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Encyclopedia > Method of least squares

Least squares is a mathematical optimization technique that attempts to find a "best fit" to a set of data by attempting to minimize the sum of the squares of the differences (called residuals) between the fitted function and the data.

It is commonly used in curve fitting. Many other optimization problems can also be expressed in a least squares form, either minimizing energy or maximizing entropy.

See linear regression and Gauss-Markov theorem. The Gauss-Markov theorem says that least-squares estimators are in a certain sense optimal.

To use the method of least squares we use a function f(x), containing some number of unknown constants (for instance f(x) = mx + b, where m and b are not yet known), and find the values of m and b that minimize the sum of the squares of the residuals (that is, the sum of terms of the form (yif(xi))2). We then have the equation for the curve, y = f(x), of the required form, that best fits the data points (xi, yi).

For linear functions f see linear least squares.

For nonlinear functions see Optimization, Gauss-Newton algorithm, Levenberg-Marquardt algorithm.

Results from FactBites:

 Method of Least Squares (475 words) The method of least squares is an alternative to interpolation for fitting a function to a set of points. The method of least squares is probably best known for its use in statistical regression, but it is used in many contexts unrelated to statistics. The polynomial of Exhibit 3 was constructed with the method of ordinary least squares.
 Least Squares Estimation Curve Fitting Program to download. Nonlinear Weighted Least Squares Regression Analysis. ... (1450 words) The method is also called: Curve Fitting, Least Squares Fitting, Least Squares Method, Least Squares Estimation, Least Squares Approximation. On the basis of input errors, the chi-sqr parameter and its standard deviation is calculated (chi-sqr expected value equals the number of degrees of freedom). Calculation time is significantly longer due to method nonlinearity caused by the assumption of input errors for all coordinates of points.
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