Editing Linear model (section)

==Time series models==

An example of a linear time series model is an [[autoregressive moving average model]]. Here the model for values {<math>X_t</math>} in a time series can be written in the form

:<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 <math>\varepsilon_i</math> are random variables representing [[Innovation (signal processing)|innovations]] which are new random effects that appear at a certain time but also affect values of <math>X</math> at later times. In this instance the use of the term "linear model" refers to the structure of the above relationship in representing <math>X_t</math> 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 <math>\phi_i</math> and <math>\theta_i</math>, as it would be in the case of a regression model, which looks structurally similar.