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.
The least squares technique is commonly used in curve fitting. Many other optimization problems can also be expressed in a least squares form, by either minimizing energy or maximizing entropy.
The Gauss-Markov theorem says that least-squares estimators are in a certain sense optimal.
To use the method of least squares one considers a function f(x) containing a number of unknown constants (for instance f(x) = mx + b, where m and b are not yet known), and then one finds 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 (yi − f(xi))2). One then obtains 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, the Gauss-Newton algorithm, and the Levenberg-Marquardt algorithm.
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