In this thesis we study the problem of finding the best linear approximation for overdetermined systems of linear equations, where we account for errors throughout the entire system. We present the classical least squares method, which minimizes the sum of squared differences between the left and right hand sides, and discuss its shortcomings. As an extension we introduce the total least squares method, which accounts for errors both in the system matrix and in the right hand side vector. We continue with the existence and uniqueness of the solution of this method and present an efficient method and its improvements for obtaining it. We also discuss extensions of the total least squares method, with particular attention to the method with fixed columns. The theoretical derivations are verified on a concrete example of predicting the annual mass balance of the Hintereisferner glacier in the Austrian Alps, where we observe differences in the predictions of the mentioned methods.
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