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Date: 15-2-2021
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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
(1) |
Since the perpendicular distance from a line to point is given by
(2) |
the function to be minimized is
(3) |
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
(4) |
is minimized instead, the problem can be solved in closed form. is a minimum when
(5) |
and
(6) |
The former gives
(7) |
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(8) |
and the latter
(9) |
But
(10) |
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(11) |
so (10) becomes
(12) |
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(13) |
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(14) |
Plugging (◇) into (14) then gives
(15) |
After a fair bit of algebra, the result is
(16) |
So define
(17) |
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(18) |
and the quadratic formula gives
(19) |
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.
REFERENCES:
Sardelis, D. and Valahas, T. "Least Squares Fitting-Perpendicular Offsets." https://library.wolfram.com/infocenter/MathSource/5292/.
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