Linear model: Difference between revisions

Revision as of 12:11, 19 October 2020

Revision as of 20:52, 18 February 2021

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{{distinguishDistinguish|linear model of innovation}}

In [[statistics]], the term '''linear model''' is used in different ways according to the context. 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.

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:<math>Y_i = \beta_0 + \beta_1 \phi_1(X_{i1}) + \cdots + \beta_p \phi_p(X_{ip}) + \varepsilon_i \qquad i = 1, \ldots, n </math>

where <math> \phi_1, \ldots, \phi_p </math> may be [[Nonlinear system|nonlinear]] functions. In the above, the quantities <math>\varepsilon_i</math> are [[Randomrandom variable|random variables]]s representing errors in the relationship. The "linear" part of the designation relates to the appearance of the [[regression coefficient]]s, <math>\beta_j</math> in a linear way in the above relationship. Alternatively, one may say that the predicted values corresponding to the above model, namely

:<math>\hat{Y}_i = \beta_0 + \beta_1 \phi_1(X_{i1}) + \cdots + \beta_p \phi_p(X_{ip}) \qquad (i = 1, \ldots, n), </math>

are linear functions of the <math>\beta_j</math>.

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==References==

{{Reflist}}

<references/>

{{Statistics}}

{{Authority control}}

[[Category:Regression models]]