represents a weakly stationary seasonal autoregressive moving-average process with ARMA coefficients and , seasonal order s, seasonal ARMA coefficients and , and normal white noise with variance v.

represents a weakly stationary vector SARMA process driven by normal white noise, with covariance matrix Σ.

represents a weakly stationary vector SARMA process with multiple seasonal orders .

represents a SARMA process with initial data init.

represents a SARMA process with a constant c.


  • SARMAProcess is a discrete-time and continuous-state random process.
  • The SARMA process is described by the difference equation , with , where is the state output, is white noise input, is the shift operator, and the constant c is taken to be zero if not specified.
  • The initial data init can be given as a list or a single-path TemporalData object with time stamps understood as .
  • A scalar SARMA process should have real coefficients , , , , and c, positive integer seasonality coefficients s, and a positive variance v.
  • An -dimensional vector SARMA process should have real coefficient matrices , , , and of dimensions ×, real vector c of length , integer positive seasonality constants or integer positive seasonality constant s, and the covariance matrix Σ should be symmetric positive definite of dimensions ×.
  • The SARMA process with zero constant has transfer function , where , , , , and is an n-dimensional unit.
  • SARMAProcess[p,q,{s,sp,sq}] represents a SARMA process with autoregressive and moving-average orders p and q, their seasonal counterparts sp and sq, and seasonality s for use in EstimatedProcess and related functions.
  • SARMAProcess can be used with such functions as CovarianceFunction, RandomFunction, and TimeSeriesForecast.
Introduced in 2012
| Updated in 2014