## 4 Discussion  
The result of simulation 2 that the LMS and MCC with a symmetric Gaussian kernel can deal with asymmetric Gaussian noise well is beyond expectation. However, that could be explained. It is because the criteria selected by the several algorithms in this simulation can well describe the second-order statistics of the error contained in the asymmetric Gaussian distribution.  
Combining the two simulation results in simulation 2 and simulation 3, one can observe that the asymmetric non-Gaussian noise is the real test of the algorithms, while the asymmetric Gaussian noise is not challenging.  
The result in simulation 3 shows the desirable performance of the asymmetric Gaussian kernel, for which the underlying reason is that the asymmetric Gaussian kernel can flexibly select the kernel widths of positive and negative axes, so it can adapt to the asymmetric noise. In simulation 3, the noise is in accordance with the right-skewed distribution, and the distribution in the positive axis has impact characteristics. Therefore, the MACC with a small positive kernel width performs well and can suppress the impact component of the noise in the positive axis.  
The result in simulation 4 shows that the shape parameter of GMACC can improve the adaptability and the robustness of the algorithm. The adjustment of the shape parameter can not only balance the steady-state performance and convergence rate of the GMACC algorithm, but also help to achieve better performance under an asymmetric and non-Gaussian noise. Furthermore, that is the advantage of GMACC.  
According to the results of simulation 3 and simulation 4, the GMACC has an enhanced adaptability because of the flexible selection of the shape and kernel width parameters. The GMACC with a small positive kernel width performs well under a right-skew-distributed system noise while the GMACC with a small negative kernel width is suitable for dealing with a left-skew-distributed system noise. This could be considered as a reference for the selection of the kernel width parameters. According to the stability analysis presented in Section 2.3.1, the step size should be set smaller than a threshold value, which is uncertain before an experiment, as the steady-state error is unknown in advance. Therefore, to make the algorithm converge to the steady state, one can just set a small step size at the beginning of the simulation, and carefully try larger step sizes to get a faster convergence speed if necessary. By running the simulations, we can provide the empirical value of the step size threshold. For example, in simulation 1, the empirical value of step size was less than 2.5. The shape parameter of the GMACC ranged from 1 to 3.5. With the other parameters unchanged, the larger the shape parameter is, the better the GMACC algorithms perform in terms of the steady-state MSD. In contract, larger shape parameters lead to a lower convergence rate. Furthermore, that could be the reference for the selection of the shape parameter. According to our experimental experience, the GMACC algorithm achieves the best performance under an asymmetric non-Gaussian noise when the shape parameter is set to around 2.25.