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BUILT-IN MATHEMATICA SYMBOL
ARIMAProcess
ARIMAProcess[{a1, ..., ap}, d, {b1, ..., bq}, v]
represents the autoregressive integrated moving-average process 
such that its 
difference is an ARMAProcess[{a1, ..., ap}, {b1, ..., bq}, v].
ARIMAProcess[{a1, ..., ap}, d, {b1, ..., bq}, 
]
represents the vector autoregressive integrated moving-average process 
such that its 
difference is a vector ARMAProcess.
ARIMAProcess[{a1, ..., ap}, {d1, ..., dn}, {b1, ..., bq}, ]
represents the vector autoregressive integrated moving-average process 
such that its 
difference is a vector ARMAProcess.
Details
- ARIMAProcess is a discrete-time and continuous-state random process.
- An ARIMAProcess[..., d, ..., v] has a polynomial trend of degree d for d≥1.
- The ARIMA process is described by the difference equation
,
where 
is the state output, 
is the white noise input, and 
is the shift operator. - The scalar ARIMA process has transfer function
, where 
. - The vector ARIMA process has transfer matrix
, where 
, and where 
is the 
×
identity matrix. - A scalar ARIMA process should have real coefficients
and 
, non-negative integer integration order d, and a positive variance v. - An
-dimensional vector ARIMA process should have real coefficient matrices 
and 
of dimensions 
×
, integer non-negative integrating orders 
or integer non-negative integrating order d, and the covariance matrix should be symmetric positive definite of dimensions 
×
. - ARIMAProcess[p, d, q] gives ARIMAProcess[{
1, ..., p}, d, {
1, ..., q}, 
] for non-negative integers p and q. - ARIMAProcess[p, q] gives ARIMAProcess[{1, ..., p},
, {1, ..., q}, ] for non-negative integers p and q. - ARIMAProcess can be used with such functions as CovarianceFunction, PDF, Probability, and RandomFunction.
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