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ARIMA Analysis
The ARIMA^{i} model is an extension of the ARMA^{i} model that applies to nonstationary time series (the kind of time series with one or more integrated unitroots).By definition, the autoregressive integrated moving average (ARIMA) process is an ARMA process for the differenced time series:.
Where:
 is the original nonstationary output at time t.
 is the observed differenced (stationary) output at time t.
 is the integration order of the time series.
 is the innovation, shock or error term at time t.
 is the order of the last lagged variables.
 is the order of the last lagged innovation or shock.
 time series observations are independent and identically distributed (i.e. i.i.d^{i}) and follow a Gaussian distribution (i.e. )
notes
 The variance of the shocks is constant or timeinvariant.
 Assuming (i.e. ) is a stationary process with a longrun mean of , then taking the expectation from both sides, we can express as follows:
 Thus, the ARIMA(p,d,q) process can now be expressed as:
 In sum, is the differenced signal after we subtract its longrun average.
 The order of an ARIMA process is solely determined by the order of the last lagged variable with a nonzero coefficient. In principle, you can have fewer number of parameters than the order of the model.

Example: Consider the following ARIMA(12,2) process:
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