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 the case, case, equal and/or error:
“To be President of the United States, sir, is to act as advocate for a blind, venomous, and ungrateful client; still, one must make the best of the case, for the purposes of Providence.”
—John Updike (b. 1932)
“Consumer wants can have bizarre, frivolous, or even immoral origins, and an admirable case can still be made for a society that seeks to satisfy them. But the case cannot stand if it is the process of satisfying wants that creates the wants.”
—John Kenneth Galbraith (b. 1908)
“I swear ... to hold my teacher in this art equal to my own parents; to make him partner in my livelihood; when he is in need of money to share mine with him; to consider his family as my own brothers and to teach them this art, if they want to learn it, without fee or indenture.”
—Hippocrates (c. 460–c. 370 B.C.)
“It is not to everyone’s taste that truth should be pronounced pleasant. But at least let no one believe that error becomes truth when it is pronounced unpleasant.”
—Friedrich Nietzsche (1844–1900)