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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.
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 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]
The true distribution is then approximated by a linear regression, and the best estimators are obtained in closed form as ^ = ((~) ~) (~) (¯), where denotes the template matrix with the values of the known or previously determined model for any of the reference values β, are the random variables (e.g. a measurement), and the matrix ~ and the ...
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 ...
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The numerical methods for linear least squares are important because linear regression models are among the most important types of model, both as formal statistical models and for exploration of data-sets. The majority of statistical computer packages contain
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]