ARProcess[{a1, ..., ap}, v]
represents an autoregressive process of order p with normal white noise that has variance v.

ARProcess[{a1, ..., ap}, ]
represents a vector autoregressive process with multinormal white noise that has covariance matrix .

ARProcess[tproc, p]
approximate a time series process tproc with an AR process of order p.


  • ARProcess is a discrete-time and continuous-state random process.
  • The AR process is described by the difference equation , where is the state output, is the white noise input, and is the shift operator.
  • The scalar AR process has transfer function , where .
  • The vector AR process has transfer matrix , where , and where is the × identity matrix.
  • A scalar AR process should have real coefficients and a positive variance v.
  • An -dimensional vector AR process should have real coefficient matrices 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.
  • ARProcess[p] gives ARProcess[{FormalA1, ..., FormalAp}, FormalV] for a non-negative integer p.
  • ARProcess can be used with such functions as CovarianceFunction, PDF, Probability, and RandomFunction.
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