Linear model: Difference between revisions

Revision as of 12:11, 19 October 2020

Latest revision as of 23:29, 17 November 2024

(6 intermediate revisions by 6 users not shown)

Line 1:

{{Short description|Type of statistical model}}

{{distinguishDistinguish|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==

Line 10 ⟶ 11:

:<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>.

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;

Line 43 ⟶ 44:

==References==

{{Reflist}}

<references/>

{{Statistics}}

{{Authority control}}

[[Category:Curve fitting]]

[[Category:Regression models]]