(1. 武漢大學(xué) 測繪學(xué)院，武漢 430079；
2. 內(nèi)蒙古科技大學(xué) 礦業(yè)與煤炭學(xué)院，包頭014010；
3. 武漢大學(xué) 地球空間環(huán)境與大地測量教育部重點(diǎn)實(shí)驗(yàn)室，武漢 430079)
摘 要: 當(dāng)誤差含變量(EIV)模型的設(shè)計(jì)矩陣病態(tài)時(shí)，采用普通整體最小二乘(TLS)算法得不到穩(wěn)定的數(shù)值解。為了減弱病態(tài)性，在整體最小二乘準(zhǔn)則的基礎(chǔ)上附加解的二次范數(shù)約束，組成拉格朗日目標(biāo)函數(shù)，推導(dǎo)EIV模型的正則化整體最小二乘解(RTLS)。然后將RTLS的求解轉(zhuǎn)換為矩陣特征向量問題，設(shè)計(jì)一個(gè)迭代方案逼近RTLS解。通過L曲線法求得正則化因子來確定正常數(shù)，從而避免人為選擇正常數(shù)的隨意性。數(shù)值實(shí)例表明，提出的迭代正則化算法是有效可行的。
關(guān)鍵字: EIV模型；病態(tài)問題；正則化整體最小二乘；L曲線法；正常數(shù)
（1. School of Geodesy and Geomatics, Wuhan University, Wuhan 430079, China;
2. Mining and Coal Institute, Inner Mongolia University of Science and Technology, Baotou 014010, China;
3. Key Laboratory of Geospace Environment and Geodesy, Ministry of Education,
Wuhan University, Wuhan 430079, China）
Abstract:When the design matrix of errors-in-variables (EIV) model was ill-conditioned, the ordinary total least squares (TLS) solution was unstable. In order to weaken the ill-conditioning, an Euclid norm constraint of the solution was added to the TLS minimization rule. Then, the Lagrange objective function was formed and the regularized total least squares (RTLS) solution was deduced. Afterwards, the RTLS was transformed to a problem of looking for a matrix’s eigenvector. An iterative program was designed to approximate the solution. The L-curve method was used to choose the regularization factor to determine the positive constant, which can avoid the subjective decision. The simulations show the efficiency and feasibility of the algorithm.
Key words: EIV model; ill-posed problem; regularized total least squares; L-curve method; positive constant
