## 6 Four-Zero Texture Model Validation
### HE-DEPSO Performance in \(\chi^{2}\) Optimization
#### 6.1.2 Experiment Results  
To evaluate the effectiveness of the proposed HE-DEPSO algorithm, comparisons and analysis of its optimization results were performed with the DE, PSO, and three advanced variants of DE algorithms applied to the \(\chi^{2}(X)\) function. Using an approach similar to that described in Section 5, Table 6 presents the statistical data for each algorithm evaluated.  
According to the data presented in Table 6, the HE-DEPSO algorithm excels in obtaining the best results in optimizing the \(\chi^{2}(X)\) function, outperforming DEPSO. Although DEPSO comes close to the results of HE-DEPSO, its accuracy is not matched. On the other hand, the DE algorithm also demonstrated acceptable accuracy, although it was lower than that achieved by HE-DEPSO and DEPSO. In contrast, the SHADE, CoDE, and PSO algorithms obtained the lowest results compared to the others. In terms of average ranking, HE-DEPSO is positioned as the leader among all the analyzed algorithms. Furthermore, the results of the Wilcoxon test confirm the superiority of HE-DEPSO over DEPSO, SHADE, CoDE, DE, and PSO in the optimization of the \(\chi^{2}(X)\) function.  
Figure 5 presents the curves obtained over 31 independent runs by the HE-DEPSO, DEPSO, SHADE, CoDE, DE, and PSO algorithms. It is observed that the HE-DEPSO algorithm exhibits excellent convergence behavior, consistently outperforming SHADE, CoDE, DE, and PSO and achieving a lower mean error value. Although DEPSO converges faster starting from the 300 iteration, the performance of HE-DEPSO stands out for its consistency and exploration capability, which could explain its more extended convergence compared to DEPSO.  
In Figure 6, box and whisker plots show the results of the HE-DEPSO, DEPSO, SHADE, CoDE, DE, and PSO algorithms, based on the best overall fits of 31 independent runs to the chi-square function. These graphs illustrate the distribution of these values into quartiles, with the median highlighted by a red horizontal line within a blue box. The box boundaries correspond to the upper (Q3) and lower (Q1) quartiles. Lines extending from the box indicate the maximum and minimum values, excluding outliers represented by the \"\(+\)\" symbol in blue. The green circles represent the mean of the overall best fits of the 31 independent runs.  
According to the box and whisker plots, the HE-DEPSO algorithm demonstrated superior performance compared with the other evaluated algorithms regarding stability and solution quality, based on 31 independent runs. The distribution of the best global fitness values achieved by HE-DEPSO presents a higher concentration around a lower optimum, reflected in a lower average than the other algorithms. Consequently, the HE-DEPSO offers a more robust and stable performance.  
\begin{table}
\begin{tabular}{l l l l l l l l} \hline \hline
**Function** & **Metrics** & **HE-DEPSO** & **DEPSO** & **SHADE** & **CoDE** & **DE** & **PSO** \\ \hline \multirow{3}{*}{\(\chi^{2}(X)\)} & Mean & **1.7267 \(\times\)10\({}^{-28}\)** & 3.4971 \(\times\)10\({}^{-28}\) & 7.2296 \(\times\)10\({}^{-3}\) & 2.3720\(\times\)10\({}^{-5}\) & 6.1610\(\times\)10\({}^{-13}\) & 7.4247\(\times\)10\({}^{0}\) \\  & Std & **1.4496 \(\times\)10\({}^{-28}\)** & 5.6400 \(\times\)10\({}^{-28}\) & 8.7102\(\times\)10\({}^{-3}\) & 7.7748\(\times\)10\({}^{-5}\) & 3.4294\(\times\)10\({}^{-12}\) & 1.6815 \(\times\)10\({}^{1}\) \\  & Classification & **1** & 2 & 5 & 4 & 3 & 6 \\  & Significance & & + & + & + & + & + \\ \hline Final & \multirow{3}{*}{\(\chi^{2}(X)\) classification} & **1** & 2 & 5 & 4 & 3 & 6 \\ \cline{1-1}  & & \(-\)/\(-\)/\(-\) & 1/0/0 & 1/0/0 & 1/0/0 & 1/0/0 & 1/0/0 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Comparison of results between HE-DEPSO, DE, PSO, and three advanced variants of the DE algorithm on the function \(\chi^{2}(X)\).  
Figure 5: Convergence curves of the error measure in the solution for the \(\chi^{2}(X)\) function. The horizontal and vertical axis represent the iterations and the mean error values for the 31 independent iterations.