Straight line fitting of an observation path by least normal squares

In curtain hydro graphic problems, and perhaps in other geophysical problems, information must be collected as profiles along straight-line courses. When the inevitable deviations from perfect linearity occur, one must then find the straight line course that best approximates the actual observation path. Classical least squares (regression) does not solve this problem, because the line thus fitted depends upon which coordinate is taken as the dependent variable.

A coordinate-free solution is obtained by minimizing the sum of squares of normal distances between the line and the observation points. As in classical least squares, the line of best fit passes through the geometrical centroid of the observation points. The slope of this line, however, is closer to 1.0 than the slope of the classical regression line. The least normal squares and classical least squares solutions coincide when the observation path is nearly perfectly straight or when it runs generally parallel to one of the coordinate axes.