In the statistical analysis of time series, an autoregressive–moving-average (ARMA) model is used to represent a (weakly) stationary stochastic process by combining two components: autoregression (AR) and moving average (MA). These models are widely used for analyzing the structure of a series and for forecasting future values.

The AR component specifies that the current value of the series depends linearly on its own past values (lags), while the MA component specifies that the current value depends on a linear combination of past error terms. An ARMA model is typically denoted as ARMA(p, q), where p is the order of the autoregressive part and q is the order of the moving-average part.

The general ARMA model was described in the 1951 thesis of Peter Whittle, Hypothesis testing in time series analysis, and it was popularized in the 1970 book by George E. P. Box and Gwilym Jenkins.

ARMA models can be estimated by using the Box–Jenkins method.

The notation AR(p) refers to the autoregressive model of order p. The AR(p) model is written as

where ${\displaystyle \varphi _{1},\ldots ,\varphi _{p}}$ are parameters and the random variable ${\displaystyle \varepsilon _{t}}$ is white noise, usually independent and identically distributed (i.i.d.) normal random variables.

In order for the model to remain stationary, the roots of its characteristic polynomial must lie outside the unit circle. For example, processes in the AR(1) model with ${\displaystyle |\varphi _{1}|\geq 1}$ are not stationary because the root of ${\displaystyle 1-\varphi _{1}B=0}$ lies within the unit circle.

The augmented Dickey–Fuller test can assesses the stability of an intrinsic mode function and trend components. For stationary time series, the ARMA models can be used, while for non-stationary series, Long short-term memory models can be used to derive abstract features. The final value is obtained by reconstructing the predicted outcomes of each time series.^{[citation needed]}