Linear model: Difference between revisions

Revision as of 05:13, 25 May 2017

Revision as of 13:01, 23 June 2017

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{{main|Linear regression}}

For the regression case, the [[statistical model]] is as follows. Given a (random) sample <math> (Y_i, X_{i1}, \ldots, X_{ip}), \, i = 1, \ldots, n </math> the relation between the observations ''Y_i'' and the [[independent variables]] ''X_ij'' is formulated as

:<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>

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:<math> X_t = c + \varepsilon_t + \sum_{i=1}^p \phi_i X_{t-i} + \sum_{i=1}^q \theta_i \varepsilon_{t-i}.\,</math>

where again the quantities ''ε_t'' are random variables representing [[Innovation (signal processing)|innovations]] which are new random effects that appear at a certain time but also affect values of ''X'' at later times. In this instance the use of the term "linear model" refers to the structure of the above relationship in representing ''X_t'' as a linear function of past values of the same time series and of current and past values of the innovations.<ref>Priestley, M.B. (1988) ''Non-linear and Non-stationary time series analysis'', Academic Press. {{ISBN |0-12-564911-8}}</ref> This particular aspect of the structure means that it is relatively simple to derive relations for the mean and [[covariance]] properties of the time series. Note that here the "linear" part of the term "linear model" is not referring to the coefficients ''φ_i'' and ''θ_i'', as it would be in the case of a regression model, which looks structurally similar.

==Other uses in statistics==

There are some other instances where "nonlinear model" is used to contrast with a linearly structured model, although the term "linear model" is not usually applied. One example of this is [[nonlinear dimensionality reduction]].

==See also==