)果表明模型的LS解和TLS解幾乎沒有差別(如大地坐標(biāo)轉(zhuǎn)換模型)。因此,我們認(rèn)為EIV模型估計(jì)的一個(gè)基本問題為：究竟在什么情況下可以采用LS方法代替TLS方法?采用數(shù)學(xué)語言描述,即系數(shù)矩陣誤差如何影響EIV模型的LS估計(jì)結(jié)果。遺憾的是,測繪領(lǐng)域尚沒有文獻(xiàn)從理論上進(jìn)行研究,僅統(tǒng)計(jì)學(xué)領(lǐng)域兩篇文獻(xiàn)(Hodges和Moore1972、Davies和Hutton1975)在非常簡單的假設(shè)下進(jìn)行了探討,且推導(dǎo)過程存在錯(cuò)誤。因此,論文的主要研究內(nèi)容之一是全面系統(tǒng)地研究系數(shù)矩陣誤差對(duì)EIV模型LS估計(jì)結(jié)果的各種影響。 平差模型的可靠性度量是平差的基本問題之一。盡管建立在Gauss-Markov模型基礎(chǔ)上的經(jīng)典可靠性理論成果豐富,但到口前為止,僅兩篇文獻(xiàn)討論了EIV模型的可靠性理論。Schaffrin和Uzun (2011/2012)在極為特殊的權(quán)陣條件下推導(dǎo)了EIV模型的可靠性度量,由于公式中包括無粗差情況下的TLS解,不能用于實(shí)際計(jì)算。Proszynski (2013)直接簡單套用經(jīng)典可靠性理論,且只討論了觀測向量的可靠性,不能視為真正意義上的EIV模型的可靠性度量。針對(duì)EIV模型可靠性理論的缺陷,論文的主要研究內(nèi)容之二是推導(dǎo)了系數(shù)矩陣誤差對(duì)經(jīng)典可靠性度量的影響,并且系統(tǒng)地發(fā)展了一般情況下EIV模型的可靠性理論和方法。 論文的主要內(nèi)容和貢獻(xiàn)如下： (1)從EIV模型的一般情況出發(fā),全面系統(tǒng)地推導(dǎo)了系數(shù)矩陣誤差引起的LS參數(shù)估計(jì)值及其方差協(xié)方差陣的偏差、觀測向最殘差偏差的嚴(yán)密計(jì)算公式。研究結(jié)果表明,系數(shù)矩陣誤差對(duì)平差結(jié)果的影響與系數(shù)矩陣量級(jí)及其方差、參數(shù)的大小有關(guān)。若定義系數(shù)矩陣的信噪比為系數(shù)矩陣量級(jí)與其中誤差之比,則參數(shù)估計(jì)值的相對(duì)偏差隨系數(shù)矩陣信噪比二次方的增大而迅速減小,參數(shù)估計(jì)值的相對(duì)中誤差隨系數(shù)矩陣信噪比的增長而減小。通常情況下,系數(shù)矩陣誤差對(duì)LS參數(shù)估計(jì)值精度的影響大于對(duì)參數(shù)估計(jì)值偏差的影響。 (2)論文推導(dǎo)了系數(shù)矩陣誤差引起的單位權(quán)方差偏差的計(jì)算公式。公式表明,單位權(quán)方差的偏差隨參數(shù)二次方的增長而迅速增長,隨系數(shù)矩陣方差協(xié)方差陣的增大而增長。公式從理論上完美地解釋了測繪領(lǐng)域有關(guān)文獻(xiàn)報(bào)道的EIV模型經(jīng)典LS單位權(quán)方差估計(jì)結(jié)果異常且顯著偏大的情況。 (3)在以上研究成果的基礎(chǔ)上,通過對(duì)LS估計(jì)結(jié)果進(jìn)行偏差改正,構(gòu)造了EIV模型偏差改正的LS參數(shù)估計(jì)值、參數(shù)的方差協(xié)方差估計(jì)值以及單位權(quán)方差估計(jì)公式。 (4)論文研究了系數(shù)矩陣誤差對(duì)經(jīng)典可靠性度量的影響,導(dǎo)出了系數(shù)矩陣誤差引起的內(nèi)部可靠性和外部可靠性偏差的計(jì)算公式。公式反映了偏差隨系數(shù)矩陣信噪比二次方的增長而減小,隨模型本身可靠性的增大而減小。利用偏差公式,論文構(gòu)造了EIV模型觀測向量偏差改正的的可靠性度最公式。 (5)以partial-EIV模型為基礎(chǔ),建立了一般權(quán)矩陣條件下EIV模型的可靠性理論和方法,推導(dǎo)了觀測向量和系數(shù)矩陣的內(nèi)部可靠性和外部可靠性的計(jì)算公式。研究結(jié)果表明,EIV模型的多余觀測數(shù)根據(jù)觀測向量和系數(shù)矩陣的方差以及參數(shù)大小在觀測的量和系數(shù)矩陣之間進(jìn)行分配。當(dāng)觀測向量方差很小(系數(shù)矩陣或參數(shù)很小)時(shí),由于觀測向最(系數(shù)矩陣)分配的多余觀測數(shù)很少,若出現(xiàn)粗差將難以發(fā)現(xiàn)。
[Abstract]:The classical Gauss-Markov model only assumes that the observation vector contains random error, the coefficient matrix is a non random fixed value. When the model is linear, the least square estimation method (LS:least squers) can obtain the optimal solution of the model parameters. However, in practical applications, the observation vector and the coefficient matrix all contain random errors in many situations. The class adjustment model is called EIV (errors-in-variables) model.
