## 8 Simulation Results  
In this section, the V2X traffic dataset was collected and processed after simulating the V2X environment. The dataset was split into 70% training, 15% validation, and 15% testing. A DNN model was built to estimate the performance of the V2X traffic. DNN trained with the robust BP algorithm uses both M-estimators as robust loss functions and the traditional MSE loss function. All the above loss functions try to find the optimal V2X performance in the presence of outliers--comparing the performance of robust and traditional DNNs using RMSE and MAPE for each model. The performance of DNNs was examined for the percentage of outliers in three cases as follows:  
1. Set A: The DNN is trained with noise-free, high-quality, clean data. 2. Set B: The network is trained with perfect data contaminated with slight Gaussian noise (GN): G2-\(N\) (0, 0.1). 3. Set C: DNN is trained with data contaminated with \(GN\), \(G2\)-\(N\) (0,0.1), in addition to very good, randomized outliers of the form: \[H_{1}\
eg N\ (+15,2),\ H_{2}\
eg N\ (-20,3),\ H_{3}\
eg N\ (+30,1.5),\ H_{4}\
eg N\ (-12,3).\]  
Figure 3: Deep neural network architecture. The training data used in this case is as follows:  
\[\text{Data}=(1-\varepsilon)G_{2}+\varepsilon(H_{1}+H_{2}+H_{3}+H_{4})\]  
Data are error distribution, and \(G_{2}\), \(H_{1}\), \(H_{2}\), \(H_{3}\), and \(H_{4}\) are probability distributions occurring on probability \(1-\varepsilon\) and \(\varepsilon\), respectively, where \(\varepsilon\) is fixed and \(H\) random. In this case, the outliers were included in the training data with a percentage \(\varepsilon\) = 10% of the data. We assigned the outliers randomly to the desired percentage of data (percent outliers) (25% of this percentage will have outlier \(H_{1}\) type, the other 25% will have \(H_{2}\) type, and so on \(\ldots\)\(H_{3}\) and \(H_{4}\)). The dataset used for this experiment was generated from the V2X network. The dataset is then contaminated in the \(x\)-\(y\) axis by Gaussian noise with a mean of zero and a standard deviation of 0.1, G2-\(N\) (0, 0.1). A variable percentage, \(\varepsilon\), of data was randomly selected and then replaced with probability, \(\varepsilon\), by background noise uniformly distributed in the specific range [24; 26; 29; 31; 32]. In this work, a DNN architecture is MFNN consisting of three layers with one hidden layer containing 100 hidden neurons. The network was trained using the robust BP algorithm based on the robust loss functions mentioned earlier. The training function is Traincgf, the max epoch for training is 1000 epochs. Moreover, the performance goal (min loss) is \(1\times 10^{-3}\). The normalization of the input data must be in the interval \([-1,1]\), which is compatible with the actual maximum to minimum values. DNN can estimate the optimal performance of a V2X network based on the collected V2X dataset; it used V2X throughput as input and packet loss rate as output (desired output). The goal is to develop a robust DNN that can estimate V2X performance when the data contains outliers. We conducted a comparative study between robust and traditional DNN performances regarding RMSE and MAPE, to prove which algorithm gives excellent results for the considered application. \[RMSE=\sqrt{\frac{1}{n}\sum_{i=1}^{n}(x_{i}-\hat{x}_{i})^{2}} \tag{2}\]  
\[MAPE=\frac{100}{n}\sum_{i=1}^{n}\left|\frac{x_{i}-\hat{x}_{i}}{x_{i}}\right| \tag{3}\]  
where \(n\) is the number of data points, \(x_{i}\) is the observed value, and \(\hat{x}_{i}\) is the estimated value. The subscript i denotes the corresponding individual values of the observed and estimated values. Table 2 shows the performance values  
function, and the maximum average improvement is 0.2% in this case. On the other hand, the Cauchy loss function provides the lowest performance compared to the other functions. Looking at the tabulated results using set B, it is found that the robust Cauchy loss function has the best performance compared to the others in both RMSE and MAPE.