What is "least squares"

Least \Least\ (l[=e]st), a. [OE. last, lest, AS. l[=ae]sast, l[=ae]sest, superl. of l[=ae]ssa less. See Less, a.] [Used as the superlative of little.] Smallest, either in size or degree; shortest; lowest; most unimportant; as, the least insect; the least mercy; the least space.

Note: Least is often used with the, as if a noun.

I am the least of the apostles.
--1 Cor. xv. 9.

At least, or At the least, at the least estimate, consideration, chance, etc.; being no less than; hence, at any rate; at all events; even. See However.

He who tempts, though in vain, at least asperses The tempted with dishonor.
--Milton.

Upon the mast they saw a young man, at least if he were a man, who sat as on horseback.
--Sir P. Sidney.

In least, or In the least, in the least degree, manner, etc. ``He that is unjust in the least is unjust also in much.''
--Luke xvi. 10.

Least squares (Math.), a method of deducing from a number of carefully made yet slightly discordant observations of a phenomenon the most probable values of the unknown quantities.

Note: It takes as its fundamental principle that the most probable values are those which make the sum of the squares of the residual errors of the observation a minimum.

n. a method of fitting a curve to data points so as to minimize the sum of the squares of the distances of the points from the curve [syn: method of least squares]

The method of least squares is a standard approach in regression analysis to the approximate solution of overdetermined systems, i.e., sets of equations in which there are more equations than unknowns. "Least squares" means that the overall solution minimizes the sum of the squares of the errors made in the results of every single equation.

The most important application is in data fitting. The best fit in the least-squares sense minimizes the sum of squared residuals, a residual being the difference between an observed value and the fitted value provided by a model. When the problem has substantial uncertainties in the independent variable (the x variable), then simple regression and least squares methods have problems; in such cases, the methodology required for fitting errors-in-variables models may be considered instead of that for least squares.

Least squares problems fall into two categories: linear or ordinary least squares and non-linear least squares, depending on whether or not the residuals are linear in all unknowns. The linear least-squares problem occurs in statistical regression analysis; it has a closed-form solution. The non-linear problem is usually solved by iterative refinement; at each iteration the system is approximated by a linear one, and thus the core calculation is similar in both cases.

Polynomial least squares describes the variance in a prediction of the dependent variable as a function of the independent variable and the deviations from the fitted curve.

When the observations come from an exponential family and mild conditions are satisfied, least-squares estimates and maximum-likelihood estimates are identical. The method of least squares can also be derived as a method of moments estimator.

The following discussion is mostly presented in terms of linear functions but the use of least-squares is valid and practical for more general families of functions. Also, by iteratively applying local quadratic approximation to the likelihood (through the Fisher information), the least-squares method may be used to fit a generalized linear model.

For the topic of approximating a function by a sum of others using an objective function based on squared distances, see least squares (function approximation).

The least-squares method is usually credited to Carl Friedrich Gauss (1795), but it was first published by Adrien-Marie Legendre.