Statistics for Environmental Engineers

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How Effectively was the Optimum Located?

Let us see how efficient the method was in this case. Figure 43.5a shows the contour plot from which the experimental data were obtained. This plot was constructed by interpolating the Hobson-Mills data with a simple contour plotting routine; no equations were fitted to the data to generate the surface. The location of their 14 runs is shown in Figure 43.5, which also shows the three-dimensional response surface from two points of view.

An experiment was run by interpolating a value of R from the Figure 43.5a contour map and adding to it an “experimental error.” Although the first 22 design was not very close to the peak, the maximum was located with a total of only 13 experimental runs (4 in iteration 1, 4 in iteration 2, plus 5 in iteration 3). The predicted optimum is very close to the peak of the contour map from which the data were taken. Furthermore, the region of interest near the optimum is nicely approximated by the contours derived from the fitted model, as can be seen by comparing Figures 43.4 and Figure 43.5.

Hobson and Mills made 14 observations covering an area of roughly C = 0.5 to 1.5 mg/L and D = 0.125 to 0.205 h-1. Their model predicted an optimum at about D = 0.15 h-1 and C = 1.1 g/L, whereas the largest removal rate they observed was at D = 0.178 h 1 and C = 1.37 g/L. Their model optimum differs from experimental observation because they tried to describe the entire experimental region using a quadratic model that could not describe the entire experimental region (i.e., all their data). A quadratic model gives a poor fit and a poor estimate of the optimum’s location because it is not adequate to describe the irregular response surface. Observations that are far from the optimum can be useful in pointing us in a profitable direction, but they may provide little information about the location or value of the maximum. Such observations can be omitted when the region near the optimum is modeled.

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