# Developing a Generalized Regression Forecasting Network for the Prediction of Human Body Dimensions
## 4 Experimental Results and Discussion
### Comparison of PSO Variant Algorithms for Benchmark Functions
#### 4.2.3 Analysis using Friedman Statistical Test  
To verify the statistical differences of the full algorithm, a nonparametric Friedman test and a Nemenyi test were used [50, 51]. Table 6 presents the Friedman test results for the eight algorithms using a significance level of \(\alpha=5\%\). Prior to the analysis, an average ranking of the algorithm was performed so that \(r_{n}^{m}\) would represent the average ranking of the \(m\)-th algorithm with the \(n\)-th benchmark function, which is expressed as follows:  
\[R_{m}=\frac{1}{N_{K}}{\sum_{n=1}^{N_{K}}{r_{n}^{m}}}, \tag{46}\]  
where \(N_{K}=25\) represents the number of benchmark functions. The results demonstrate that the MMPSO algorithm ranks the highest on the benchmark test set and significantly outperforms other algorithms with unimodal, multimodal, and hybrid composition functions. In this experiment, there were 25 benchmark functions, eight algorithms, and seven degrees of freedom. The square value of the 30-dimensional Friedman test card was 47.160, and the \(p\)-value was less than 0.05. According to the critical table of the chi-square distribution, the critical value was 14.07, and the actual chi-square value was larger than the critical value, indicating that there were obvious differences in the overall performance of each algorithm.  
The Nemenyi rank sum test was used to achieve pairwise comparisons among multiple samples. Figure 4 presents the final rank in 30 dimensions, visually indicating the Friedman statistical test's critical difference (\(CD\)). If the discrepancy between the mean rank of the two  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Final Rank**} & \multirow{2}{*}{**Algorithm**} & **Avg** & \multicolumn{2}{c}{**UN Functions**} & \multicolumn{2}{c}{**MN, EF Functions**} & \multicolumn{2}{c}{**CF Functions**} \\ \cline{3-8}  & & **Rank** & **Algorithm** & **Rank** & **Algorithm Rank** & **Algorithm** & **Rank** \\ \hline
1 & MMPSO & 2.2 & MMPSO & 2.0 & MMPSO & 2.4 & MMPSO & 2.1 \\
2 & HCLDMSPSO & 3.2 & EPSO & 3.2 & HCLDMSPSO & 3.2 & HCLDMSPSO & 2.6 \\
3 & EPSO & 4.2 & HIDMSPSO & 3.6 & HCLPSO & 4.0 & HCLPSO & 4.1 \\
4 & HCLPSO & 4.4 & HCLDMSPSO & 4.4 & HIDMSPSO & 4.7 & EPSO & 4.2 \\
5 & HIDMSPSO & 4.5 & FDRPSO & 4.8 & EPSO & 4.9 & HIDMSPSO & 4.8 \\
6 & DMSPSO & 5.2 & DMSPSO & 5.2 & DMSPSO & 5.6 & DMSPSO & 5.0 \\
7 & FDRPSO & 6.0 & HCLPSO & 5.6 & FDRPSO & 5.6 & CLPSO & 6.3 \\
8 & CLPSO & 6.2 & CLPSO & 7.2 & CLPSO & 5.7 & FDRPSO & 6.9 \\ chi-value & 47.160 & & 14.250 & & 14.872 & & 30.964 & \\ \(p\)-value & 0.000 & & 0.046 & & 0.037 & & 0.000 & \\ \hline \hline \end{tabular}
\end{table}
Table 6: Mean values of the benchmark under the Friedman test (30-dim).  
Figure 3: The average ranking of PSO variants: (**A**) average ranking in unimodal functions of PSO variants; (**B**) average rank in multimodal and expanded functions of PSO variants.  
algorithms exceeds the value domain's \(CD\), the null hypothesis is rejected at the designated level of confidence. The \(CD\) calculation formula is built as follows:  
\[CD=q_{\alpha}\sqrt{\frac{K(K+1)}{6N_{K}}}, \tag{47}\]  
where \(q_{\alpha}\) = 3.031, and \(K=8\) represents the number of the algorithm's critical value range \(CD=1.863687\).  
There were obvious differences among the performances of MMPSO, EPSO, HCLPSO, HIDMSPSO, DMSPSO, FDRPSO, and CLPSO, as shown in Figure 4. In our qualitative analysis, there were no significant differences between the proposed algorithm and the HCLDMSPSO algorithm; however, the proposed algorithm showed superior performance according to the overall average rank.