# Local Coupled Extreme Learning Machine Based on Particle Swarm Optimization
## 4 Simulations and Performance Verification  
In this section, the proposed LC-PSO-ELM learning algorithm and three alternative ELM algorithms in the aspect of four function approximation (regression) and four classification benchmark problems, the original ELM, LC-ELM [10] PSO-ELM [17], are conducted in the MATLAB R16a environment running with 3.4 GHz CPU and 16 G RAM. The parameters specification of the benchmarks problems is shown in Table 1. The experimentally well-characterized datasets were chosen for good comparison in this paper [28; 29], in which the Box and Jenkins gas furnace data were sourced from the reference [30], the Calhousing data came from the StatLib dataset [31] and the other dataset was derived from the UCI (University of California, Irvine, CA, USA) Machine Learning Repository [32], respectively. For each dataset, the input sequence of the data was changed randomly and then the data were divided into two groups of training data and testing data for experiments based approximately on a 70-30 ratio. The number of the two groups is shown in Table 1.  
The number of the population of the PSO algorithm is 200 and the maximum iterative number is 50. The configurations of the ELM, PSO-ELM, LC-ELM and the LC-PSO-ELM are listed in Table 2. For simplicity, RN is the abbreviation for random number and NDRN is the abbreviation for normally distributed random numbers. As shown in Table 2, the sigmoid function is selected as the activation function of the four learning algorithms. The wave kernel \(S(x,y)=(\theta/||x-y||)\sin(||x-y||/\theta)\) is selected as the similarity function and the reversed sigmoid function Equation (13) \(F(x)=\frac{2}{1+\exp(x/r)}\) is selected as the fuzzy membership function in the LC-ELM and the LC-PSO-ELM algorithms, respectively. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Problem** & **Dataset** & **Attributes** & **Class** & **Training Data** & **Testing Data** \\ \hline \multirow{4}{*}{Regression} & Box and Jenkins gas furnace data & 10 & 1 & 203 & 87 \\  & Autompg & 7 & 1 & 279 & 119 \\  & Abalone & 8 & 1 & 2923 & 1254 \\  & Calhousing & 8 & 1 & 14,448 & 6192 \\ \hline \multirow{4}{*}{Classification} & Wine & 13 & 3 & 124 & 54 \\  & Iris & 4 & 3 & 95 & 42 \\ \cline{1-1}  & Diabetes & 8 & 2 & 537 & 231 \\ \cline{1-1}  & Satimage & 36 & 6 & 4504 & 1931 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters specification of the benchmark problems. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Configurations** & **ELM** & **PSO-ELM** & **LC-ELM** & **LC-PSO-ELM** \\ \hline Input weight and hidden layer biases & RN in \([-1,1]\) & RN in \([-1,1]\) & RN in \([-1,1]\) &
\begin{tabular}{c} NDRN (normally distributed random numbers) \\ \end{tabular} \\ \hline Activation function & sigmoid & sigmoid & sigmoid & sigmoid \\ \hline Hidden node address & – & – & RN in [0,1] \& 0.4 & NDRN \\ and window radius & – & – & Wave kernel & Wave kernel \\ \hline Similarity & – & – & Equation (13) & Equation (13) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Configurations of the ELM (extreme learning machine), PSO-ELM (extreme learning machine based on particle optimization), LC-ELM (local coupled extreme learning machine) and LC-PSO-ELM algorithms. In order to increase the persuasion of different algorithms in terms of validity, 10 trials of the average simulation results (root mean square error (RMSE) is the abbreviation) for regression benchmarks and classification accuracy for classification (pattern classification) are given in the following tables. The training and testing subsets of each experiment in the 10 trials are created by randomly choosing samples of the datasets based on a 70-30 ratio renewedly, the robustness of the algorithms is compared using the standard deviation (STD is the abbreviation) of the 10 trials. The CPU time of training is used to evaluate the computational complexity of the algorithms.