# 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
#### 4.2.2 Numerical Experimental Results  
The performance of the MMPSO and several well-known PSO variants was evaluated using a benchmark test suite. Following the acquisition of the results from 30 independent runs of each algorithm, the evaluation metrics were determined based on the average value (mean) and standard deviation (Std) of each benchmark function. The mean reflects the overall trend of convergence and optimization ability, whereas the Std indicates the stability of the algorithm and its capacity to evade local optima. The dimensions needed to solve the problem were 30, the upper limit for the total number of fitness evaluations was 300,000, the population size was set to 40, and the sizes of the three heterogeneous subgroups (i.e., teacher, independent learner, and student) were 12, 16, and 12, respectively. The MMPSO algorithm in F04, F06, F08, F10, F11, F14, F17-F20, and F22-F25 benchmark functions were compared with the alternative PSO variants, revealing clearly improved solution accuracy, as demonstrated in Table 5. Figure 3A illustrates how the MMPSO algorithm demonstrated excellent performance in terms of finding the global optimum using unimodal functions. Furthermore, the MMPSO algorithm performed well when solving multimodal functions and expanded functions, as illustrated in Figure 3B. In terms of algorithm stability, the standard deviations of the F04, F05, F06-F08, F10, F17-F18, and F23-F25 benchmark functions exhibited superior performance over the alternative PSO variants. The final ranking algorithms are arranged based on overall averages in the following order: MMPSO, HCLDMSPSO, EPSO, HCLPSO, HIDMSPSO, \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Algorithms** & **Year** & **Parameter Settings** & **Reference** \\ \hline FDRPSO & 2003 & \(w:0.9\sim 0.4,c_{1}=c_{2}=1.0,c=2.0\) & [PERSON] et al. [45] \\ DMSPSO & 2005 & \(w=0.729,c_{1}=c_{2}=1.49445\) & [PERSON] et al. [24] \\ CLPSO & 2006 & \(w:0.9\sim 0.4,c_{1}=c_{2}=1.49445,m=7\) & [PERSON] et al. [26] \\ HCLPSO & 2015 & \(w:0.99\sim 0.2,c_{1}:2.5\sim 0.5,c_{2}:0.5\sim 2.5\) & [PERSON] et al. [46] \\ EPSO & 2017 & \(w:0.9\sim 0.4,c_{1}=c_{2}=2.0\) & [PERSON] et al. [47] \\ HCLDMSPSO & 2020 & \(w_{1}:0.99\sim 0.29,w_{2},c_{1}:2.5\sim 0.5,c_{2}:0.5\sim 2.5\), \(P_{m}=0.1\) & [PERSON] et al. [48] \\ HIDMSPSO & 2020 & \(w_{1}:0.99\sim 0.2,w_{2}:1:2.5\sim 0.5,c_{2}:0.5\sim 2.5\) & [PERSON] et al. [49] \\ MMPSO & – & \(w_{1}:0.9\sim 0.4,w_{2}:0.99\sim 0.2,w_{c}=0.8,c_{1}:2.5\sim 0.5\), & \\  & & \(c_{2}:0.5\sim 2.5,c_{3}:2.0\sim 0.4,c_{4}=c_{5}=1.49445\) & – \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameters settings for PSO variants. DMS-PSO, FDRPSO, and CLPSO. This study's experimental findings provide evidence that the proposed algorithm exhibits effective global convergence capability, particularly when dealing with multimodal and compound test functions. In summary, the MMPSO algorithm shows a superior optimization effect, and experimental results clearly validate that the presence of diverse populations aids in the avoidance of local optima within the model (i.e., locating the global optimum).