model. When two out of the three terms are zeros, the model may be referred to based on the non-zero parameter, dropping "", "" or "" from the acronym describing the model. For example, is , is , and is . ARIMA models can be estimated following the Box–Jenkins approach. Definition Given time series data Xt where t is an integer index and the Xt are real numbers, an model is given by or equivalently by where is the lag operator, the are the parameters of the autoregressive part of the model, the are the parameters of the moving average part and the are error terms. The error terms are generally assumed to be independent, identically distributed variables sampled from a normal distribution with zero mean. Assume now that the polynomial has a unit root (a factor ) of multiplicity d. Then it can be rewritten as: An ARIMA(p,d,q) process expresses this polynomial factorisation property with p=p'−d, and is given by: and thus can be thought as a particular case of an ARMA(p+d,q) process having the autoregressive polynomial with d unit roots. (For this reason, no process that is accurately described by an ARIMA model with d > 0 is wide-sense stationary.) The above can be generalized as follows. This defines an ARIMA(p,d,q) process with drift . Other special forms The explicit identification of the factorization of the autoregression polynomial into factors as above can be extended to other cases, firstly to apply to the moving average polynomial and secondly to include other special factors. For example, having a factor in a model is one way of including a non-stationary seasonality of period s into the model; this factor has the effect of re-expressing the data as changes from s periods ago. Another example is the factor , which includes a (non-stationary) seasonality of period 2. The effect of the first type of factor is to allow each season's value to drift separately over time, whereas with the second type values for adjacent seasons move together. Identification and specification of appropriate factors in an ARIMA model can be an important step in modeling as it can allow a reduction in the overall number of parameters to be estimated while allowing the imposition on the model of types of behavior that logic and experience suggest should be there. Differencing A stationary time series's properties do not depend on the time at