# ARIMAProcess

ARIMAProcess[{a1,,ap},d,{b1,,bq},v]

represents an autoregressive integrated moving-average process such that its  difference is a weakly stationary ARMAProcess[{a1,,ap},{b1,,bq},v].

ARIMAProcess[{a1,,ap},d,{b1,,bq},Σ]

represents a vector ARIMA process (y1(t), ,yn(t)) such that its (d,,d) difference is a vector weakly stationary ARMAProcess.

ARIMAProcess[{a1,,ap},{d1,,dn},{b1,,bq},Σ]

represents a vector ARIMA process (y1(t), ,yn(t)) such that its (d1,,dn) difference is a vector weakly stationary ARMAProcess.

ARIMAProcess[{a1,,ap},d,{b1,,bq},v,init]

represents an ARIMA process with initial data init.

ARIMAProcess[c,]

represents an ARIMA process with a constant c.

# Details • ARIMAProcess is a discrete-time and continuous-state random process.
• An ARIMAProcess[,d,,v] has a polynomial trend of degree d for d1.
• The ARIMA process is described by the difference equation , where is the state output, is the 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 ARIMA process should have real coefficients ai, bj, and c, non-negative integer integration order d, and a positive variance v.
• An -dimensional vector ARIMA process should have real coefficient matrices ai and bj of dimensions × , real vector c of length , integer non-negative integrating orders di or integer non-negative integrating order d, and the covariance matrix Σ should be symmetric positive definite of dimensions × .
• The ARIMA process with zero constant has transfer function , where , , and where is an -dimensional unit.
• ARIMAProcess[p,d,q] represents an ARIMA process with autoregressive and moving average orders p and q and integration order d for use in EstimatedProcess and related functions.
• ARIMAProcess can be used with such functions as CovarianceFunction, RandomFunction, and TimeSeriesForecast.

# Examples

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## Basic Examples(2)

Simulate an ARIMA process with a linear trend:

 In:= Out= In:= Out= Simulate an ARIMA process with a quadratic trend:

 In:= Out= In:= Out= ## Neat Examples(2)

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