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m Task 18 (cosmetic): eval 11 templates: del empty params (2×); hyphenate params (1×); del |ref=harv (11×); |
Removed 192.147.66.4's comment that the theorem from Minda and Phelps is "wrong". The problem with their supposed counterexample is that the sum of the squares of four such points is not 0, but a positive real number. |
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For the case of equal error variances, i.e., when <math>\delta=1</math>, Deming regression becomes ''orthogonal regression'': it minimizes the sum of squared [[distance from a point to a line|perpendicular distances from the data points to the regression line]]. In this case, denote each observation as a point ''z''<sub>''j''</sub> in the complex plane (i.e., the point (''x''<sub>''j''</sub>, ''y''<sub>''j''</sub>) is written as ''z''<sub>''j''</sub> = ''x''<sub>''j''</sub> + ''iy''<sub>''j''</sub> where ''i'' is the [[imaginary unit]]). Denote as ''Z'' the sum of the squared differences of the data points from the [[centroid]] (also denoted in complex coordinates), which is the point whose horizontal and vertical locations are the averages of those of the data points. Then:<ref>Minda and Phelps (2008), Theorem 2.3.</ref>
*If ''Z'' = 0, then every line through the centroid is a line of best orthogonal fit
*If ''Z'' ≠ 0, the orthogonal regression line goes through the centroid and is parallel to the vector from the origin to <math>\sqrt{Z}</math>.
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