In this experiment, the IGHS is used for Example 1awith N = 5.
In addition, the minimal mean square error (MSE) obtained by the IGHS is equal to 5.4594e-003, which is a satisfactory result.
It is clear from Figure 3 that a satisfactory approximation can be attained by utilizing the IGHS. Additionally, the minimal MSE yielded by the IGHS is equal to 3.7074e - 003 for Example 1a with N = 8, which provides better modeling capacity.
In addition to random signal, another testing input signal x[n] = 0.8cos(([pi]/9)n) is used to investigate the performance of second-order Volterra filter model using the IGHS. For Example 1b with N = 5 and N = 8, the IGHS parameters are the same as those for Example 1a, and Figures 4 and 5 display the comparisons of results for Example 1b with N = 5 and N = 8.
Moreover, the IGHS parameters used for Example 2 are the same as those of Example 1.
Table 4 gives the comparison of the results obtained by the IGHS, against the other three methods including the HS , the IHS , and the NGHS , and the best performance is reported in boldface.
Additionally, Mann-Whitney U test [27, 28], also known as "Mann-Whitney Wilcoxon test," is used to ensure a statistical significant difference between the IGHS and any of the other three HSs.
In order to compare the IGHS with the other three HSs in a statistical way, three groups of Mann-Whitney U tests are executed, and they are ([U.sub.HS], [U.sub.IGHS]), ([U.sub.IHS], [U.sub.IGHS]), and ([U.sub.NGHS], [U.sub.IGHS]), respectively.
From Table 5, it is evident that the IGHS completely dominates the HS for solving all problems, because the values of [U.sub.HS] are all equal to 400.
In addition, both the IHS and the NGHS converge faster than the HS but slower than the IGHS. Obviously, the IGHS has the fastest convergence rate in each case.