Linear model: Difference between revisions

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{{Short description|Type of statistical model}}

{{Distinguish|linear model of innovation}}

In [[statistics]], the term '''linear model''' isrefers usedto inany differentmodel wayswhich accordingassumes to[[linearity]] in the contextsystem. The most common occurrence is in connection with regression models and the term is often taken as synonymous with [[linear regression]] model. However, the term is also used in [[time series analysis]] with a different meaning. In each case, the designation "linear" is used to identify a subclass of models for which substantial reduction in the complexity of the related [[statistical theory]] is possible.

==Linear regression models==

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Given that estimation is undertaken on the basis of a [[least squares]] analysis, estimates of the unknown parameters <math>\beta_j</math> are determined by minimising a sum of squares function

:<math>S = \sum_{i = 1}^n \varepsilon_i^2 = \sum_{i = 1}^n \left(Y_i - \beta_0 - \beta_1 \phi_1(X_{i1}) - \cdots - \beta_p \phi_p(X_{ip})\right)^2 .</math>

From this, it can readily be seen that the "linear" aspect of the model means the following:

:*the function to be minimised is a quadratic function of the <math>\beta_j</math> for which minimisation is a relatively simple problem;

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{{Authority control}}

[[Category:Curve fitting]]

[[Category:Regression models]]