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In general, the coefficients of the matrices , and can be complex. By using a Hermitian transpose instead of a simple transpose, it is possible to find a vector β ^ {\displaystyle {\boldsymbol {\widehat {\beta }}}} which minimizes S ( β ) {\displaystyle S({\boldsymbol {\beta }})} , just as for the real matrix case.
IRLS is used to find the maximum likelihood estimates of a generalized linear model, and in robust regression to find an M-estimator, as a way of mitigating the influence of outliers in an otherwise normally-distributed data set, for example, by minimizing the least absolute errors rather than the least square errors.
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 .
Multiple linear regression is a generalization of simple linear regression to the case of more than one independent variable, and a special case of general linear models, restricted to one dependent variable. The basic model for multiple linear regression is
While the identity is primarily used on matrices, it holds in a general ring or in an Ab-category. The Woodbury matrix identity allows cheap computation of inverses and solutions to linear equations. However, little is known about the numerical stability of the formula.
Fermat's little theorem and some proofs; Gödel's completeness theorem and its original proof; Mathematical induction and a proof; Proof that 0.999... equals 1; Proof that 22/7 exceeds π; Proof that e is irrational; Proof that π is irrational; Proof that the sum of the reciprocals of the primes diverges
In statistics, the Gauss–Markov theorem (or simply Gauss theorem for some authors) [1] states that the ordinary least squares (OLS) estimator has the lowest sampling variance within the class of linear unbiased estimators, if the errors in the linear regression model are uncorrelated, have equal variances and expectation value of zero. [2]
Proof: Note that is ... Similar formulas arise when applying general formulas for statistical influences functions in the regression context. [9] ...