# Statistics for Environmental Engineers

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Somehow, allowance must be made for the seemingly random drift in influent P load that is unrelated to the ban. Intuition suggests that to minimize the influence of random drift or trends on the estimation, the averages of observations made to estimate the effect of the ban should not stretch too far away from time of the intervention. On the other hand, relying on too few observations around the intervention ban might impair the estimation because there was too little averaging of random fluctuations. It would be appealing if the observations closest to the expected shift were given more weight than observations more distant in time.

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0    12    24    36    48    60    72    84    96    108

Month (Month 1 — January 1975)

FIGURE 54.1 Monthly average mass flow of total phosphorus for the Milwaukee South Shore and Jones Island plants combined.

Before presenting a model that is useful for analysis of many environmental series, a brief discussion is given of two other models that have often been relied upon (explicitly or implicitly) to estimate the effects of change. These are the (1) white noise model and the (2) random walk model. The white noise model is the basis for estimating the change in level when the data series consists of random independent variation about a fixed mean level. The intervention in the fixed-mean-levels case would be estimated by taking the difference of the mean levels before and after the shift. The random walk model is used when the data series consists of pure random walk in which there is no fixed level. In the random walk case, only the observations immediately before and after the intervention would be used to estimate the shift. In the first case, all the data contribute to the estimation; in the second case, almost all the data are disregarded in making the estimation. In both cases, we assume that the intervention shifts the data series upward or downward and without changing the pattern of variation.

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