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== Generalizations ==
There are various generalizations of ARMA. '''Nonlinear''' MAAR (NMANAR), nonlinear ARMA (NARNMA), and nonlinear ARMA (NARMA) model nonlinear dependence on past values and error terms. [[vector autoregression|Vector AR]] (VAR) and vector ARMA (VARMA) model '''multivariate''' time series. [[Autoregressive fractionally integrated moving average]] (or Fractional ARIMA, FARIMA, or ARFIMA) model time-series that exhibits '''long memory'''. [[Autoregressive integrated moving average]] (ARIMA) models non-stationary time series (that is, whose mean changes over time). [[Autoregressive conditional heteroskedasticity]] (ARCH) models time series where the variance changes. Seasonal ARIMA (SARIMA or periodic ARMA) models '''periodic''' variation. [[Autoregressive fractionally integrated moving average]] (ARFIMA, or Fractional ARIMA, FARIMA) model time-series that exhibits '''long memory'''. Multiscale AR (MAR) is indexed by the nodes of a [[Tree (graph theory)|tree]] instead of integers.
=== {{anchor|ARMAX}}Autoregressive–moving-average model with exogenous inputs model (ARMAX model) === <!-- This section is linked from [[ARMAX]], so if you change the title, please also change the corresponding link in the ARMAX page -->
The notation ARMAX(''p'', ''q'', ''b'') refers to thea model with ''p'' autoregressive terms, ''q'' moving average terms and ''b'' exogenous inputs terms. ThisThe modellast containsterm the AR(''p'') and MA(''q'') models andis a linear combination of the last ''b'' terms of a known and external time series <math>d_t</math>. It is given by:
:<math> X_t = \varepsilon_t + \sum_{i=1}^p \varphi_i X_{t-i} + \sum_{i=1}^q \theta_i \varepsilon_{t-i} + \sum_{i=1}^b \eta_i d_{t-i}.\,</math>
Some nonlinear variants of models with exogenous variables have been defined: see for example [[Nonlinear autoregressive exogenous model]].
Statistical packages implement the ARMAX model through the use of "exogenous" (that is, independent,) variables. Care must be taken when interpreting the output of those packages, because the estimated parameters usually (for example, in [[R (programming language)|R]]<ref name="R.stats.arima">[http://search.r-project.org/R/library/stats/html/arima.html ARIMA Modelling of Time Series], R documentation</ref> and [[gretl]]) refer to the regression:
: <math> X_t - m_t = \varepsilon_t + \sum_{i=1}^p \varphi_i (X_{t-i} - m_{t-i}) + \sum_{i=1}^q \theta_i \varepsilon_{t-i}.\,</math>
where <math>m_t</math> incorporates all exogenous (or independent) variables:
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