## 9 Experimental Assessment and Analysis
### Experiment 1: Analysis of the Transitions Generated by the Fuzzy-Genetic Model
\[\epsilon_{F1}^{BuGT}=\frac{\sum_{n=0}^{N}\left(F_{1}^{Pr}(n)-F_{1}^{GT}(n) \right)^{2}}{N} \tag{4}\]  
\[\epsilon_{F2}^{BuGT}=\frac{\sum_{n=0}^{N}\left(F_{2}^{Pr}(n)-F_{2}^{GT}(n) \right)^{2}}{N} \tag{5}\]  
\[\epsilon_{F3}^{BuGT}=\frac{\sum_{n=0}^{N}\left(F_{3}^{Pr}(n)-F_{3}^{GT}(n) \right)^{2}}{N}, \tag{6}\]  
where \(Pr\) stands for \(Praat\) and \(GT\) for \(GroundTruth\), \(N\) is the length of the alignement transitions, which is the number of tested frequency values, \(F_{1}^{Pr}\), \(F_{2}^{Pr}\), \(F_{3}^{Pr}\) are the thirst, second and third formant obtained by \(Praat\) and \(F_{1}^{GT}\), \(F_{2}^{GT}\), \(F_{3}^{GT}\) are the thirst, second and third formant of the \(GroundTruth\). These three errors describe the mean squared deviations of the formant trajectories computed with the \(Burg\) algorithm from the \(GroundTruth\). Likewise, other three error measures derive from the comparison of the formant trajectories obtained with the proposed \(Fuzzy-Genetic\) algorithm from the \(GroundTruth\). \[\epsilon_{F1}^{Fg-GT}=\frac{\sum_{n=0}^{N}\left(F_{1}^{Fg}(n)-F_{1}^{GT}(n) \right)^{2}}{N} \tag{7}\]\[e_{F2}^{Fg-GT} =\frac{\sum_{n=0}^{N}\left(F_{2}^{Fg}(n)-F_{2}^{GT}(n)\right)^{2}}{N} \tag{8}\] \[\epsilon_{F3}^{Fg-GT} =\frac{\sum_{n=0}^{N}\left(F_{3}^{Fg}(n)-F_{3}^{GT}(n)\right)^{2}} {N}, \tag{9}\]  
where \(Fg\) stands for \(FuzzyGenetic\) and \(GT\) for \(GroundTruth\). Each set of three errors must be then combined to obtain the final Mean Squared Errors with weighted sums:  
\[MSE^{BuGT}=\sqrt{w_{11}\epsilon_{F1}+w_{21}\epsilon_{F2}+w_{31}\epsilon_{F3}} \tag{10}\]  
\[MSE^{Fg-GT}=\sqrt{w_{12}\epsilon_{F1}+w_{22}\epsilon_{F2}+w_{32}\epsilon_{F3}}. \tag{11}\]  
In (10) and (11) the indexes \((. )^{BuGT}\) and \((. )^{Fg-GT}\), are omitted for simplicity. Of course it should be clear that the \(\epsilon^{BuGT}\) must be used for computing \(MSE^{BuGT}\) and the \(\epsilon^{Fg-GT}\) must be used for computing \(MSE^{Fg-GT}\). The weights \(w_{11},w_{21},w_{12},w_{22}\) are computed such that the variances of the weighted sums is minimized [65]. More precisely, calling \(\sigma_{F1}^{2}\) the variance of \(\epsilon_{F1}^{BuGT}\), \(\sigma_{F2}^{2}\) the variance of \(\epsilon_{F2}^{BuGT}\) and \(\sigma_{F3}^{2}\) the variance of \(\epsilon_{F1}^{BuGT}\) (note that we omit again the indexes \((. )^{BuGT}\) for simplicity), and under the assumption of statistical independence among the three errors, the variance of \(MSE^{BuGT}\) is minimized with  
\[w_{11}=\frac{\sigma_{F1}^{-2}}{\sum_{n=1}^{3}\sigma_{Fn}^{-2}},\]  
\[w_{21}=\frac{\sigma_{F2}^{-2}}{\sum_{n=1}^{3}\sigma_{Fn}^{-2}}\]  
and  
\[\epsilon_{F2}^{BuGT}=\frac{\sum_{n=0}^{N}\left(F_{2}^{Pr}(n)-F_{2}^{GT}(n) \right)^{2}}{N} \tag{5}\]0  
The same considerations apply to the minum variance of \(MSE^{Fg-GT}\). The goal of this comparison is to measure the deviation of \(FuzzyGenetic\) algorithm from the \(Burg\) algorithm. This comparison is reported in \[\epsilon_{F2}^{BuGT}=\frac{\sum_{n=0}^{N}\left(F_{2}^{Pr}(n)-F_{2}^{GT}(n) \right)^{2}}{N} \tag{5}\]1 In this table, the pair of numbers corresponding to each transition represent the average deviation of \(FuzzyGenetic\) algorithm from the \(Burg\) algorithm respectively. From Table 1 we see the \(Fuzzy-Genetic\) algorithm give costantly very similar results to the \(GroundTruth\). \begin{table}
\begin{tabular}{c c c c c c c c} \hline  & **VO** & **PL** & **FR** & **AF** & **LI** & **NA** & **Sil** \\ \hline  & **FG Bu** & **FG Bu** & **FG Bu** & **FG Bu** & **FG Bu** & **FG Bu** & **FG Bu** \\ \hline VO & 630 844 & 504 610 & 432 645 & 605 505 & 744 634 & 560 450 & 334 234 \\ \hline PL & 217 196 & & 8888 655 & 1032 670 & 976 546 & & \\ \hline FR & 249 150 & 945 754 & & & & & \\ \hline AF & 855 745 & & & & & & \\ \hline LI & 640 780 & 1020 569 & 923 866 & & & \\ \hline NA & 670 567 & 1200 754 & 234 378 & & & \\ \hline Sil & 155 96 & 167 223 & & 389 222 & \\ \hline \end{tabular}
\end{table}