A Direct-indirect Procedure for Estimation of Kinetic Parameters

The direct integral method (a generalization of the Himmelblau-Jones-Bischoff technique) and the conventional least squares approach are compared in the solution of kinetic estimation problems. Some examples demonstrate that the former method may be preferable in terms of mean square error of the estimates. Recent results on biased estimators are used to explain this fact. A three-stage estimation procedure is presented which preserves the advantages of its stages, thereby combining robustness and numerical efficiency of the direct method with unbiasedness of the indirect estimates. Scope-Methods for estimation of the parameters in dynamic models are either direct or indirect, involving the set of differential equations or its solution, respectively. Although the direct approach is particularly important and well-established in discrete-time process identification, it is at present somewhat overlooked in reaction engineering. Indeed, it yields biased estimates, whereas the indirect (conventional) least squares estimates are unbiased. According to the recent results on biased estimators, however, unbiasedness by no means implies superiority, since biased methods can improve the mean square error of the parameters. In view of these theoretical expectations, statistical properties of the direct integral method are studied by solution of simulation examples. An efficient estimation procedure involving spline-approximation is used to solve the test problems. Empirical properties of the estimates are compared to the expected ones, derived from an eigenvalue analysis. The well-known instrumental variable method of discrete-time process identification is extended to continuous models for removing biasedness of direct estimates, if necessary. The two-step (i.e. indirect least squares and instrumental variable) procedure is extended by the indirect least squares method as its third stage. Advantageous properties of the combined procedure are empirically checked by performing special tests, involving elimination of data points and addition of further “measurement” errors. A simple method is proposed to answer the question whether the indirect or the direct estimates are preferable in the actual problem. Conclusions and Significance-As has been previously shown[l], the direct integral method for estimation of kinetic parameters reduces computer time requirements up to 100 times and yields reasonable, though biased estimates. Our present analysis and examples, however, show that the method usually improves the mean square error and reduces the joint confidence region of the parameters. Thus, in kinetic estimation problems with moderate measurement errors and sufficiently dense sample points, the direct method may be preferable to the indirect (conventional) least squares approach, even from a statistical point of view. Biasedness of estimates induced by large measurement errors and/or sparsely allocated sample points may outweigh the advantages of the direct method, stemming from a more uniform eigenvalue spectrum of the corresponding Hessian matrix. Biasedness, however, can be removed by a continuous extension of the instrumental variable method, well-known in discrete-time process identification. The direct integral least squares (DILS) method, the instrumental variable (IV) method and the indirect least squares (ILS) method are regarded as the different stages of increasing computational complexity of the parameter estimation procedure ROBUST. The combined method preserves the separate advantages of its three stages. The DILS approach is a robust method of obtaining reasonable estimates with low computational effort. In a number of cases these results may be even preferable to the ones obtained in the subsequent, more involved stages. The IV method removes biasedness and minimizes the indirect sum of squares objective function with a reduced number of equivalent function evaluations. Finally, the ILS stage can further reduce the objective function and also furnishes the approximate covariance matrix of the estimates. It is emphasized that the ILS approach is not necessarily superior to the simpler direct methods-as usually assumed in the literature[l]. Therefore, a simple simulation method is proposed to select the best estimates for the true parameters in case of significant deviations between the DILS and the ILS results. INTRODUCIION dxldt = f(x,k), x(O) = x0 , To determine kinetic rate constants from integral rewhere x represents the n-dimensional concentration actor data, one should estimate the parameters of orvector, and k is a p-vector of unknown parameters. dinary differential equations We will consider the case in which all concentrations