Statistics for Environmental Engineers

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FIGURE 35.2 Monod model fitted to the original five points plus two more at higher substrate concentrations (left) and the resulting joint confidence region of the parameters (right).

Substrate Concentration (mg/L)    01

FIGURE 35.3 Monod model fitted to four of the original five points and one additional point at a higher substrate concentration (left) and the resulting joint confidence region of the parameters (right).

The Problem of Parameter Correlation

Parameter correlation means that the estimate of one parameter is related to the estimate of another. Parameter correlation is what causes elongated joint confidence regions. Here we look at the importance of experimental design in reducing parameter correlation.

The location of observations is crucial and making a large number of observations at the wrong locations does not overcome the weakness of a bad experimental design. A great many articles on the effect of temperature, pH, metal concentration, etc. on reactions contain parameters estimated from weak designs (Berthouex and Szewczyk, 1984). The efficiency of aeration equipment was sometimes estimated using experimental designs that could not yield precise parameter estimates (Boyle et al., 1974).

Asymptotic functions, which are common in environmental modeling, present a particular problem in parameter estimation that we will illustrate with the classical first-order model for long-term BOD:

yi = 0j[ 1 — exp(-02ti)] + ei

where yi is the BOD measured at time ti. The ultimate BOD (0j) is approached asymptotically as time goes on. 02 is the first-order reaction rate coefficient. The reaction is slow and t is measured in days. Each observation of y comes from incubating a test specimen for time t and, as a result, the y values and their errors (e) are independent. We further assume that the errors are normally distributed and have constant variance.

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