method of least squares - meaning and definition. What is method of least squares
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What (who) is method of least squares - definition

APPROXIMATION METHOD IN STATISTICS
Method of least squares; Least-squares method; Least-squares estimation; Least-Squares Fitting; Least squares fitting; Sum of Squared Error; Least-squares; Least squares approximation; Least-squares approximation; Least squares method; Least-squares analysis; Least squares fit; Least squares problem; Least-squares problem; LSQF; Principle of least squares; Least-squares fit; Method of Least Squares; Least Squares
  • [[Carl Friedrich Gauss]]
  • "Fanning Out" Effect of Heteroscedasticity
  • 251x251px
  • The result of fitting a set of data points with a quadratic function
  • The residuals are plotted against the corresponding <math>x</math> values. The parabolic shape of the fluctuations about <math>r_i=0</math> indicates a parabolic model is appropriate.
  • Conic fitting a set of points using least-squares approximation

Least squares         
The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the residuals (a residual being the difference between an observed value and the fitted value provided by a model) made in the results of each individual equation.
least squares         
¦ noun a method of estimating a quantity or fitting a graph to data so as to minimize the sum of the squares of the differences between the observed values and the estimated values.
Ordinary least squares         
  • Fitted regression
  • Residuals plot
  • [[Scatterplot]] of the data, the relationship is slightly curved but close to linear
  • OLS estimation can be viewed as a projection onto the linear space spanned by the regressors. (Here each of <math>X_1</math> and <math>X_2</math> refers to a column of the data matrix.)
  • [[Okun's law]] in [[macroeconomics]] states that in an economy the [[GDP]] growth should depend linearly on the changes in the unemployment rate. Here the ordinary least squares method is used to construct the regression line describing this law.
METHOD FOR ESTIMATING THE UNKNOWN PARAMETERS IN A LINEAR REGRESSION MODEL
Ordinary least squares regression; Normal equations; Ordinary Least Squares; Ordinary list squares; OLS Regression; Ordinary least square; Standard error of the equation; OLS regression; Ordinary Least Squares Regression; Partitioned regression; Least-squares normal matrix; Large Sample Properties
In statistics, ordinary least squares (OLS) is a type of linear least squares method for choosing the unknown parameters in a linear regression model (with fixed level-one effects of a linear function of a set of explanatory variables) by the principle of least squares: minimizing the sum of the squares of the differences between the observed dependent variable (values of the variable being observed) in the input dataset and the output of the (linear) function of the independent variable.

Wikipedia

Least squares

The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the residuals (a residual being the difference between an observed value and the fitted value provided by a model) made in the results of each individual equation.

The most important application is in data fitting. 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 nonlinear 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 nonlinear 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 with identity as its natural sufficient statistics and mild-conditions are satisfied (e.g. for normal, exponential, Poisson and binomial distributions), standardized 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.

The least-squares method was officially discovered and published by Adrien-Marie Legendre (1805), though it is usually also co-credited to Carl Friedrich Gauss (1795) who contributed significant theoretical advances to the method and may have previously used it in his work.