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SARIMA Analysis
The SARIMA^{i} model is an extension of the ARIMA model, often used when we suspect a model may have a seasonal effect.
By definition, the seasonal autoregressive integrated moving average  SARIMA(p,d,q)(P,D,Q)s  process is a multiplicative of two ARMA^{i} processes of the
differenced time series.
Where:
 is the original nonstationary output at time t.
 is the differenced (stationary) output at time t.
 is the nonseasonal integration order of the time series.
 is the order of the nonseasonal AR component.
 is the order of the seasonal AR component.
 is the order of the nonseasonal MA component.
 is the order of the seasonal MA component.
 is the seasonal length.
 is the seasonal integration order of the time series.
 is the innovation, shock or the error term at time t.
 time series observations are independent and identically distributed (i.e. i.i.d^{i}) and follow a Gaussian distribution (i.e. )
Assuming follows a stationary process with a longrun mean of , then taking the expectation from both sides, we can express as follows:
Thus, the SARIMA(p,d,q)(P,D,Q)s process can now be expressed as:
In sum, is the differenced signal after we subtract its longrun average.
Notes: The order of the seasonal or nonseasonal AR (or MA) component is solely determined by the order of the last lagged variable with a nonzero coefficient. In principle, you can have fewer parameters than the order of the component.
notes
 The variance of the shocks is constant or timeinvariant.
 The order of the seasonal or nonseasonal AR (or MA) component is solely determined by the order of the last lagged variable with a nonzero coefficient. In principle, you can have fewer parameters than the order of the component.

Example: Consider the following SARIMA(0,1,1)(0,1,1)12 process:
Note: This is the AIRLINE model, a special case of the SARIMA model.
Files Examples
References
 Hamilton, J .D.; Time Series Analysis , Princeton University Press (1994), ISBN 0691042896
 Tsay, Ruey S.; Analysis of Financial Time Series John Wiley & SONS. (2005), ISBN 0471690740