1044
Ind. Eng. Chem. Process Des. Dev. iS86, 25, 1044
Literature Cited
Rippin, D. W. T.; Rose, L. M.;Schifferii, C. Chem. Eng. Sci. 1980, 35, 356.
Agarwai, A. K.; Brisk, M. L. Ind. Eng. Chem. Process D e s . Dev. 1985, 2 4 , 203. Federov. V. V. Theorv of O ~ t i ~ l E x ~ e r i t ~Academic: ~ n t s : New York, 1972; p 80. Klaus, R. Doctoral Dlssertation 6866, ETH, Zurich, Switzerland. 1981. Klaus, R.; Rlppin, D. W. T. Comp. Chem. €ng. 1979, 3 , 105. N R Numerical Optimization Software Library, National Physical Laboratory, Teddington. England, 1978.
Technisch-Chemisches Laboratorium E.T.H. Zurich CH-8092 Zurich, Switzerland
T. Rimensberger D. W. T. Rippin*
Received for review July 8, 1985 Revised manuscript received March 3, 1986
Response to Comments on “Sequential Experlmental Design for Precise Parameter Estimation. 1. Use of Reparameterizatlon” Sir: Rimensberger and Rippin correctly point out that the volume of the confidence region of transformed parameters will theoretically differ from that for untransformed parameters by a constant factor. Hence, successive sequentially designed experiments will be placed at the same locations for both “normal” and reparameterized models-in theory. The difficulty of course is that the factor /GI, the determinant of the matrix of partial derivatives of the transformed function with respect to the parameters, will depend on the accuracy with which the best parameter values are estimated. If the parameters are not estimated sufficiently accurately, IGl will not be constant. In our case, the first block of experiments gave the same parameter estimates for each model, as Rimensberger and Rippin found. These estimates were used to design the next experiment, giving identical experimental conditions for each model. These experiments were used (with the first block) to obtain the next estimate of parameters, but now the two sets were not identical. From this point on, the two models diverge. Theory and Rimensberger and Rippin’s evidence indicate otherwise. Why? We believe the explanation is that the search routine (optimizer)used in our case failed to converge to the %rue” values of the parameters in the estimation stage. The reparameterized model produces a better response surface, so the routine was far more likely to approach the best estimates. The non reparameterized model gives rise to the well-known elongated valley surface characteristic of highly correlated parameters. Such a surface will cause problems for some optimizers, and these problems increase
0196-4305/86/1125-1044$01.50/0
as the true values are approached in successive experiments. Whilst it can be argued with hindsight that we should have used a better optimizer, we believe our original claims still have some merit. The use of reparameterization, because it gives one a better chance of success in the parameter estimation stages, leads ultimately to a more effective sequential design. In cases of high experimental noise, or inadequate convergence of the parameter fitting stage, reparameterization does help. It does not and cannot affect the design stages as such, but our empirical evidence is that it provides a more robust approach to the combined design-parameter estimation problem when one does not have total success with optimization routines. We agree that the reparameterized model is not “better” than the original model, per se. Its smaller volume confidence region arises from the scale factor /GI which is approximately constant. Finally, in response to the comment that the values in Table I of our paper %e not exactly consistent with the transformation procedure reported”, we note that the values were calculated using R = 1.987 19 and T (K) = t (“C) + 273.16 and are somewhat sensitive to small variations in these constants. School of Chemical Engineering & Industrial Chemistry University of New South Wales Kensington, N S W 2033, Australia ICI Australia Operations Pty. Ltd. Matraville, NSW 2036, Australia
0 1986 American
Chemical Society
Ani1 K. Agarwal*
Michael L. Brisk
