## 3 Results
The highest reduction in correlated feature removal is exhibited by the LR classifier, with 2.31%, whereas the CART classifier has the lowest reduction of 0.01% in the \(PD\). The GNB classifier has a 0.09% lower \(PD\) after removing the correlated features. Other models, ANN and RF, also slightly have reduced \(PD\) by 0.5% and 0.27%, respectively. The classifiers have a higher \(Pmd\) after removing correlations. The highest \(Pmd\) with correlations is exhibited by the LR classifier, at 3.1%, while the CART classifier has slightly higher results after removing correlations, with a 0.2% lower \(Pmd\). GNB and AN, have an approximately 2% higher \(Pmd\) without correlations, while ANN and RF have an increase of Pmd by 1% after removing correlations. Removing correlations also impacts the probability of false alarm. The SVM classifier has the highest reduction in \(P\!\!/\)\(\!a\), at 1.3%, after discarding the correlated features. The CART classifier has a higher \(P\!\!/\)\(\!a\), 0.02%, after removing correlated features. The RF classifier has a slightly higher \(P\!\!/\)\(\!a\) after removing the correlated features of 0.01%. Other models, such as GNB, have an approximately 1% higher \(P\!\!/\)\(\!a\) without correlations. Using a dataset with  
Figure 8: Impact of class imbalance on model performance. correlated features increases model _ACC_ and _PD_ and decreases the _Pfa_ and _Pmd_. Eliminating correlated features will ensure more optimal learning and accurate performance for the final ML models. Figure 7 illustrates the impact of dataset size on model performance for 1000, 2000, 4000, 6000, 8000, 10,000, 12,000, and 14,000 samples. All created datasets were balanced, that is, we used equal numbers of both spoofed and normal GPS signal samples from the original dataset. This investigation evaluates two scenarios: one with correlated features and the other after removing the correlated features. Figure 7A illustrates the performance of the six models with correlations in terms of the four metrics. As it can be observed, all the models exhibit an increase in ACC and PD and a decrease in the probabilities of false alarm and misdetection ranging from 2% to 10% as the dataset size increases. Figure 7B illustrates the results without correlation. Like in the previous case (Figure 7A), the accuracy and probability of detection increase as the dataset size increases for all models, while the probabilities of false alarm and misdetection decrease. Therefore, with and without correlations, the performance of machine learning models improves when more training data are used. We also evaluated the effect of dataset imbalance on ML model performance by examining different percentages of normal and malicious data samples in the dataset size of 14,000 samples (without correlation) by using the following ratios: 10% malicious signals to 90% normal signals, 20% malicious signals to 80% normal signals, 30% malicious signals to 70% normal signals, and 40% malicious signals to 60% normal signals. The results of these four scenarios were compared with those of the original balanced dataset (50% normal samples and 50% malicious samples). Figure 8 presents the performance of the six models for different percentages of malicious signals. As it can be seen, the accuracy and probability increase with the rate of malicious signals and reach their best values at a rate of 50%. In addition. the probability of misdetection and false alarm decrease as functions of malicious signals rate and reach their best values at 50% rate. Therefore, an imbalance dataset causes degradation in classifiers' performance. In summary, we can conclude that:  
* Tuned hyperparameters and regularized learning parameters improve models' performance in terms of the selected metrics. * The presence of correlated features in a dataset degrades models' performance. Identifying and removing redundant features from the dataset improves significantly this performance. * Dataset size plays an important role in models' performance which increases as the size increases.