"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 made in the results of each individual equation."
Least squares approximation is a method for solving an over-determined system of linear equations. It finds the vector that minimizes the sum of the squares of the differences between the equations and the corresponding values.
Matrix Algebra: Understanding basic operations on matrices and matrix algebra is important in linear algebra.
Linear Regression: Linear regression is a statistical tool used to analyze the relationship between dependent and independent variables with the help of a best-fit line.
Vector Spaces: Vector spaces are mathematical structures where vectors and scalars can be added, multiplied, and scaled. They are an essential part of linear algebra, which is the backbone of least squares approximation.
Matrix Inversion: Inverting matrices is an important operation in solving linear equations and finding solutions to the problem of least squares approximation.
Eigenvalues and Eigenvectors: Eigenvalues and eigenvectors are important concepts in linear algebra that play a crucial role in many applications such as image processing, signal analysis, and robotics.
Singular Value Decomposition: Singular value decomposition is an important technique used in linear algebra that enables us to understand the structure of matrices and analyze their properties.
Orthogonal Projection: Orthogonal projection is the process of projecting one vector onto another at right angles. It is used in least squares approximation to find the best-fit line or plane that minimizes the distance between the data points and the model.
Matrix Factorization: Factorization of matrices is a powerful technique used in many applications of linear algebra. It helps in understanding the structure of large matrices and enables us to solve complex problems.
QR Factorization: QR factorization is a technique used to factor a matrix into an orthogonal matrix and an upper-triangular matrix. It is a useful tool in least squares approximation to compute the coefficients of the best-fit line or plane.
Gauss-Markov Theorem: The Gauss-Markov theorem is a fundamental result in the theory of linear regression. It states that under some assumptions, the method of least squares gives unbiased estimates of the coefficients of the best-fit line with the minimum variance.
Ordinary Least Squares: This is the most common type of least squares approximation in which we fit a linear model that minimizes the sum of the squared errors between the predicted and actual values.
Partial Least Squares: This method is used when we have a large number of predictor variables and some of them are highly correlated. It involves finding a low-dimensional subspace that captures the most variation in the data, and fitting a linear model in this subspace.
Ridge Regression: This is a modification of ordinary least squares that adds a penalty term to the sum of the squared errors in order to prevent overfitting. The penalty term is proportional to the square of the magnitude of the regression coefficients.
Lasso Regression: This is another modification of ordinary least squares that also adds a penalty term to the sum of the squared errors, but the penalty term is proportional to the absolute value of the regression coefficients. This has a sparsity-inducing effect, meaning that some of the coefficients are forced to zero, resulting in a simpler model.
Elastic Net: This is a combination of ridge regression and lasso regression, which combines their strengths to obtain a more robust model. The penalty term is a linear combination of the ridge and lasso penalties.
Non-negative Least Squares: This is a special case of least squares where all the regression coefficients are constrained to be non-negative. This is useful when we want to ensure that the regression coefficients represent additive contributions to the response variable.
Weighted Least Squares: This is a modification of ordinary least squares that assigns weights to each observation based on its relative importance or reliability. This can be useful when there are outliers or when the variance of the errors is not constant across observations.
Total Least Squares: This is a variant of ordinary least squares in which both the response variable and predictor variables contain measurement errors. The goal is to find a linear model that minimizes the sum of the squared distances between the observed data and the fitted model.
"When the problem has substantial uncertainties in the independent variable (the x variable), then simple regression and least-squares methods have problems."
"Least squares problems fall into two categories: linear or ordinary least squares and nonlinear least squares."
"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."
"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."
"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)."
"The least-squares method is usually also co-credited to Carl Friedrich Gauss (1809)."
"[Gauss] contributed significant theoretical advances to the method and may have also used it in his earlier work (1795)."
"The method of least squares [...] minimiz[es] the sum of the squares of the residuals made in the results of each individual equation."
"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)."
"A residual being the difference between an observed value and the fitted value provided by a model."
"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."
"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."