A standard approach for approximating solutions to overdetermined systems by finding model parameters that minimize the sum of the squared differences between observed and predicted values. This sum is known as the sum of squared residuals (SSR). The goal is to find the parameters [latex]\hat{\beta}[/latex] that minimize the function [latex]S(\beta) = \sum_{i=1}^{n} (y_i – x_i^T \beta)^2[/latex].
这 方法 of ordinary least squares is a cornerstone of regression analysis. It provides a direct way to estimate the unknown parameters in a linear model. The principle is to find the line (or hyperplane in multiple regression) that is closest to all the data points simultaneously. ‘Closest’ is defined in terms of minimizing the vertical distances from each point to the line, specifically, the sum of the squares of these distances (residuals).
This minimization problem can be solved using calculus. By taking the derivative of the sum of squared residuals function [latex]S(\beta)[/latex] with respect to the parameter vector [latex]\beta[/latex] and setting it to zero, we derive a set of equations known as the ‘normal equations’. In matrix form, these are expressed as [latex]X^T X \hat{\beta} = X^T y[/latex], where [latex]X[/latex] is the matrix of independent variables and [latex]y[/latex] is the vector of the dependent variable.
The solution for the estimated coefficient vector is then given by [latex]\hat{\beta} = (X^T X)^{-1} X^T y[/latex]. This closed-form solution is computationally efficient and provides a unique estimate, provided that the matrix [latex]X^T X[/latex] is invertible (i.e., there is no perfect multicollinearity among the independent variables). Geometrically, the OLS solution corresponds to an orthogonal projection of the outcome vector [latex]y[/latex] onto the vector subspace spanned by the columns of the predictor matrix [latex]X[/latex]. While powerful, OLS is sensitive to outliers, as squaring the residuals gives large errors a disproportionately large influence on the final fit.
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