Normalengleichungen

Normalengleichungen, also known as normal equations, are a set of linear equations derived from the method of least squares. They are used to find the parameters of a linear regression model that minimize the sum of the squared differences between the observed and predicted values of the dependent variable. The process involves forming a matrix X, which contains the independent variables and a column of ones for the intercept, and a vector y, which contains the dependent variable. The normal equations are then given by the matrix equation X^T X beta = X^T y, where beta is the vector of unknown coefficients. Solving this system of equations for beta yields the least-squares estimates for the regression parameters. This method is computationally efficient for smaller datasets but can become computationally expensive and numerically unstable for very large numbers of predictors. Alternative methods like gradient descent are often preferred in such cases. The derivation relies on calculus, specifically finding the minimum of the sum of squared errors by setting its partial derivatives with respect to each coefficient to zero.