Wolfram Language & System 10.3 (2015)|Legacy Documentation

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represents a weakly stationary autoregressive moving-average process with AR coefficients , MA coefficients , and normal white noise variance v.

represents a weakly stationary vector ARMA process with coefficient matrices and and covariance matrix Σ.

represents an ARMA process with initial data init.

represents an ARMA process with a constant c.


  • ARMAProcess is also known as ARMA and VARMA (vector ARMA).
  • ARMAProcess is a discrete-time and continuousstate random process.
  • The ARMA process is described by the difference equation , 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 ARMA process should have real coefficients , , and c, and a positive variance v.
  • An -dimensional vector ARMA process should have real coefficient matrices and of dimensions ×, real vector c of length n, and the covariance matrix Σ should be symmetric positive definite of dimensions ×.
  • The ARMA process with zero constant has transfer function , where equals:
  • scalar process
    vector process; is the × identity matrix
  • ARMAProcess[tproc,{p,q}] for a time series process tproc gives an ARMA process of orders p and q, such that its transfer function agrees with PadeApproximant about zero with degrees of the transfer function of tproc.
  • ARMAProcess[tproc] attempts to return an ARMA process such that its transfer function is the same as the one of tproc.
  • Possible time series processes tproc include ARProcess, SARMAProcess, and SARIMAProcess.
  • ARMAProcess[p,q] represents an ARMA process of orders p and q for use in EstimatedProcess and related functions.
  • ARMAProcess can be used with such functions as CovarianceFunction, RandomFunction, and TimeSeriesForecast.
Introduced in 2012
| Updated in 2014