## 3 Results
### Effect of DFs on the PE Problem  
As a first group of tests, we compared the performance of the PE using the analytical anti-log DF and an evolved DF. In the latter case, we automatically evolved DFs composed of up to 5 BFs by means of the two-layered algorithm presented in Section 2.3. The population of the evolutionary algorithm was formed by 10 individuals, and we evolved the population for 11 generations. Each process to determine the best parameterization of the BFs was performed with 10 particles for 10 iterations. The number of sampled points \(I\) to compute \(\widehat{f}\) was determined with the heuristic proposed in Section 2.3, which corresponds to \(I=23\) in the following experiments. The overall budget to evolve the DF with such settings is \(253,000\) fitness evaluations. The actual DFs used here--i.e., the anti-log and the evolved one--are shown in Figure 5. It is worth noting that the optimal DF evolved by our approach for the PE of the MM model, despite being \"stronger\" than the analytical DF, applies a similar dilation to the fitness landscape, i.e., the region corresponding to the lowest orders of magnitude is expanded. To be more specific, the optimal DF obtained for the PE is a composition of three BFs (two out of five BFs were identity functions that can be ignored), as follows: \(l_{0.09}(\mathcal{F}_{\frac{1}{2}}(l_{4.65}(l_{0.02}(x_{d}))))\), for all \(d=1,\ldots,D\). Figure 6 shows the fitness landscapes for the PE of the MM model (restricted to the parameters \(c_{2}\) and \(c_{3}\) in order to represent the 3D landscape) corresponding to the original problem (left), the landscape dilated with the analytical anti-log function (center), and the landscape dilated using the evolved DF (right). The effect of these DFs is to shift away the global optimum from the lowest orders of magnitude. The shift exerted by the evolved function seems to be stronger, revealing a larger region characterized by sub-optimal fitness values (yellow points), even though the intensity also depends on the magnitude of the original parameters. We compared the performance of FST-PSO, executed considering the three aforementioned fitness landscapes to estimate the parameters of the MM model. In each case, we performed 30 independent runs using the default functioning settings that correspond to 14 individuals and 100 iterations. In the first case (i.e., the original fitness landscape), the fuzzy reasoning that governs the \(v_{min}\) of FST-PSO is disabled because it is known to affect the performance in the original formulation of the PE problem [22]. Figure 7 shows the convergence plot (left) and the distributions of the best fitness values found over 30 runs of each methodology (right). These results show that the analytic DF (green dashed line) is the best option for the PE problem: the average best fitness (ABF) is indeed much lower with respect to the original landscape (coral solid line) and slightly lower than in the case of the evolved DF (blue dotted line). Although the best individual in the first iteration of the optimization with the evolved DF is, on average, worse than in the case of the analytical anti-log DF, all runs based on the evolved DF are characterized by a very fast convergence: as a matter of fact, after 20 iterations, the average quality of the identified solutions is comparable (Figure 7, left), as also confirmed by the boxplots (Figure 7, right). The distributions reported by means of the boxplot were compared by leveraging the Mann-Whitney U rank test. The \(p\)-values confirmed that the differences are statistically significant. It is worth noting that the original landscape induced by the PE problem is difficult to explore, even with a low number of dimensions (3, in these  
Figure 5: Graphical representation of the DF used in the PE problem. (**Left**): anti-log function analytically designed to expand the fitness landscape in the lowest orders of magnitude. (**Right**): evolved DF that substantially performs the same dilation of the anti-log DF.