# Statistics for Environmental Engineers

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An interesting variation on nonlinear least squares is fitting kinetic models that cannot be solved explicitly for the measured response. For example, the change in substrate concentration for Michaelis-Menten biokinetics in a batch reactor is Kmln(Sf)(S0 — S) = vmt. There is no analytic solution for S, so the predicted value of S must be calculated iteratively and then compared with the observed value. Goudar and Devlin (2001) discuss this problem.

### References

Berthouex, P. M. and J. E. Szewczyk (1984). “Discussion of Influence of Toxic Metals on the Repression of Carbonaceous Oxygen Demand,” Water Res., 18, 385-386.

Boyle, W. C., P. M. Berthouex, and T. C. Rooney (1974). “Pitfalls in Parameter Estimation for Oxygen Transfer Data,” J. Envir. Engr. Div., ASCE, 100, 391-408.

Draper, N. R. and H. Smith (1998). Applied Regression Analysis, 3rd ed., New York, John Wiley.

Goudar, C. T. and J. F. Devlin (2001), “Nonlinear Estimation of Microbial and Enzyme Kinetic Parameters from Progress Curve Data,” Water Envir. Res., 73, 260-265.

### Exercises

35.1 Soybean Oil. The data below are from a long-term BOD test on soybean oil, where y = g BOD per g of soybean oil, and t = time in days. There are four replicates at each time. (a) Fit the first-order BOD model y = в1[1 — exp(—02t)] to estimate the ultimate BOD 61 and the rate coefficient, 62. (b) Plot the data and the fitted model. (c) Plot the residuals as a function of the predicted values and as a function of time. (d) Estimate a
11 from s11 = SR/(n — 2) and from the replicate observations. Compare the two values obtained. (e) Map the sum of squares surface and indicate the approximate 95% joint confidence region.

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