# Fingerprint Classification Based on Multilayer Extreme Learning Machines
## 5 Results and Discussion
### Complexity Analysis
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & \begin{tabular}{c} **Hong®8 and Original** \\ **ELM Original** \\ \end{tabular} & \begin{tabular}{c} **Hong08 and ELM-M2** \\ \end{tabular} & \begin{tabular}{c} **Hong08 and ELM-M3** \\ \end{tabular} & \
\begin{tabular}{c} **Zabala-Blanco20 [13]** \\ \end{tabular} &
\begin{tabular}{c} **Peralta18 [10]** \\ \end{tabular} \\ \hline  & **Exac** & \(\mid\)_PR_ \(\mid\) & **Exac** & \(\mid\)_PR_ \(\mid\) & **Exac** & \(\mid\)_PR_ \(\mid\) & **Exac** & \(\mid\)_PR_ \(\mid\) \\ \hline Default & 0.9302 & 0.0523 & 0.9604 & 0.0466 & 0.9634 & 0.0426 & 0.9300 & 0.0388 & 0.9807 & 0.0153 \\ \hline HQNoPert & 0.9512 & 0.0381 & 0.9730 & 0.0378 & 0.9740 & 0.0335 & 0.9400 & 0.0299 & 0.9960 & 0.0031 \\ \hline VQAndPert & 0.9044 & 0.0736 & 0.9418 & 0.0591 & 0.9471 & 0.0575 & 0.8800 & 0.0533 & 0.9640 & 0.0279 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Performance comparison between the best identified methodologies and the other considered approaches. posed model is 0.0036 lower than that reported in [13]. This discrepancy can be attributed to differences in the learning approaches: the current work does not account for unbalanced learning where classes have varying probabilities of occurrence, whereas the approach proposed in [13] does. Furthermore, the optimization criterion in this study is based on raw accuracy, while that proposed in [13] optimizes performance using the geometric mean, considering class-specific averages. When comparing training times, the proposed ELM-M3 demonstrates substantial improvements, underscoring its computational efficiency. This is further validated by the number of hidden nodes in the architectures: the ELM-M3 employs 1250 hidden nodes (\(N1=250\), \(N2=250\), \(N3=750\)), whereas the model proposed in [13] utilizes 5000 hidden nodes. Thus, the reduced complexity of ELM-M3 offers a notable advantage in terms of computational cost. Table 6 presents a comparison of testing times for the best proposed methodology and state-of-the-art approaches. Unfortunately, [PERSON]'s manuscript does not report these times, and replicating his work is challenging due to the use of a high-performance computational architecture. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Hong08 and ELM-M3** & **Zabala-Blanco20 [13]** & **Peralta18 [10]** \\ \hline HQNoPert & 1.5827 & 2.6321 & not reported \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Evaluation of the computational testing times of the best methodologies found in this work and other noteworthy approaches in the literature. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Hong08 and Original ELM** & **Hong08 and ELM-M2** & **Hong08 and ELM-M3** & **Zabala-Blanco20 [13]** & **Peralta18 [10]** \\ \hline Default & 1.0259 (\(\pm\)0.0633) & 3.5320 (\(\pm\)0.2037) & 2.5688 (\(\pm\)0.1205) & 18.7819 (\(\pm\)0.3542) & 957 \\ \hline HQNoPert & 1.0545 (\(\pm\)0.0757) & 3.6884 (\(\pm\)0.1506) & 1.5860 (\(\pm\)0.0757) & 18.8498 (\(\pm\)0.6013) & 960 \\ \hline VQAndPert & 1.0963 (\(\pm\)0.0531) & 3.3228 (\(\pm\)0.1027) & 3.0118 (\(\pm\)0.2172) & 15.5062 (\(\pm\)0.3279) & 960 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Evaluation of the computational times of the training of the best methodologies found in this work and other noteworthy approaches in the literature. Figure 14: Confusion matrices for the ELM-M2 and ELM-M3 models. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Hong08 and ELM-M2** & **Hong08 and ELM-M3** & **Zabala-Blanco20 [13]** & **Peralta18 [10]** \\ \hline Default & 2000 (N1 = 2000) & 2000 (N1 = 1000 + N2 = 1000) & 1500 (N1 = 250 + N2 = 250 + N3 = 1000) & 5000 (N1 = 5000) & 14,875 \\ \hline HQNoPert & 2000 (N1 = 2000) & 2000 (N1 = 1000 + N2 = 1000) & 1250 (N1 = 250 + N2 = 250 + N3 = 750) & 5000 (N1 = 5000) & 14,875 \\ \hline VQAndPert & 2000 (N1 = 2000) & 2000 (N1 = 1000 + N2 = 1000) & 2250 (N1 = 750 + N2 = 750 + N3 = 750) & 5000 (N1 = 5000) & 14,875 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Number of total hidden neurons.