# Wolfram Language & System 10.3 (2015)|Legacy Documentation

This is documentation for an earlier version of the Wolfram Language.
BUILT-IN WOLFRAM LANGUAGE SYMBOL

# ARMAProcess

ARMAProcess[{a1,,ap},{b1,,bq},v]
represents a weakly stationary autoregressive moving-average process with AR coefficients , MA coefficients , and normal white noise variance v.

ARMAProcess[{a1,,ap},{b1,,bq},Σ]
represents a weakly stationary vector ARMA process with coefficient matrices and and covariance matrix Σ.

ARMAProcess[{a1,,ap},{b1,,bq},v,init]
represents an ARMA process with initial data init.

ARMAProcess[c,]
represents an ARMA process with a constant c.

## DetailsDetails

• ARMAProcess is also known as ARMA and VARMA (vector ARMA).
• ARMAProcess is a discrete-time and continuousstate random process.
• The ARMA process is described by the difference equation , where is the state output, is 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 ARMA process should have real coefficients , , and c, and a positive variance v.
• An -dimensional vector ARMA process should have real coefficient matrices and of dimensions ×, real vector c of length n, and the covariance matrix Σ should be symmetric positive definite of dimensions ×.
• The ARMA process with zero constant has transfer function , where equals:
•  scalar process vector process; is the × identity matrix
• ARMAProcess[tproc,{p,q}] for a time series process tproc gives an ARMA process of orders p and q, such that its transfer function agrees with PadeApproximant about zero with degrees of the transfer function of tproc.
• ARMAProcess[tproc] attempts to return an ARMA process such that its transfer function is the same as the one of tproc.
• Possible time series processes tproc include ARProcess, SARMAProcess, and SARIMAProcess.
• ARMAProcess[p,q] represents an ARMA process of orders p and q for use in EstimatedProcess and related functions.
• ARMAProcess can be used with such functions as CovarianceFunction, RandomFunction, and TimeSeriesForecast.

## ExamplesExamplesopen allclose all

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

Simulate an ARMA process:

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Covariance function:

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Correlation function:

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Partial correlation function:

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