Least Squares Fitting--Perpendicular Offsets

In practice, the vertical offsets from a line (polynomial, surface, hyperplane, etc.) are almost always minimized instead of the perpendicular offsets. This provides a fitting function for the independent variable that estimates for a given (most often what an experimenter wants), allows uncertainties of the data points along the - and -axes to be incorporated simply, and also provides a much simpler analytic form for the fitting parameters than would be obtained using a fit based on perpendicular offsets.

The residuals of the best-fit line for a set of points using unsquared perpendicular distances of points are given by

Since the perpendicular distance from a line to point is given by

the function to be minimized is

Unfortunately, because the absolute value function does not have continuous derivatives, minimizing is not amenable to analytic solution. However, if the square of the perpendicular distances

is minimized instead, the problem can be solved in closed form. is a minimum when

and

The former gives

and the latter

But

so (10) becomes

Plugging (◇) into (14) then gives

After a fair bit of algebra, the result is

So define

and the quadratic formula gives

with found using (◇). Note the rather unwieldy form of the best-fit parameters in the formulation. In addition, minimizing for a second- or higher-order polynomial leads to polynomial equations having higher order, so this formulation cannot be extended.