总体最小二乘导论
An Introduction to Total Least Squares
AI总结：
本文以统一视角介绍总体最小二乘（TLS）方法，通过列空间与行空间几何解释其与普通最小二乘的联系，推导简单及多元回归的TLS解，并讨论其存在性及推广。
摘要：
“总体最小二乘”方法被提出作为一种更自然的方式（相对于普通最小二乘）来逼近数据，当矩阵和右端项都被“误差”污染时。在本教程笔记中，我们给出了普通最小二乘和总体最小二乘问题及其解的一个初等统一视角。由于问题设置背后的几何结构极大地有助于理解解，我们通过列空间和（对偶）行空间中的解释来引入最小二乘问题及其推广，并将使用这两种方法来阐明解。在研究简单回归的最小二乘逼近之后，我们为此问题引入“总体最小二乘（TLS）”意义上的逼近概念，并以自然方式推导其解。接下来，我们考虑多元回归问题的普通和总体最小二乘逼近，并研究一般超定方程组在TLS意义上的解。在最后一节中，我们考虑具有多个右端项和“冻结”列的推广。我们注意到，TLS逼近一般不一定存在；然而，对于回归问题，在TLS意义上的最佳逼近直线（或超平面）总是存在的。
英文摘要：
The method of ``Total Least Squares'' is proposed as a more natural way (than ordinary least squares) to approximate the data if both the matrix and and the right-hand side are contaminated by ``errors''. In this tutorial note, we give a elementary unified view of ordinary and total least squares problems and their solution. As the geometry underlying the problem setting greatly contributes to the understanding of the solution, we introduce least squares problems and their generalization via interpretations in both column space and (the dual) row space and we shall use both approaches to clarify the solution. After a study of the least squares approximation for simple regression we introduce the notion of approximation in the sense of ``Total Least Squares (TLS)'' for this problem and deduce its solution in a natural way. Next we consider ordinary and total least squares approximations for multiple regression problems and we study the solution of a general overdetermined system of equations in TLS-sense. In a final section we consider generalizations with multiple right-hand sides and with ``frozen'' columns. We remark that a TLS-approximation needs not exist in general; however, the line (or hyperplane) of best approximation in TLS-sense for a regression problem does exist always.
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