BUILT-IN MATHEMATICA SYMBOL

# SARIMAProcess

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

SARIMAProcess[{a1, ..., ap}, d, {b1, ..., bq}, {s, {1, ..., m}, , {1, ..., r}}, ]
represents a vector SARIMA process with coefficient matrices , , , and and covariance matrix .

SARIMAProcess[{a1, ...}, {d1, ...}, {b1, ...}, {{s1, ...}, {1, ...}, {1, ...}, {1, ...}}, ]
represents a vector SARIMA process with multiple integration orders , seasonal orders , and seasonal integration orders .

## DetailsDetails

• SARIMAProcess is a discrete-time and continuous-state random process.
• The SARIMA process is effectively the composition of an ARIMA process and a seasonal version of an ARIMA process.
• The SARIMA process is described by the difference equations , , where is the state output, is white noise input, and is the shift operator.
• The scalar SARIMA process has transfer function , where , .
• The vector SARIMA process has transfer matrix , where , , and where is the × identity matrix.
• A scalar SARIMA process should have real coefficients , , , and , positive integer seasonality order s, non-negative integer integration orders d and , and a positive variance v.
• An -dimensional vector SARIMA process should have real coefficient matrices , , , and of dimensions ×; positive integer seasonality orders or s; non-negative integer integration orders or d, as well as or ; and symmetric positive definite covariance matrix of dimension ×.
• SARIMAProcess can be used with such functions as CovarianceFunction, PDF, Probability, and RandomFunction.

## ExamplesExamplesopen allclose all

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

Simulate a SARIMA process:

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Simulate SARIMA with seasonal trend:

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Simulate SARIMA with linear trend:

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