# Fingerprint Classification through Standard and Weighted Extreme Learning Machines
## 5 Results and Discussion
### Estimation of Optimal Hyper-Parameters of the ELMs  
In order to maximize the G-means metric, the hyper-parameters of ELMs to be estimated are the regularization parameter (\(C\)) and the number of neurons in the hidden layer (\(N\)) in the validation set. Remember that the G-means makes sense in unbalanced datasets because it allows the normalization of the accuracies of each class. As mentioned in the Section 4.2, the dataset was divided into three subsets: training, validation, and testing. Figures 6-8 show the validation results using G-mean in terms of these hyper-parameters for the standard ELM, W-ELM1, and W-ELM2, respectively. Aiming to establish overall observations, the regularization parameter varied from \(2^{-12}\) to \(2^{12}\) (very-small and very-large positive numbers based on its definition, see Equation (2)), and the number of hidden nodes gradually increased. All studied feature extractors (Capelli02, Hong08, and Liu10) and datasets (HQNoPert, Default, and VQAndPert) were taking into account. According to each subfigure, an ELM could achieve higher performance for some values of the hyper-parameters that mostly form a continuous and irregular region. For resolution reasons, determining the relationship of \(C\) and \(N\) that maximizes the G-mean was unfeasible. Fortunately, the ELM performance within the maximization zone could be considered as invariable. Note that this brute-force optimization procedure was only feasible in ELM algorithms since parameters of neurons were arbitrarily generated. Instead, the input weights and biases of other learning algorithms required iterative processes and/or high-performance computing methods [15; 17]. Table 2 illustrates the G-means metric with the optimal values of the number of hidden nodes and the regularization parameter for all artificial neural networks. Once again in the study, the results for each dataset and feature extractor were exposed. The best values of \(N\) and \(C\) were determined through the intersection of the best performing areas on the datasets. This procedure was carried out aiming for the optimization of hyper-parameters that did not depend on the fingerprint quality. For all types of ELMs, the highest performance was obtained on the best quality database, resulting in the best feature extractor corresponds to Hong08. In a general sense, it appears that the W-ELM2 (refer to Table 2c) achieved the best classification performance. This result is explained by the fact that studied databases were comprised of unbalanced data (i.e., they did not follow a uniform distribution), and W-ELM2 could effectively classify this kind of data thanks to considering the golden ratio in its matrix of weights (see Equation (5)). On the other hand, as expected, the basic ELM was the worst classifier because it was prone to outlier interference, which naturally occurred for fingerprint datasets (refer to Table 2a). Finally, it can be seen that the adoption of the Hong08 and W-ELM2 as the feature extractor and classifier, respectively, produced the highest G-mean for any fingerprint quality. Figure 6: Validation results obtained by the standard ELM as a function of the regularization parameter and number of hidden neurons. Each subfigure illustrates the database and feature extractor. Figure 7: Validation results obtained by the weighted ELM (W-ELM1) as a function of the regularization parameter and number of hidden neurons. Each subfigure illustrates the database and feature extractor. Figure 8: Validation results obtained by the W-ELM2 as a function of the regularization parameter and number of hidden neurons. Each subfigure illustrates the database and feature extractor. Figure 9 and Table 3 present the previous study for the DW-ELM. This network has three hyper-parameters, the extra degree of freedom comes to be \(d\) (see expression 6), which is related to the weights of the misclassification cost matrix in the ELM algorithm.