Ind. Eng. Chem. Process Des. Dev. 1085, 2 4 , 891
80 1
Response to Comments on “Comparison of Methods for Nonlinear Parameter Estimation”
Sir: Kvaalen’s remarks contain two misconceptions regarding my paper. The first is that the paragraph in question refers to the Deming method. While that method is mentioned, the subject of the paragraph is methods that are similar but not identical with Deming’s. The second and more important is the interpretation of the phrase “linear Constraints”. I agree that my paper is ambiguous on this point. As Kvaalen shows, the various methods can give different parameter estimates for constrainta that would normally be considered linear. However, as stated in my paper and covered in detail by the authors to whom I referred (e.g., Reilly and Patino-Leal, 1981),the maximum likelihood method treats both the measured variables, z, and the parameters, 8, as estimates to be adjusted in the optimization procedure. Therefore, in this case the constraints must be linear with respect to both z and 8, i.e. f(z,e) = o = AZ + Be + c where A, B, and c are constant matrices. Kvaalen’s example does not satisfy this requirement. His example does, however, illustrate the main point of my paper, which was that one should regard the output of any parameter estimation method with healthy skep-
ticism. It is true that computing time should not be overly emphasized, but the fact that the maximum likelihood fit gives a lower sum of squares does not mean that the maximum likelihood fit is “correct” or that the Deming method “does not work”. It is clear from his figure that the (hypothetical) data and/or the model are suspect, as are the assumptions implicit in the maximum likelihood approach. I would not put much faith in the parameter estimates produced by either method in this case. If, on the other hand, the (hypothetical) experimentalist had spent less time on the computer but had obtained more precise data covering a wider range of conditions (and had derived the true theoretical relationship between x and y ) the difference in the parameter estimates would have been insignificant. Literature Cited Rellly, P. M.; Patlndeal, H. Technometrlca 1981, 23(3),221.
Department of Chemical Engineering, BF-10 University of Washington Seattle, Washington 98195
CORRECTION Flash Calculations for Continuous or Semicontinuous Mixtures Using an Equation of State, Ronald L. Cotterman and John M. Prausnitz, Ind. Eng. Chem. F’rocess Des. Dev. 1985, 24, 434. Page 435. Equation 7 should read a(?? p = - -R- T u - b U(U + b)
0196-4305/85/1124-0891$01.50/0
0 1985 American Chemlcal Society
N. Lawrence Ricker
