Least squares, also known as ordinary least squares analysis, is a method for linear regression that determines the values of unknown quantities in a statistical model by minimizing the sum of the squared residuals (the difference between the predicted and observed values). This method was first described by Carl Friedrich Gauss. It can be shown that the least-squares approach to regression analysis is optimal in the sense that it satisfies the Gauss-Markov theorem.
Many other types of optimization problems can be expressed in a least squares form, by either minimizing energy or maximizing entropy. The least squares method is particularly important in estimation of model parameters from measured data.
Consider the problem of adjusting a model function to best fit a data set. The chosen model function has adjustable parameters. The data set consist of n points
The model function has the form
where y is the dependent variable, x is the vector of independent variables, and a are the adjustable parameters of the model. We wish to find the values of these parameters such that the model best fits the data according to a defined error criterion. The least squares method minimizes the sum of squares of errors,
with respect to the adjustable parameters of the model a.