病态整体最小二乘的迭代正则化算法

Iterative and regularized algorithm to ill-posed total least squares

当误差含变量(EIV)模型的设计矩阵病态时,采用普通整体最小二乘(TLS)算法得不到稳定的数值解。为了减弱病态性,在整体最小二乘准则的基础上附加解的二次范数约束,组成拉格朗日目标函数,推导EIV模型的正则化整体最小二乘解(RTLS)。然后将RTLS的求解转换为矩阵特征向量问题,设计一个迭代方案逼近RTLS解。通过L曲线法求得正则化因子来确定正常数,从而避免人为选择正常数的随意性。数值实例表明,提出的迭代正则化算法是有效可行的。

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.