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 ai, d, and bj; seasonal order s; seasonal ARIMA coefficients αi, δ, and βj; 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 ai, bj, αi, and βj and covariance matrix Σ.

SARIMAProcess[{a1,},{d1,},{b1,},{{s1,},{α1,},{δ1,},{β1,}},Σ]

represents a vector SARIMA process with multiple integration orders di, seasonal orders sj, and seasonal integration orders δk.

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.

Details

• 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 {,y[-2],y[-1]} or a single-path TemporalData object with time stamps understood as {,-2,-1}.
• A scalar SARIMA process should have real coefficients ai, bj, αi, βj, 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 ai, bj, αi, and βj of dimensions ×; vector c of length ; positive integer seasonality orders si or s; non-negative integer integration orders di or d, as well as δi 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.

Examples

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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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