# Statistics for Environmental Engineers

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The forecasts for nonstationary processes (i.e., MA processes) do not converge to a mean value because there is no long-term mean for a nonstationary process). The forecast from origin t for the useful IMA(0,1,1) model (EMWA forecasts) is a horizontal line projected from the forecast origin. The forecast variance increases as the forecasts are extended farther into the future.

Box et al. (1994) gives details for several process models. It also gives general methods for deriving forecasting weights, similar to the exponential decaying weights of the EWMA. Minitab is a convenient statistical software package for fitting time series models and forecasting.

### References

Box, G. E. P. (1991). “Understanding Exponential Smoothing: A Simple Way to Forecast Sales and Inventory,” Quality Engineering, 3, 561-566.

Box, G. E. P. and L. Luceno (2000). “Six Sigma, Process Drift, Capability Indices, and Feedback Adjustment,” Quality Engineering, 12(3), 297-302.

Box, G. E. P., G. M. Jenkins, and G. C. Reinsel (1994). Time Series Analysis, Forecasting and Control, 3rd ed., Englewood Cliffs, NJ, Prentice-Hall.

### Exercises

53.1 AR(1) Forecasting. Assume the current value of Zt is 165 from a process that has the AR(1) model zt = 0.4zt-1 + a, and mean 162. (a) Make one-step-ahead forecasts for the 10 observations.

 t 121 122 123 124 125 126 127 128 129 130 Zt 2.1 2.8 1.5 1.2 0.4 2.7 1.3 -2.1 0.4 0.9

(b)    Calculate the 50 and 95% confidence intervals for the forecasts in part (a).

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