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:
“I do not allow myself to be moved by anything except the law. If there has been a mistake in the law, or if I think there has been perjury or injustice, I will weigh the petition most carefully, but I do not permit myself to be moved by more harrowing details, and I try to treat each case as if I was reviewing it or hearing it for the first time from the bench.”
—William Howard Taft (1857–1930)
“The life of a good man will hardly improve us more than the life of a freebooter, for the inevitable laws appear as plainly in the infringement as in the observance, and our lives are sustained by a nearly equal expense of virtue of some kind. The decaying tree, while yet it lives, demands sun, wind, and rain no less than the green one. It secretes sap and performs the functions of health. If we choose, we may study the alburnum only. The gnarled stump has as tender a bud as the sapling.”
—Henry David Thoreau (1817–1862)
“It is the very error of the moon,
She comes more near the earth than she was wont,
And makes men mad.”
—William Shakespeare (1564–1616)