ARIMAProcess

ARIMAProcess[{a1, ..., ap}, d, {b1, ..., bq}, v]
represents the autoregressive integrated moving-average process such that its ^(th) difference is an ARMAProcess[{a1, ..., ap}, {b1, ..., bq}, v].

ARIMAProcess[{a1, ..., ap}, d, {b1, ..., bq}, ]
represents the vector autoregressive integrated moving-average process such that its ^(th) difference is a vector ARMAProcess.

ARIMAProcess[{a1, ..., ap}, {d1, ..., dn}, {b1, ..., bq}, ]
represents the vector autoregressive integrated moving-average process such that its ^(th) difference is a vector ARMAProcess.

DetailsDetails

  • ARIMAProcess is a discrete-time and continuous-state random process.
  • An ARIMAProcess[..., d, ..., v] has a polynomial trend of degree d for d≥1.
  • The ARIMA process is described by the difference equation , where is the state output, is the white noise input, and is the shift operator.
  • The scalar ARIMA process has transfer function , where .
  • The vector ARIMA process has transfer matrix , where , and where is the × identity matrix.
  • A scalar ARIMA process should have real coefficients and , non-negative integer integration order d, and a positive variance v.
  • An -dimensional vector ARIMA process should have real coefficient matrices and of dimensions ×, integer non-negative integrating orders or integer non-negative integrating order d, and the covariance matrix should be symmetric positive definite of dimensions ×.
  • ARIMAProcess[p, d, q] gives ARIMAProcess[{FormalA1, ..., FormalAp}, d, {FormalB1, ..., FormalBq}, FormalV] for non-negative integers p and q.
  • ARIMAProcess[p, q] gives ARIMAProcess[{FormalA1, ..., FormalAp}, FormalD, {FormalB1, ..., FormalBq}, FormalV] for non-negative integers p and q.
  • ARIMAProcess can be used with such functions as CovarianceFunction, PDF, Probability, and RandomFunction.
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