The EIV model has been put forward in the late nineteenth Century, and a small amount of research work was carried out in the field of statistics before 1980s. Golub and van Loan published a famous singular value decomposition algorithm (this method is essentially the same as the orthogonal regression method proposed in 1901). Then the EIV model has aroused wide attention in various fields. The EIV model has become one of the basic mathematical models, and is widely used in many scientific research and engineering applications such as signal processing, communication engineering, computer vision, etc.
The simplest algorithm of the EIV model is to ignore the error of the coefficient matrix and use the least square method to solve it, but the solution is an approximate solution and no longer has the optimal statistical property (Xu et al. 2012). In order to obtain the optimal solution of the EIV model, the least second total of the error of the observation vector and the coefficient matrix can be considered at the same time by extending the least square criterion. The multiplicative estimation method (TLS:total least squares) is the strict estimation method of the EIV model. However, the calculation of TLS is far greater than the LS method. In the case of large observation and parameter, TLS can not be solved even. At the same time, some reference examples show that the LS solution of the model and the TLS solution are almost no difference (such as the geodetic coordinate transformation model). We think one of the basic problems of EIV model estimation is: in what case can we use the LS method instead of the TLS method? The mathematical language is used to describe how the coefficient matrix error affects the LS estimation results of the EIV model. Unfortunately, there is no literature in the field of Surveying and mapping, but only two literature in the field of Statistics (Hod) Ges and Moore1972, Davies and Hutton1975) are discussed under very simple assumptions and there are errors in the derivation process. Therefore, one of the main contents of this paper is to systematically study the influence of the coefficient matrix error on the LS estimation results of the EIV model.
The reliability measurement of the adjustment model is one of the basic problems of the adjustment. Although the classical reliability theory based on the Gauss-Markov model is rich, only two papers discuss the reliability theory.Schaffrin and Uzun (2011/2012) of the EIV model so far. The reliability of the model is derived from the EIV model under the extremely special weight matrix condition. Because the formula includes the TLS solution in the case of no gross error, it can not be used to calculate the reliability theory of.Proszynski (2013) directly and simply, and only discusses the reliability of the observation vector. It can not be regarded as the reliability measure of the real EIV model. The main research of the thesis is the defect of the reliability theory of the EIV model. The two content is to deduce the influence of the coefficient matrix error on the classical reliability measurement, and systematically develop the reliability theory and method of the EIV model in general.
The main contents and contributions of the paper are as follows:
(1) from the general situation of the EIV model, the error of the estimation value of LS parameters and the covariance covariance matrix caused by the coefficient matrix error and the rigorous calculation formula of the observation to the most residual deviation are derived. The results show that the magnitude and variance of the coefficient matrix error to the result of the coefficient matrix error and the coefficient matrix are the size of the parameter. If the signal-to-noise ratio of the coefficient matrix is defined as the ratio of the coefficient matrix to the error, the relative deviation of the parameter estimation decreases rapidly with the increase of the signal to noise ratio of the coefficient matrix, and the relative error of the parameter estimation decreases with the increase of the signal to noise ratio of the coefficient matrix. In the general case, the coefficient matrix error is estimated for the LS parameter. The effect of value accuracy is greater than that on parameter estimation bias.
(2) the paper derives the calculation formula of the deviation of unit weight variance caused by the coefficient matrix error. The formula shows that the deviation of the unit weight variance increases rapidly with the increase of the parameter of the two order square, and increases with the increase of the covariance matrix of variance covariance. The formula explains perfectly the classical EIV model L in the field of Surveying and mapping in the field of Surveying and mapping. S unit weight variance estimation results are abnormal and significantly larger.
(3) on the basis of the above research results, by correcting the deviation of the LS estimation results, the LS parameter estimation value of the EIV model deviation correction, the variance covariance estimation value of the parameter and the formula of the unit weight variance estimation are constructed.
(4) the paper studies the influence of the coefficient matrix error on the classical reliability measurement, derives the calculation formula of the internal reliability and the external reliability deviation caused by the coefficient matrix error. The formula reflects the decrease of the deviation with the increase of the signal to noise ratio of the coefficient matrix and the increase of the reliability of the model with the increase of the reliability of the model, and the use of the deviation formula. The most reliable formula for the deviation correction of the EIV model is constructed.
(5) based on the partial-EIV model, the reliability theory and method of the EIV model under the condition of the general weight matrix are established. The calculation formulas for the internal reliability and external reliability of the observation vector and the coefficient matrix are derived. The results show that the superfluous observations of the EIV model are based on the variance of the observation vector and the coefficient matrix and the size of the parameters. When the observation vector is very small (the coefficient matrix or the parameter is very small), the number of superfluous observations allocated to the most (coefficient matrix) is very few, and it is difficult to find out if the gross error occurs, when the observation vector is very small (the coefficient matrix or the parameter is very small).
【學(xué)位授予單位】：武漢大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2013
【分類號(hào)】：P207

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