## 4 Proposed Approach
### Threshold \(\mu\)  
An artificial neural network serves as a mathematical model mapping input \(\mathbf{x}\) to output \(\mathbf{z}\), typically represented by a function \(f(W)\), where \(f:\mathbb{R}^{N}\rightarrow\mathbb{R}^{C}\). Here, \(\mathbf{x}\in\mathbb{R}^{N}\) and \(\mathbf{z}\in\mathbb{R}^{C}\) denote the input and output vectors, respectively [39].  
Vector \(\mathbf{z}\) contains the neural network outputs, and its dimensionality corresponds to the number of classes \(C\). In this work, the dimensionality of \(\mathbf{z}\) is two because we classify only two classes: physical violence (\(PV\)) or non-physical violence (\(NPV\)). Each position \(c\) in \(\mathbf{z}_{q}\) is assigned to a particular class; in our case, \(\mathbf{z}_{q}[0]\) corresponds to class \(PV\) and \(\mathbf{z}_{q}[1]\) to the \(NPV\) class. Thus, \(\mathbf{z}_{q}[c]\) is the likelihood of a sample \(\mathbf{x}_{q}\in c\), where \(c=0,1,2,3,...,C\). The classifier assigns the class \(c\) to sample \(\mathbf{x}_{q}\) if \(\mathbf{z}_{q}[c]\) contains the maximum value of \(\mathbf{z}_{q}\)[40].  
In binary classification tasks (\(C=2\)), if (\(\mathbf{z}_{q}[0]<<\mathbf{z}_{q}[1]\)) or (\(\mathbf{z}_{q}[0]>>\mathbf{z}_{q}[1]\)), it is clear the class that the classifier should assign to \(\mathbf{x}_{q}\), but when \(\mathbf{z}_{q}[0]\approx\mathbf{z}_{q}[1]\), the classifier decision could be wrong. When the outputs \(\mathbf{z}_{q}[0]\) and \(\mathbf{z}_{q}[1]\) are significantly skewed towards one class, the classifier's decision is straightforward. However, when these outputs are close, the classifier may struggle to make a definitive decision. This uncertainty motivates the use of a threshold to identify ambiguous samples, where the difference between the outputs is negligible, potentially leading to misclassifications.  
Prior studies have explored the effectiveness of employing thresholds to improve classification accuracy in scenarios where ambiguous samples are prevalent [53; 54]. Determining an appropriate threshold value \(\mu\) involves an iterative process, typically conducted using the initial dataset \(DB\). Through trial and error, a threshold value that effectively distinguishes between unambiguous and ambiguous samples can be set, aiding in the refinement of classification models. This iterative process is described as follows:  
1. The model \(\mathrm{M}_{j}\) is built using \(DB\).
2. \(\mathrm{M}_{j}\) is tested with \(DB\).
3. We choose the outputs from the samples of \(DB\) classified incorrectly by \(\mathrm{M}_{j}\), and we use Equation (4) to find the initial threshold \(\mu_{ini}\). \[\mu_{ini}=\frac{\sum_{q=0}^{Q}(\mathbf{z}_{q}(0)-\mathbf{z}_{q}(1))^{2}}{Q},\] (4) where \(Q=\|DB\|\).
4. Equation (5) obtains the final threshold \(\mu\), in which the initial \(\mu_{ini}\) is increased by a constant \(\vartriangle\) (obtained in a trial-and-error process performed by a human expert), to give a certain level of tolerance to threshold \(\mu\). The process for setting the \(\vartriangle\) value is performed only one time, where \(\vartriangle\) is a small value (\(0<\vartriangle\) and \(\vartriangle<<1\)). Algorithm 4 summarizes the process employed to obtain the best value for the \(\mu\) threshold. \[\mu=\mu_{ini}+\vartriangle\,.\] (5)  
Using a threshold \(\mu\) takes advantage of the neural network output meaning to identify ambiguous images, i.e., those images where the classifier could misclassify. Each neural network output can be seen as the likelihood that a specific image or sample belongs to a determined class [39; 40]. Thus, if the likelihoods of each output are very different between them, then the likelihood of error in the classifier may be reduced, as has been studied in Refs. [53; 54].