# Developing a Generalized Regression Forecasting Network for the Prediction of Human Body Dimensions
## 3 Methodology of the GRFN
### Construction of MMPSO Algorithm for Body Dimension Prediction
#### 3.3.2 Teaching and Learning Co-Optimization Strategy  
Inspired by the literature [31], the teaching and learning co-optimization strategy for the teacher and student subpopulations includes two stages: global and local tuning. With global tuning, the teacher subpopulation uses a ring-topological comprehensive learning strategy to search based on self-cognition and social learning behaviors to obtain information from neighbors, enhance search efficiency and stability, and accelerate convergence. Social learning consists of group induction guidance and ring neighborhood particle learning:  
\[v_{id}^{(t+1)}=w_{1}v_{id}^{(t)}+c_{1}r_{1}(pb_{fi(d)}^{(t)}-x_{id}^{(t)})+c_{2 }r_{2}\beta^{ST}(pb_{gb,d}^{(t)}-x_{id}^{(t)})+c_{3}r_{3}(rb_{id}^{(t)}-x_{id}^{ (t)}), \tag{20}\]  
where the linear decreasing inertia weight is expressed as \(w_{1}:0.9\sim 0.4\), and the first acceleration factor is \(c_{1}:2.5\sim 2.0\), which is linearly decreasing. The second acceleration factor is \(c_{2}:0.5\sim 2.0\), which is linearly increasing, and the third acceleration factor is \(c_{3}:2.0\sim 0.0\), which is linearly decreasing. Function \(f_{i}(d)=[f_{i}(1),f_{i}(2),\ldots,f_{i}(D)]\) represents \(pb\) of the \(i\)-th particle and must conform to requirements in the \(d\)-th dimension, \(rb_{i}=[rb_{i1},rb_{i2},\ldots,rb_{iD}]\) is the \(i\)-th particle's closest neighbor in the ring topology, and \(\beta^{ST}=0.5\) is the social learning tendency coefficient. These items are used to improve the performance of particles and enhance their self-search. To strike a balance between exploring and exploiting particles across different dimensions, the CLS is used to define the individual learning probability, \(pc\), of each particle, which generates a \([0,1]\)-interval random number for each dimension. A random number performs better than learning probability in solving the multipeak function. CLS adopts an aging strategy to regenerate the historical optimal solution, \(pb_{fi(d)}\), of individual nearest neighbors when the number of stagnant neighbors exceeds the threshold of seven. During global tuning, a subgroup stagnation state-checking mechanism is employed to detect whether the subpopulations of teachers and students are stagnant. Then, the fitness deviation of the optimal solution is calculated for generations \(t\) and \(t-1\) subgroups to judge whether any are stagnant. \[\delta\textit{sbest}(t+1)=fit(pb_{sb}^{(t+1)})-fit(pb_{sb}^{(t)})\leq 0, \tag{21}\]  
where the fitness function is \(fit(\cdot)\), and \(pb_{sb}\) represents the locally optimal solution within the subgroup. When \(\delta\textit{sbest}(t+1)<0\), the capacity for global search of the subgroup is normal, and the stagnant state counter, \(\textit{Stag}_{s}\) is zero. When \(\delta\textit{sbest}(t+1)=0\), the capacity for global search of the subpopulation weakens and \(\textit{Stag}_{s}=\textit{Stag}_{s}+1\). If \(\textit{Stag}_{s}\geq C_{t1}\), where \(C_{t1}\) is the stagnation threshold [37] selected as \(T_{\max}/10\), where \(T_{\max}\) denotes the fixed number of iterations, the subgroup is identified as having entered stagnation. Then, the local tuning operation is employed to reinvigorate the search capability and counteract stagnation. During local tuning, particle disturbance update and refresh compensation operations are used. Inspired by the literature [38], during the particle disturbance update, a search operator guided by the historical extremum of the induction group is used to remove bias from the search and improve population diversity as follows:  
\[\text{new}X_{i}=X_{i}+\zeta_{1}(X_{i}-X_{k})+\zeta_{2}(pb_{gb}-X_{i}), \tag{22}\]  
where \(\zeta_{1}\), \(\zeta_{2}\) are random numbers in the range of \([-1,1]\) and \([0,1.5]\), respectively. \(k\) is a random index in \([1,N]\), and \(k\
eq i\). The new search operator guides particle-balanced exploration and exploitation by inducing historical extrema in the group. During the particle disturbance update operation, an individual particle adopts an optimal greedy strategy.