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BUILT-IN MATHEMATICA SYMBOL
SARMAProcess
SARMAProcess[{a1, ..., ap}, {b1, ..., bq}, {s, {
1, ..., m}, {
1, ..., r}}, v]
represents a seasonal integrated autoregressive moving-average process with ARMA coefficients 
and 
, seasonal order s, seasonal ARMA coefficients 
and 
, and normal white noise with variance v.
SARMAProcess[{a1, ..., ap}, {b1, ..., bq}, {s, {1, ..., m}, {1, ..., r}}, 
]
represents a vector SARMA process driven by normal white noise, with covariance matrix .
SARMAProcess[{a1, ..., ap}, {b1, ..., bq}, {{s1, ...}, {1, ..., m}, {1, ..., r}}, ]
represents a vector SARMA process with multiple seasonal orders 
.
Details
- SARMAProcess is a discrete-time and continuous-state random process.
- The SARMA process is described by the difference equations
, 
, where 
is the state output, 
is white noise input, and 
is the shift operator. - The scalar SARMA process has transfer function
, where 
, 
. - The vector SARMA process has transfer matrix
, where 
, 
, and where 
is the 
×
identity matrix. - A scalar SARMA process should have real coefficients
, 
, 
, and 
, positive integer seasonality coefficients s, and a positive variance v. - An
-dimensional vector SARMA process should have real coefficient matrices 
, 
, 
, and 
of dimensions 
×
, integer positive seasonality constants 
or integer positive seasonality constant s, and the covariance matrix should be symmetric positive definite of dimensions 
×
. - SARMAProcess can be used with such functions as CovarianceFunction, PDF, Probability, and RandomFunction.
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