## 6 Moduli Estimation with Partial Experimental Data  
In the previous section, it has been demonstrated that the FDGM is an analytical model able to fit the real viscoelastic behavior with the minimum necessary number of parameters. Furthermore, the pole-zero formulation ensures a robust and fast parameter identification process starting from the experimental data of moduli in the frequency domain. Hence, the question arises about the minimum number of experimental data needed to reproduce the materials' behavior in a wide frequency range. To carry out this study, five frequency zones have been identified in the range in which the storage modulus and the loss tangent were defined for compound A, as depicted in Figure 5. Zones 1 and 5 are, respectively, representative of the lower and upper frequencies plateau of the storage modulus. Zone 3 identifies the peak of the loss tangent curve, while zones 2 and 4 describe the curvature changing of both curves.  
Figure 5: Definition of the five frequency zones.  
The aim of the following analysis is to test the capability of the FDGM model in moduli estimation starting from a reduced set of experimental data on which to make the fitting. To pursue this aim, six different combinations of the zones defined in Figure 5 have been considered. For each combination, the parameters of the three-element FDGM have been identified and models results have been compared with the experimental ones in the entire frequency range. In Table 4, we summarize the zone combination and the relative fitting results in terms of NRMSE on the storage modulus and loss tangent while Figure 6 depicts the curves comparisons.  
Adopting all the five zones to identify the parameters of the three-element FDGN model, it is possible to obtain an equally good fitting of the case in which all experimental data are take into account. This condition points out that the five zones chosen represent remarkable points of the curves that unequivocally determine the characteristics of a compound. Combination B results are totally insufficient to describe compound behavior, while combinations D and E fail principally in the fitting of the loss tangent. The latter result was predictable considering that, in these combinations, we did not take into account the information on the peak of the loss tangent. Combination F gives good results in the transition area of the curves, failing in both plateaus at lower and upper frequencies. Satisfying results can be achieved by combination C that, using only the data of the upper and low frequency plateaus plus the ones at the loss tangent peak, is able to fit the global compound behavior.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Case** & **ZONES** & **NRMSE E\({}^{\prime}\) (MPa)** & **NRMSE tan\(\delta\) (-)** \\ \hline Combination A & 1,2,3,4,5 & 0.004 & 0.035 \\ Combination B & 1,5 & 0.595 & 2.395 \\ Combination C & 1,3,5 & 0.004 & 0.066 \\ Combination D & 1,2,5 & 0.085 & 0.1695 \\ Combination E & 1,4,5 & 0.007 & 0.1640 \\ Combination F & 2,3,4 & 0.052 & 0.127 \\ \hline \hline \end{tabular}
\end{table}
Table 4: NRMSE for Fractal derivative Generalized Maxwell models composed by 3, 4 and 5 elements.  
Figure 6: _Cont._