# ARMAProcess

ARMAProcess[{a1,,ap},{b1,,bq},v]

represents a weakly stationary autoregressive moving-average process with AR coefficients ai, MA coefficients bj, and normal white noise variance v.

ARMAProcess[{a1,,ap},{b1,,bq},Σ]

represents a weakly stationary vector ARMA process with coefficient matrices ai and bj and covariance matrix Σ.

ARMAProcess[{a1,,ap},{b1,,bq},v,init]

represents an ARMA process with initial data init.

ARMAProcess[c,]

represents an ARMA process with a constant c.

# Details

• ARMAProcess is also known as ARMA and VARMA (vector ARMA).
• ARMAProcess is a discrete-time and continuousstate random process.
• The ARMA process is described by the difference equation , 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 ARMA process should have real coefficients ai, bj, and c, and a positive variance v.
• An -dimensional vector ARMA process should have real coefficient matrices ai and bj of dimensions ×, real vector c of length n, and the covariance matrix Σ should be symmetric positive definite of dimensions ×.
• The ARMA process with zero constant has transfer function , where equals:
•  scalar process vector process; is the × identity matrix
• ARMAProcess[tproc,{p,q}] for a time series process tproc gives an ARMA process of orders p and q, such that its transfer function agrees with PadeApproximant about zero with degrees {q,p} of the transfer function of tproc.
• ARMAProcess[tproc] attempts to return an ARMA process such that its transfer function is the same as the one of tproc.
• Possible time series processes tproc include ARProcess, SARMAProcess, and SARIMAProcess.
• ARMAProcess[p,q] represents an ARMA process of orders p and q for use in EstimatedProcess and related functions.
• ARMAProcess can be used with such functions as CovarianceFunction, RandomFunction, and TimeSeriesForecast.

# Examples

open allclose all

## Basic Examples(3)

Simulate an ARMA process:

 In[1]:=
 Out[1]=
 In[2]:=
 Out[2]=

Covariance function:

 In[1]:=
 Out[1]=
 In[2]:=
 Out[2]=

Correlation function:

 In[1]:=
 Out[1]=

Partial correlation function:

 In[2]:=
 Out[2]=