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In those cases where a past history of 50 or more observations are not available, one proceeds by using experience and past information to yield a preliminary model. This model may be updated from time to time as more data become available. In fitting dynamic models, a theoretical analysis can sometimes tell us not only the appropriate form for the model, but may also provide us with good estimates of the numerical values of its parameters. These values can then be checked later by analysis of data.

If the set is continuous, the time series is said to be continuous. If the set is discrete, the time series is said to be discrete. Thus, the observations from a discrete time series made at times τ 1 , τ 2 , . . , τ t , . . , τ N may be denoted by z(τ 1 ), z(τ 2 ), . . , z(τ t ), . . , z(τ N ). In this book we consider only discrete time series where observations are made at a fixed interval h. When we have N successive values of such a series available for analysis, we write z1 , z2 , . .

In practice, d is usually 0, 1, or at most 2, with d = 0 corresponding to stationary behavior. 6) provides a powerful model for describing stationary and nonstationary time series and is called an autoregressive integrated moving average (ARIMA) process, of order (p, d, q). 7) with wt = ∇ d zt . 3). The reason for inclusion of the word “integrated” (which should perhaps more appropriately be “summed”) in the ARIMA title is as follows. 6), is zt = S d wt , where S = ∇ −1 = (1 − B)−1 = 1 + B + B 2 + · · · is the summation operator defined by ∞ Swt = wt−j = wt + wt−1 + wt−2 + · · · j =0 Thus, the general ARIMA process may be generated by summing or “integrating” the stationary ARMA process wt , d times.

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