BUILT-IN MATHEMATICA SYMBOL

# ARIMAProcess

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

ARIMAProcess[{a1, ..., ap}, {d1, ..., dn}, {b1, ..., bq}, ]
represents the vector autoregressive integrated moving-average process such that its 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[{1, ..., p}, d, {1, ..., q}, ] for non-negative integers p and q.
• ARIMAProcess[p, q] gives ARIMAProcess[{1, ..., p}, , {1, ..., q}, ] for non-negative integers p and q.
• ARIMAProcess can be used with such functions as CovarianceFunction, PDF, Probability, and RandomFunction.

## ExamplesExamplesopen allclose all

### Basic Examples (2)Basic Examples (2)

Simulate an ARIMA process with a linear trend:

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Simulate an ARIMA process with a quadratic trend:

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