ARMAProcess[{a1, ..., ap}, {b1, ..., bq}, v]
represents an autoregressive moving-average process with AR coefficients , MA coefficients , and normal white noise variance v.

ARMAProcess[{a1, ..., ap}, {b1, ..., bq}, ]
represents the vector autoregressive moving-average process with coefficient matrices and and covariance matrix .

attempts to give the ARMA representation of tproc.

ARMAProcess[tproc, {p, q}]
approximates a time series process tproc with an ARMA process of orders p and q.


  • ARMAProcess is a discrete-time and continuous-state random process.
  • The ARMA process is described by the difference equation , where is the state output, is white noise input, and is the shift operator.
  • The scalar ARMA process has transfer function , where .
  • The vector ARMA process has transfer matrix , where , and where is the × identity matrix.
  • A scalar ARMA process should have real coefficients and and a positive variance v.
  • An -dimensional vector ARMA process should have real coefficient matrices and of dimensions ×, and the covariance matrix should be symmetric positive definite of dimensions ×.
  • The following time series processes tproc can be approximated: MAProcess, ARProcess, ARMAProcess, ARIMAProcess, FARIMAProcess, and SARIMAProcess.
  • ARMAProcess[p, q] gives ARMAProcess[{FormalA1, ..., FormalAp}, {FormalB1, ..., FormalBq}, FormalV] for non-negative integers p and q.
  • ARMAProcess can be used with such functions as CovarianceFunction, PDF, Probability, and RandomFunction.
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