SARMAProcess

SARMAProcess[{a1,,ap},{b1,,bq},{s,{α1,,αm},{β1,,βr}},v]
represents a weakly stationary seasonal autoregressive moving-average process with ARMA coefficients ai and bj, seasonal order s, seasonal ARMA coefficients αi and βj, and normal white noise with variance v.

SARMAProcess[{a1,,ap},{b1,,bq},{s,{α1,,αm},{β1,,βr}},Σ]
represents a weakly stationary vector SARMA process driven by normal white noise, with covariance matrix Σ.

SARMAProcess[{a1,,ap},{b1,,bq},{{s1,},{α1,,αm},{β1,,βr}},Σ]
represents a weakly stationary vector SARMA process with multiple seasonal orders si.

SARMAProcess[{a1,,ap},{b1,,bq},{s,{α1,,αm},{β1,,βr}},v,init]
represents a SARMA process with initial data init.

SARMAProcess[c,]
represents a SARMA process with a constant c.

DetailsDetails

  • SARMAProcess is a discrete-time and continuous-state random process.
  • The SARMA 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 SARMA process should have real coefficients ai, αi, bj, βj, and c, positive integer seasonality coefficients s, and a positive variance v.
  • An -dimensional vector SARMA process should have real coefficient matrices ai, αi, bj, and βj of dimensions ×, real vector c of length , integer positive seasonality constants si or integer positive seasonality constant s, and the covariance matrix Σ should be symmetric positive definite of dimensions ×.
  • The SARMA process with zero constant has transfer function , where , , , , and is an n-dimensional unit.
  • SARMAProcess[p,q,{s,sp,sq}] represents a SARMA process with autoregressive and moving-average orders p and q, their seasonal counterparts sp and sq, and seasonality s for use in EstimatedProcess and related functions.
  • SARMAProcess can be used with such functions as CovarianceFunction, RandomFunction, and TimeSeriesForecast.

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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Introduced in 2012
(9.0)
| Updated in 2014
(10.0)
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