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The design matrix contains data on the independent variables (also called explanatory variables), in a statistical model that is intended to explain observed data on a response variable (often called a dependent variable). The theory relating to such models uses the design matrix as input to some linear algebra : see for example linear regression.
Unlike Tikhonov regularization, this scheme does not have a convenient closed-form solution: instead, the solution is typically found using quadratic programming or more general convex optimization methods, as well as by specific algorithms such as the least-angle regression algorithm.
Spectral Regularization is also used to enforce a reduced rank coefficient matrix in multivariate regression. [4] In this setting, a reduced rank coefficient matrix can be found by keeping just the top n {\displaystyle n} singular values, but this can be extended to keep any reduced set of singular values and vectors.
Linear least squares (LLS) is the least squares approximation of linear functions to data. It is a set of formulations for solving statistical problems involved in linear regression, including variants for ordinary (unweighted), weighted, and generalized (correlated) residuals.
Consider a set of data points, (,), (,), …, (,), and a curve (model function) ^ = (,), that in addition to the variable also depends on parameters, = (,, …,), with . It is desired to find the vector of parameters such that the curve fits best the given data in the least squares sense, that is, the sum of squares = = is minimized, where the residuals (in-sample prediction errors) r i are ...
Regression analysis – use of statistical techniques for learning about the relationship between one or more dependent variables (Y) and one or more independent variables (X). Overview articles [ edit ]
In a regression context, we combine leverage and influence functions to compute the degree to which estimated coefficients would change if we removed a single data point. Denoting the regression residuals as ^ = ^, one can compare the estimated coefficient ^ to the leave-one-out estimated coefficient ^ using the formula [6] [7]
In machine learning, a key challenge is enabling models to accurately predict outcomes on unseen data, not just on familiar training data. Regularization is crucial for addressing overfitting—where a model memorizes training data details but can't generalize to new data. The goal of regularization is to encourage models to learn the broader ...