Autoregressive Moving Average (ARMA)
The autoregressive-moving average model can be used to describe a (weekly) stationary process. In digital signal processing, ARMA represents a filter with white noise as input and the process \(X_{t}\) as output. It models the process in terms of two polynomials, derived from the two models for \(X_t\):
- Moving-average model: \(X_t=\sum\limits_{i=1}^q\theta_i\epsilon_{t-i}+\epsilon_t+\mu\)
- Autoregressive model: \(X_t=\sum\limits_{i=1}^{p}\upvarphi_iX_{t-i}+\epsilon_t\) They combine to the ARMA model:
$$
X_t=\epsilon_t+\sum\limits_{i=1}p\upvarphi_iX_{t-i}+\sum\limits_{i=1}q\theta_i\epsilon_{t-i}
$$ The autoregressive part is explaining the current value of \(X_t\) based on its past realizations. The moving average incorporates past prediction errors (noise) to better estimate \(X_t\). ARMA is useful for predicting signals that have some sort of autocorrelation and noise.