## 3 Experiments and Results
### Satellite Network Delay Prediction Based on GMACC
The algorithms compared in the simulation included the LMS based on the minimum mean square error criterion, the MCC based on the maximum correntropy criterion, the MACC based on the maximum asymmetric correntropy criterion, and the GMACC based on the generalized maximum asymmetric correntropy criterion. The performance was measured by the mean squared deviation (MSD):  
\[\text{MSD}(\Phi_{k})=\mathbf{E}\Big{\{}\|\Phi_{o}-\Phi_{k}\|^{2}\Big{\}}, \tag{40}\]  
where \(\Phi_{k}\) is the estimated AR weight vector at time \(k\), and \(\Phi_{o}\) is the optimal value of the parameter vector. Figure 6: Queue length at one node. To discuss the performance of the proposed algorithm under a symmetric and asymmetric noise, we first carried out a simulation under a non-Gaussian noise and asymmetric Gaussian noise, then conducted a simulation with an asymmetric non-Gaussian noise. The noise parameters settings of the simulations are shown in Table 2. The parameters selection of simulation 1 and simulation 2 are the same and are all shown in Table 3. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Simulation 1** & **Simulation 2** & **Simulation 3, 4, and 5** \\ \hline Variance & 1 & 1 & 0.25 \\ Shape parameter & 1.98 & 2 & 1.98 \\ Skewness parameter & 0 & 0.4 & 0.3 \\ Scale parameter & 0.2 & 0.2 & 0.2 \\ Location parameter & 0 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of alpha stable noise. Figure 8: Statistical distribution of first-order difference of queue length. Figure 7: The first-order difference of queue length. As can be seen from Figure 11, all algorithms including the LMS show good convergence and steady-state performance when the disturbance term follows an asymmetric Gaussian distribution. The steady-state MSD in simulation 1 and simulation 2 are presented in Table 3, from which one can observe that the asymmetric-cornertropy-criteria-based algorithms deal with asymmetric Gaussian noise and non-Gaussian noise well, and the LMS as well as MCC algorithms also perform well in an asymmetric Gaussian noise. Simulation 3: Next, we simulated the comparison among the MCC, MACC, and LMS when the disturbance term followed an asymmetric non-Gaussian distribution. It has been described that the queue delay is asymmetric due to the sudden increase of user request and the limited processing speed of the network nodes. According to the above characteristics of the queue delay, an asymmetric alpha-stable noise [42] with a lower  
Figure 11: Steady-state performance comparison of different algorithms under asymmetric Gaussian distribution disturbance. Figure 10: Steady-state performance comparison of different algorithms under the perturbation term of a symmetric non-Gaussian distribution close to a Gaussian distribution. bound was generated in this simulation, so as to simulate the disturbance term in the network delay time series. The noise parameters are shown in Table 2. The performance of the MACC and MCC with different kernel widths was compared under the condition of asymmetric noise. For a fair comparison, the initial convergence speed of the algorithms were set to be the same, which is similar to the simulation in [43]. For one algorithm, the faster the initial convergence speed was, the less optimal the convergence step was. The parameters of the algorithms are presented in Table 4. We took the mean MSD of the last 1000 iterations as the steady-state MSD and present the results in Table 4. The convergence process and the steady-state performance results are shown in Figure 12.  
It can be observed from Figure 12 that the asymmetric Gaussian kernel with properly selected kernel widths has better steady-state performance than the symmetric Gaussian kernel when dealing with asymmetric noise, which has a heavy tail component in the positive range and a lower limit in the negative range. However, the asymmetric Gaussian kernel with wrong kernel widths has even worse steady-state performance than the symmetric Gaussian kernel.