# Local Coupled Extreme Learning Machine Based on Particle Swarm Optimization
## 4 Simulations and Performance Verification
### Performance Comparison of Regression Benchmark Problems  
This section mainly shows the comparison results of the original ELM, LC-ELM, PSO-ELM and LC-PSO-ELM four algorithms on the function approximation datasets. The average simulation results of 10 experiments are shown in Tables 5 and 6. From these tables, we may see that the training time of the proposed algorithm consumed much more than the other ones, which means that the adaptive PSO algorithm needs more time for searching the global optimal solution of the parameters \((\mathbf{w},b,\mathbf{d},r)\) in the LC-PSO-ELM algorithm.  
Although the training error is higher than the other algorithms in the proposed algorithm in terms of the Autompg problem, the proposed algorithm in this paper focuses on superiority in terms of improved generalization performance, the fact that the testing time of all of them is almost equivalent and the proposed algorithm has better generalization performance with fewer parameters and compact network configuration, which shows that the proposed algorithm has good generalization value and real applicability.  
Moreover, the proposed LC-PSO-ELM and PSO-ELM learning algorithms have relatively less value of STD in the experiments, which means that the algorithms have stable performance with parameters optimized by means of the PSO algorithm, although searching the optimal parameters needs much time in the training process.  
Except for the STD value of the Autompg, the other problems of LC-ELM are bigger than the ELM, PSO-ELM and LC-PSO-ELM algorithms. The results show that the LC-ELM is the most unstable learning algorithm out of the four, and they are also the same as the simulation results in Figures 1 and 2.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Dataset** & **Box and Jenkins** & **Autompg** & **Abalone** & **Calhousing** & **Wine** & **Iris** & **Diabetes** & **Satimage** \\
**Algorithms** & **Gas Furnace Data** & & & & & & & \\ \hline ELM & 15 & 72 & 25 & 10 & 15 & 10 & 38 & 30 \\ LC\_ELM & 15 & 57 & 25 & 10 & 15 & 10 & 26 & 30 \\ PSO-ELM & 15 & 40 & 25 & 10 & 15 & 10 & 38 & 30 \\ LC-PSO-ELM & 15 & 27 & 25 & 10 & 15 & 10 & 17 & 30 \\ \hline \hline \end{tabular}
\end{table}
Table 4: The number of hidden neurons in the algorithms for different benchmark problems.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{3}{*}{**Algorithms**} & \multicolumn{4}{c}{**Box and Jenkins Gas Furnace Data**} & \multicolumn{4}{c}{**Autompg**} \\ \cline{2-9}  & **Training** **(s)** & **Testing** **(s)** & **Training** **Error** & **Testing** **(s)** & **Training** **Time (s)** & **Testing** **Error** & **Testing** **Error** \\  & **STD** & **STD** & **STD** & **STD** & **STD** & **STD** & **STD** & **STD** & **STD** \\ \hline ELM & 0 & 0 & 0.0187 & 0.0214 & 0.0125 & 0.0062 & 0.0533 & 0.0866 \\  & 0 & 0 & \(\begin{array}{c}6.9674\times\\ 10^{-4}\end{array}\) & 0.0015 & 0.0263 & 0.0197 & 0.0030 & 0.0108 \\ \hline LC-ELM & 0.0094 & 0.0035 & 0.0183 & 0.0262 & 0.0187 & 0.0156 & 0.0635 & 0.0885 \\  & 0.0211 & 0.0075 & 0.0019 & 0.0060 & 0.0301 & 0.0255 & 0.0031 & 0.0074 \\ \hline PSO-ELM & \(\begin{array}{c}116.2472\\ 2.3068\end{array}\) & 0.0337 & 0.0011 & \(\begin{array}{c}0.0161\\ \mathbf{0.0337}\end{array}\) & 0.0184 & \(\begin{array}{c}221.9707\\ \mathbf{10^{-4}}\end{array}\) & 0 & 0.0601 & 0.0662 \\ \hline LC-PSO-ELM & \(\begin{array}{c}155.2323\\ 4.6814\end{array}\) & 0.0312 & 0.0178 & 0.0182 & \(\begin{array}{c}296.2370\\ 0.0353\end{array}\) & 0.0203 & 0.0663 & \(\begin{array}{c}\mathbf{0.0653}\\ 0.0018\end{array}\) \\  & 4.6814 & 0.0353 & 0.0017 & 0.0018 & \(\begin{array}{c}5.1307\\ \mathbf{5.1307}\end{array}\) & 0.0148 & 0.0027 & 0.0043 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Performance comparison of different algorithms on regression problems of Box and Jenkins gas furnace data and Autompg.