BUILT-IN MATHEMATICA SYMBOL

# SARMAProcess

SARMAProcess[{a1, ..., ap}, {b1, ..., bq}, {s, {1, ..., m}, {1, ..., r}}, v]
represents a seasonal integrated autoregressive moving-average process with ARMA coefficients and , seasonal order s, seasonal ARMA coefficients and , and normal white noise with variance v.

SARMAProcess[{a1, ..., ap}, {b1, ..., bq}, {s, {1, ..., m}, {1, ..., r}}, ]
represents a vector SARMA process driven by normal white noise, with covariance matrix .

SARMAProcess[{a1, ..., ap}, {b1, ..., bq}, {{s1, ...}, {1, ..., m}, {1, ..., r}}, ]
represents a vector SARMA process with multiple seasonal orders .

## DetailsDetails

• SARMAProcess is a discrete-time and continuous-state random process.
• The SARMA process is described by the difference equations , , where is the state output, is white noise input, and is the shift operator.
• The scalar SARMA process has transfer function , where , .
• The vector SARMA process has transfer matrix , where , , and where is the × identity matrix.
• A scalar SARMA process should have real coefficients , , , and , positive integer seasonality coefficients s, and a positive variance v.
• An -dimensional vector SARMA process should have real coefficient matrices , , , and of dimensions ×, integer positive seasonality constants or integer positive seasonality constant s, and the covariance matrix should be symmetric positive definite of dimensions ×.
• SARMAProcess can be used with such functions as CovarianceFunction, PDF, Probability, and RandomFunction.

## ExamplesExamplesopen allclose all

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

Simulate a SARMA process:

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Covariance function:

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Correlation function:

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Partial correlation function:

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