# Fingerprint Classification through Standard and Weighted Extreme Learning Machines
## 4 Methodology
### Performance Metrics  
To assess the fingerprint classification of the new approach (feature extractors along with ELMs) and the comparative CNNs-based methods [3,53], the geometric mean (G-mean), root mean square error (Acc), and the absolute error of the Penetration Rate (PR) metrics are utilized as evaluation criteria. While values of G-mean and Accuracy near to 1 indicate that the corresponding model has  
Figure 4: Representation of the five-fold cross-validation scheme, validation set is discarded.  
Figure 3: Fingerprint examples of the different studied databases.  
better classification performance, it happens for the PR closest to 0. These metrics are defined as follows [1,10,54]:  
\[\text{G-mean}=\sqrt{\frac{TP}{TP+TN}\times\frac{TN}{TN+FP}}\quad, \tag{7}\]  
\[\text{Acc}=\sqrt{\frac{1}{K}\sum_{i=1}^{K}(t_{i}-\overline{t_{i}})^{2}}\quad, \tag{8}\]  
\[\text{PR}=\mid 0.2948-\sum_{i=1}^{M}p_{i}[1+\text{Acc}_{i}(p_{i}-1)]\mid\quad, \tag{9}\]  
where \(TP\), \(TN\), \(FP\), and \(FN\) in Equation (7) respectively stand for true positive, true negative, false positive, and false negative in a binary classification problem for example purposes. In other words, it can be interpreted as the square root of majority class accuracy times minority class accuracy. For the multi(M)-classification context, the G-mean becomes the \(M\)th root of the product of the accuracies within each class, which are denoted as \(\text{Acc}_{i}\) in the following. In Equation (8), \(K\) denotes the number of fingerprint samples, while \(t_{i}\) and \(\overline{t_{i}}\) denote the real and the prediction values of the fingerprint classification process, respectively. Finally, \(M\) means the number of classes and \(p_{i}\) is the relationship of fingerprint belonging to the \(i\)th class in Equation (9). We adopt the absolute error of the penetration rate (termed as PR in the rest of the manuscript) to easily contrast classifiers. The 0.2948 constant results from the ideal penetration rate by following the natural class distribution of the SFINGE fingerprints [3,5].  
As mentioned, the accuracy calculates the total deviation between the real and estimated values, being the most preferred criteria for assessing the performance of classifiers in the literature [55,56]. Indeed, the overall accuracy can be considered as de facto metric adopted by fingerprint recognition systems, as can be seen in all works of Table 1 as instances. Nevertheless, the G-mean is more suitable than the accuracy for unbalance data classification due to the possible presence of significant class unbalanced (minority samples is more numerous than majority samples by a large margin) [19], as it happens in fingerprint databases (see Figure 1). Namely, the accuracy can be affected by the class distribution and could give misleading results in certain cases. Instead, it does not occur for the G-mean metric (refer to observations provided in Section 5.2 for demonstration purposes). Consequently, we also consider the G-mean as many studies regarding weighted ELMs oriented to regression and classification problems also do [10,20,49,50]. Finally, in the context of fingerprint classification, the PR metric has been recently adopted [1,3] in order to give propitious information regarding the effectiveness of CNNs along with unbalance datasets.  
To summarize, all materials and methods (Section 3 along with Section 4) are depicted in Figure 5 where the fingerprint classification system (the combination of a feature extractor and ELM) is proposed for the first time.