# Least squares

From AMS Glossary

## least squares

Any procedure that involves minimizing the sum of squared differences.

For example, the deviation of the mean from the population is less, in the square sense, than any other linear combination of the population values. This procedure is most widely used to obtain the constants of a representation of a known variable The and minimizing the sum with respect to the

*Y*in terms of others*X*_{i}. Let*Y*(*s*) be represented by*a*_{n}'s are the constants to be determined, the*f*_{n}'s are arbitrary functions, and*s*is a parameter common to*Y*and*X*_{i}.*N*is usually far less than the number of known values of*Y*and*X*_{i}. The system of equations being overdetermined, the constants*a*_{n}must be "fitted." The least squares determination of this "fit" proceeds by summing, or integrating when*Y*and*X*_{i}are known continuously,*a*_{n}'s. In particular, for example, if*f*_{n}[*X*_{i}(*s*)] ≡*X*_{i}(*s*), then the regression function is being determined; and when*f*_{n}[*X*_{i}(*s*)] ≡ cos*n'***X**_{i}(*s*), or sin*n'**X*_{i}(*s*), then*Y*is being represented by a multidimensional Fourier series. Least squares is feasible only when the unknown constants*a*_{n}enter linearly. The method of least squares was described independently by Legendre in 1806, Gauss in 1809, and Laplace in 1812.