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ARIMA Definition
An autoregressive integrated moving average model can be defined as a generalization of an auto-regressive moving average model (ARMA) as these models are fitted to time series data. This is to better understand the data or to predict future points in the series (forecasting) and they are applied in some cases where data show evidence of non-stationarity. Then an initial differencing step corresponding to the "integrated" part of the models can be applied to remove the non-stationarity.
The model is generally referred to as an ARIMA(p,d,q) models where p and d as well as q are non-negative integers that refer to the order of the autoregressive and integrated as well as moving average parts of the models respectively. Autoregressive integrated moving average models form a vital part of the Box-Jenkins approach to the time-series modeling.
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When one of the three terms is zero, it is usual to drop "AR" , "I" or " MA" and for example, ARIMA(0,1,0) is I(1) , and ARIMA(0,0,1) is MA(1).
Given a time series of data that is where is an integer index and the are real numbers, then an ARMA(p' ,q) model is given as follows :
where is the lag operator and are the parameters of the auto-regressive part of the model and the are the parameters of the moving average part and the are described as the error terms. The error terms are generally assumed to be independent as well as identically-distributed random variables. These variables are sampled from a normal distribution with zero mean.
Imagine that the polynomial has a unitary root of multiplicity d and so this can be rewritten as follows :
An ARIMA(p,d,q) process expresses the polynomial factorization property with p=p'−d , and this is given as follows as
and thus can be thought as a particular case of an ARMA(p+d,q) process having the auto-regressive polynomial with d unit roots and for this reason, every Autoregressive integrated moving average model with d>0 is not the wide sense stationary and the above can be generalized as follows:
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The above generalization can define an ARIMA(p,d,q) process with drift δ/(1−Σφi). A number of variations on the Autoregressive integrated moving average models are commonly employed and sometimes a seasonal effect is suspected in the model and in that case, it is generally better to use a SARIMA
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