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.