# Directional Support Vector Machines
## 9 Experiments  
In this section, we detail the experimental evaluation of the proposed directional support vector machines against two state-of-the-art directional classifiers: the von Mises naive Bayes [8] and the directional logistic regression [4]. Following [4], the \(\kappa\) parameter of the von Mises distribution was approximated by 100 iterations (a much larger number of iterations than required to have good convergence values) of [PERSON]'s method proposed by [PERSON] [9]. The SVM regularization constant \(C=\lambda^{-1}\) was chosen using a stratified 3-fold cross-validation strategy. The range of explored values was \(10^{-3},\ldots,10^{3}\). The concentration parameter \(\kappa\) of the directional RBF kernel was also selected through 3-fold cross-validation in the range \(10^{-3},\ldots,10^{1}\). The primal directional SVM with fixed margin was randomly initialized and optimized using Adam [10] for 500 iterations. On the other hand, we initialized the asymmetric primal SVM with the fixed margin marginparameters after 400 iterations. Then, all the parameters were fine-tuned for an additional 100 iterations. Using pre-trained parameters for the SVM with an asymmetric margin facilitates convergence given the coupled effect of the \(\theta\) and \(\zeta\) parameters. The kernelized directional version with cosine and triangular kernels was optimized using the standard libsvm [11], which implements an SMO-type algorithm proposed in [12]. For the asymmetric kernel, we used the aforementioned fine-tuning approach in order to fine-tune the \(\alpha\) coefficients obtained by the standard toolkit. We validated the advantages of the proposed approach using 12 publicly-available datasets: Arrhythmia [13], Behavior [14], Characters [15], Colposcopy, Continents, eBay [16], MAGIC [17], Megaspores [18], OnlineNews [19], Temperature1, Temperature2, and Wall [20]. Relevant properties about these datasets (e.g., number of directional and non-directional features, number of classes, dimensionality) are presented in Appendix B. Experiments in previous works [4; 8] have shown that directional classifiers outperform traditional ones in these datasets, proving that directionality is an important attribute to exploit. Further details about the datasets, including their acquisition and preprocessing, were presented in [4]. Additionally, in order to facilitate the convergence of the SVM-based models, all the non-directional features were scaled to the range 0-1. Multiclass instances were handled using a one-versus-one approach for all the binary models (i.e., logistic regression and support vector machines). All the experiments detailed below were executed with a 3-fold stratified cross-validation technique (i.e., by preserving the percentage of samples for each class), selecting the best model in terms of accuracy, and the results of 30 different runs were averaged. Specifically, for each model and dataset, we have evaluated the accuracy and the macro F\({}_{1}\)-score, which corresponds to the unweighted mean value of the individual F\({}_{1}\)-scores of each class. Results of these experiments are summarized in Tables 1 and 2, exhibiting average accuracy and macro F\({}_{1}\)-score, respectively, for 30 independent runs. The best results for each dataset are marked in bold. For reproducibility purposes, the source code and the training-testing partitions are made available ([[https://github.com/dpernes/dirsvm](https://github.com/dpernes/dirsvm)]([https://github.com/dpernes/dirsvm](https://github.com/dpernes/dirsvm))). The results achieved by the von-Mises naive Bayes (vMNB) and directional Logistic Regression (dLR) align with the results reported in the literature [4].