Monte Carlo Simulation Results and Discussion

Case I: Results when informative parameter initial guesses are used

In this Case, 100 initial guesses for the seven parameters were selected as described in the first row of Table 8. Using each set of initial guesses and the simulated old data in Figure 2, preliminary values of the model parameters were estimated using both Bayesian and LO approaches. All seven parameters were estimated using the Bayesian approach, whereas subsets of parameters were estimated using the LO approach, with remaining parameters fixed at their initial guesses. Using the LO approach, parameter \(k_{2}\) was always ranked as the most-estimable parameter, followed by \(K_{\text{eq}}\), \(k_{1f}\) and\(k_{3}\) (using the ranking algorithm in Table 2). Parameters\(k_{4}\), \(k_{5}\) and \(k_{6}\) were always left out of the ranked list. Using Wu’s \(r_{\text{cc}}\) criterion, the parameter subset\(\mathbf{\theta}_{\mathbf{\text{sub}}}=\left[k_{2},\ K_{\text{eq}}\right]^{T}\)was selected for estimation in all 100 simulated old data sets. Parameters\(k_{1f}\), \(k_{3}\), \(k_{4}\), \(k_{5}\) and \(k_{6}\) were fixed at their initial values.
The preliminary parameter estimates obtained via Bayesian and LO estimation were then used to design sequential A-optimal experiments using Bayesian and LO approaches. Details concerning how many and which parameters tended to be estimated after each stage of sequential experimentation are provided in the Supplementary Information.
Figure 3 provides boxplots for 100 values of the scaled sum of squared deviations between the estimated and true parameter values:
\(\text{SS}D_{\theta}=\left(\ \hat{\mathbf{\theta}}-\mathbf{\theta}^{\text{true}}\right)^{T}\mathbf{\Sigma}_{\mathbf{0}}^{\mathbf{-1}}\left(\ \hat{\mathbf{\theta}}-\mathbf{\theta}^{\text{true}}\right)\)(19)
for all four approaches when selecting three new A-optimal experiments, one at a time. The Bayes-LO approach is the superior approach on average, resulting in the smallest mean and median for\(\text{SS}D_{\theta}\) after each round of experimentation. The results in Figure 3 indicate that designing experiments using the proposed modified Bayesian approach (i.e., with equation (16) as the objective function) is superior to designing experiments using the LO approach (i.e., the Bayes-Bayes results are better than LO-Bayes, and the Bayes-LO results are better than LO-LO). In addition, parameter estimation using the LO approach is superior to the Bayesian approach (i.e., Bayes-LO is better than Bayes-Bayes, and LO-LO is better than LO-Bayes).
Figure 4 shows boxplots for 100 values of \(\text{SS}D_{\theta}\) when designing three new A-optimal experiments all at once for Case I. Bayes-LO is the superior approach and LO-Bayes is the worst approach. Comparing these results with the results in Figure 3, it can be concluded that designing experiments one-at-a-time resulted in better final parameter values than designing all three new experiments at once.