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)
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Since the perpendicular distance from a line to point is given by
(2)
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the function to be minimized is
(3)
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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)
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is minimized instead, the problem can be solved in closed form. is a minimum when
(5)
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and
(6)
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The former gives
(7)
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(8)
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and the latter
(9)
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But
(10)
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(11)
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so (10) becomes
(12)
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(13)
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(14)
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Plugging (◇) into (14) then gives
(15)
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After a fair bit of algebra, the result is
(16)
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So define
(17)
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(18)
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and the quadratic formula gives
(19)
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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.