represents a weakly stationary autoregressive process of order p with normal white noise variance v.

represents a weakly stationary vector AR process with multinormal white noise covariance matrix Σ.

represents an AR process with initial data init.

represents an AR process with a constant c.


  • ARProcess is also known as AR or VAR (vector AR).
  • 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, 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 AR process can have real coefficients and c, a positive variance v, and a non-negative integer order p.
  • An -dimensional vector AR process can have real coefficient matrices of dimensions ×, real vector c of length , and the covariance matrix Σ should be symmetric positive definite of dimensions ×.
  • The AR process with zero constant has transfer function , where:
  • scalar process
    vector process; is the × identity matrix
  • ARProcess[tproc,p] for a time series process tproc gives an AR process of order p such that the series expansions about zero of the corresponding transfer functions agree up to degree p.
  • Possible time series processes tproc include ARProcess, ARMAProcess, and SARIMAProcess.
  • ARProcess[p] represents an autoregressive process of order p for use in EstimatedProcess and related functions.
  • ARProcess can be used with such functions as CovarianceFunction, RandomFunction, and TimeSeriesForecast.
Introduced in 2012
| Updated in 2014
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