BUILT-IN MATHEMATICA SYMBOL

# ARProcess

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.

## DetailsDetails

• 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[{1, ..., p}, ] for a non-negative integer p.
• ARProcess can be used with such functions as CovarianceFunction, PDF, Probability, and RandomFunction.

## ExamplesExamplesopen allclose all

### Basic Examples (3)Basic Examples (3)

Simulate an AR process:

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Covariance function:

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Correlation function:

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Partial correlation function:

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