Note on the total least squares problem for coplanar points

摘要

The Total Least Squares (TLS) fit to the points (x(sub k), y(sub k)), k = 1, (hor ellipsis), n, minimizes the sum of the squares of the perpendicular distances from the points to the line. This sum is the TLS error, and minimizing its magnitude is appropriate if x(sub k) and y(sub k) are uncertain. A priori formulas for the TLS fit and TLS error to coplanar points were originally derived by Pearson, and they are expressed in terms of the mean, standard deviation and correlation coefficient of the data. In this note, these TLS formulas are derived in a more elementary fashion. The TLS fit is obtained via the ordinary least squares problem and the algebraic properties of complex numbers. The TLS error is formulated in terms of the triangle inequality for complex numbers.