The Case of Equal Error Variances
When, Deming regression becomes orthogonal regression: it minimizes the sum of squared perpendicular distances from the data points to the regression line. In this case, denote each observation as a point zj in the complex plane (i.e., the point (xj, yj) is written as zj = xj + iyj 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:
- 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 .
A trigonometric representation of the orthogonal regression line was given by Coolidge in 1913.
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