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.
Read more about this topic: Deming Regression
Famous quotes containing the words case, equal and/or error:
“Sculpture and painting are very justly called liberal arts; a lively and strong imagination, together with a just observation, being absolutely necessary to excel in either; which, in my opinion, is by no means the case of music, though called a liberal art, and now in Italy placed even above the other two—a proof of the decline of that country.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)
“Just as the queen bee, the highest-ranking, peerless creature of her hive, is surrounded by lowly drones to please her, whereas the workers produce honey, the same way is the one who sits on the throne an equal only to himself, and no one’s companion.”
—Franz Grillparzer (1791–1872)
“There are strange flowers of reason to match each error of the senses.”
—Louis Aragon (1897–1982)