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

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

# SARIMAProcess

SARIMAProcess[{a1,,ap},d,{b1,,bq},{s,{α1,,αm},δ,{β1,,βr}},v]
represents a seasonal integrated autoregressive moving-average process with ARIMA coefficients , d, and ; seasonal order s; seasonal ARIMA coefficients , δ, and ; seasonal integration order δ; and normal white noise with variance v.

SARIMAProcess[{a1,,ap},d,{b1,,bq},{s,{α1,,αm},δ,{β1,,βr}},Σ]
represents a vector SARIMA process with coefficient matrices , , , and and covariance matrix Σ.

SARIMAProcess[{a1,},{d1,},{b1,},{{s1,},{α1,},{δ1,},{β1,}},Σ]
represents a vector SARIMA process with multiple integration orders , seasonal orders , and seasonal integration orders .

SARIMAProcess[{a1,,ap},d,{b1,,bq},{s,{α1,,αm},δ,{β1,,βr}},v,init]
represents a SARIMA process with initial data init.

SARIMAProcess[c,]
represents a SARIMA process with constant c.

## 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 equation , with , 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 SARIMA process should have real coefficients , , , , and c, positive integer seasonality order s, non-negative integer integration orders d and δ, and a positive variance v.
• An -dimensional vector SARIMA process should have real coefficient matrices , , , and of dimensions ×; vector c of length ; positive integer seasonality orders or s; non-negative integer integration orders or d, as well as or δ; and symmetric positive definite covariance matrix Σ of dimension ×.
• The SARIMA process with zero constant has transfer function , where , , , , , and is an n-dimensional unit.
• SARIMAProcess[p,d,q,{s,sp,sd,sq}] represents a SARIMA process with autoregressive and moving-average orders p and q and integration order d, their seasonal counterparts sp, sq, and sd, and seasonality s for use in EstimatedProcess and related functions.
• SARIMAProcess can be used with such functions as CovarianceFunction, RandomFunction, and TimeSeriesForecast.

## 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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