## 4 Methods
### Connected and Automated Vehicles (CAVs) Data Characteristics and Anomaly Model
To make them smoother and accelerate the convergence of the loss function during the training process, the input data is denoised. Table 1 demonstrates the anomalous and denoised plots of BSMs attributes. From the simulation carried out at \(d=3\) in various network sizes, \(m\) in Table 1, indicates finer coefficients of the noise measurement with decrease in standard deviation as indicated in Table 1,for instant anomaly case. Essentially, in Table 1, we equally conduct analysis on Bias anomalous BSMs, we observed a smoother noise coefficient with DWT. The results in Table 1 of the different simulation carried out on different network size \(m\), with \(d=3\), presents a noise amplitude value reduction as shown in Table 1. A similar simulation is carried out on Gradual Drift anomaly on the BSMs as shown in Table 1, the result shows that the magnitude of the noise level of Gradual Drift anomaly is smaller as compared with Instant and Bias anomaly. We note that the DWT, in which in this context, Daubechies 12 (dB12), consistently maintained a finer coefficient that shows closer representation of the original feature of BSMs. Lastly, analysis of the noise distribution on the mixed anomaly data is carried out on network size \(m=10{,}000\), and \(d=7\). From the observations in Figure 6, we find out a significant decrease in the amplitude of the noise distributions of the three BSMs attributes selected in this paper, namely Speed, \(Ax\) and \(RoC\). The values of the noise amplitude decreases by \(\sigma=0.731\) for Figure 6a, \(\sigma=0.437\) of Figure 6b and \(\sigma=335{. }858\) for Figure 6c, respectively. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Instant Anomaly (Network Size \(m\))** & \(\boldsymbol{\mu}\) **(Anomaly)** & \(\boldsymbol{\sigma}\) **(Anomaly)** & \(\boldsymbol{\mu}\) **(Wavelet dB(12))** & \(\boldsymbol{\sigma}\) **(Wavelet dB(12))** \\ \hline
2000 & 10.307401 & 6.170231 & 10.300251 & 5.7634621 \\
4000 & 10.307407 & 6.170261 & 10.300258 & 5.763563 \\
6000 & 10.307414 & 6.170258 & 10.300262 & 5.7635907 \\
8000 & 10.307417 & 6.1702566 & 10.300263 & 5.7635937 \\
10,000 & 10.307415 & 6.170257 & 10.300264 & 5.7635903 \\ \hline
**Bias Anomaly (Network Size \(m\))** & \(\boldsymbol{\mu}\) **(Anomaly)** & \(\boldsymbol{\sigma}\) **(Anomaly)** & \(\boldsymbol{\mu}\) **(Wavelet dB(12))** & \(\boldsymbol{\sigma}\) **(Wavelet dB(12))** \\ \hline
2000 & 0.5829332 & 1.5610949 & 0.5813745 & 1.1946311 \\
4000 & 0.53638434 & 1.581967 & 0.53705674 & 1.0952997 \\
6000 & 0.6300388 & 1.7301117 & 0.6321792 & 1.0724595 \\
8000 & 0.64839834 & 1.6975222 & 0.65079004 & 1.0431184 \\
10,000 & 0.7417457 & 1.7784712 & 0.74250895 & 1.0478663 \\ \hline
**Gradual Drift Anomaly (Network Size \(m\))** & \(\boldsymbol{\mu}\) **(Anomaly)** & \(\boldsymbol{\sigma}\) **(Anomaly)** & \(\boldsymbol{\mu}\) **(Wavelet dB(12))** & \(\boldsymbol{\sigma}\) **(Wavelet dB(12))** \\ \hline
2000 & 0.07693122 & 0.910007 & 0.076246604 & 0.4898912 \\
4000 & 0.07699456 & 0.9098382 & 0.0763265 & 0.49061427 \\
6000 & 0.07699457 & 0.908578 & 0.07632571 & 0.49055964 \\
8000 & 0.07699094 & 0.9098535 & 0.07632209 & 0.49055964 \\
10,000 & 0.07699085 & 0.9098535 & 0.07632202 & 0.49055642 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Selected basic safety messages (BSMs) variables Descriptive Statistics. ### Bayesian Deep Learning (BDL)

Imposing of BDL framework is required to overcome the challenges in a NN. BDL incorporates as illustrated in Figure 7, the transformation of NN from point to probabilistic estimation is achieved by first establishing series of functional transformation in different correlated layers.