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The extension to multiple and/or vector-valued predictor variables (denoted with a capital X) is known as multiple linear regression, also known as multivariable linear regression (not to be confused with multivariate linear regression). [10] Multiple linear regression is a generalization of simple linear regression to the case of more than one ...
Although polynomial regression fits a nonlinear model to the data, as a statistical estimation problem it is linear, in the sense that the regression function E(y | x) is linear in the unknown parameters that are estimated from the data. For this reason, polynomial regression is considered to be a special case of multiple linear regression. [1]
The general linear model or general multivariate regression model is a compact way of simultaneously writing several multiple linear regression models. In that sense it is not a separate statistical linear model. The various multiple linear regression models may be compactly written as [1]
Commonality analysis is a statistical technique within multiple linear regression that decomposes a model's R 2 statistic (i.e., explained variance) by all independent variables on a dependent variable in a multiple linear regression model into commonality coefficients.
Mathematically, linear least squares is the problem of approximately solving an overdetermined system of linear equations A x = b, where b is not an element of the column space of the matrix A. The approximate solution is realized as an exact solution to A x = b' , where b' is the projection of b onto the column space of A .
Using the same technique as with Bayesian linear regression, we decompose the exponential term using a matrix-form of the sum-of-squares technique. Here, however, we will also need to use the Matrix Differential Calculus ( Kronecker product and vectorization transformations).
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.
The prediction is obtained by adding these products along with a constant. When the weights are chosen to give the best prediction by some criterion, the model referred to as a proper linear model. Therefore, multiple regression is a proper linear model. By contrast, unit-weighted regression is called an improper linear model.