# Local Coupled Extreme Learning Machine Based on Particle Swarm Optimization
## 3 Local Coupled Extreme Learning Machine Based on the PSO Algorithm  
Based on the optimization technique of the above PSO algorithm with self-adaptive parameters \(w\) and \(c\), the parameter values \(w\), \(b\), \(\mathbf{d}\) and \(r\) of the LC-ELM are optimized for improving the generalization performance in this work. In the LC-ELM learning algorithm, the decoupling of the input layer and the hidden layer is determined by the address parameter \(\mathbf{d}\) and the radius parameter \(r\). However, these parameter values are randomly determined. In other words, they might not be suitable for the algorithm, resultingin the poor performance of the algorithm. In addition, the hidden biases \(b\) and input weights \(\mathbf{w}\) are also set randomly in the LC-ELM. Therefore, for improving the performance of the LC-ELM algorithm, the four parameters \((\mathbf{w},b,\mathbf{d},r)\) of the LC-ELM are optimized based on the above adaptive PSO algorithm simultaneously. When the optimal parameters of the LC-ELM algorithm are established, the t weights between the hidden layer and the output layer of the LC-ELM are determined analytically based on the Equation (5) of the ELM, which is called the LC-PSO-ELM algorithm in this paper. Therefore, the particles in the searching space of the LC-PSO-ELM are composed of a set by the parameter values of input weights, hidden biases, address and radius, which can be defined as:  
\[ \begin{array}{c}\theta\in[w_{11},w_{12},\ldots w_{1N},w_{21},w_{22},\ldots w _{2N},\ldots w_{p1},w_{p2},\ldots w_{pN},\ldots,b_{1},b_{2},\ldots b_{N}\\ d_{11},d_{12},\ldots d_{1N},d_{21},d_{22},\ldots d_{2N},\ldots d_{p1},d_{p2} \ldots d_{pN},\ldots,r_{1},r_{2},\ldots r_{N}]\end{array}  \tag{11}\]  
where \(\mathbf{w}=\{\mathbf{w}_{i}|\mathbf{w}_{i}\in R^{p},i=1,\ldots,N\}\), \(b\in R^{N}\), \(\mathbf{d}=\{\mathbf{d}_{i}|\mathbf{d}_{i}\in R^{p},i=1,\ldots,N\}\) and \(r\in R^{N}\). Based on the global searching capability of the above PSO algorithm and the universal approximation performance of the LC-ELM learning algorithm, the detailed steps of the LC-PSO-ELM algorithm (Algorithm 2) are described as follows:  
The parameters in the algorithm are defined as: the training set is denoted as \(\{(x_{i},t_{i})x_{i}\in R^{p},t_{i}\in R^{q},i=1,\ldots,M\}\), \(g(x)\) is the output function \(g(\mathbf{w}_{i}\mathbf{x}_{j}+b_{i})\) of the hidden neuron, \(N\) is the number of the hidden neurons, \(F\) and \(S\) are fuzzy membership and similarity function, respectively. \(max\_iter\) is the preset maximum learning epoch of the PSO algorithm. \(w_{max}\) and \(w_{min}\) are the initial and terminal values of inertia weight in the iterative stage. \(c_{max}\) and \(c_{min}\) are the initial and final values of the acceleration constants. ```
(1) Initiate the population (particle). Each particle in the generation is composed of a set of the input weights \(\mathbf{w}\), biases \(b\), address \(\mathbf{d}\) and radius \(r\), as is shown in Equation (11). The initialization value of all of the components of the particle are set from \(-1\) to \(1\) randomly. (2) Iter = \(1\) (3) While Iter \(\leq max\_iter\) (4) (1) Evaluate the fitness function of each particle (the root means standard error for regression problems and the classification accuracy for classification problems). (2) Modify the position of the particle according to Equations (4)-(8). (3) Iter = Iter +1 (5) endwhile (6) The optimal parameters of the LC-ELM can be determined. Then, based on the optimized parameters: (1) The output matrix \(\mathbf{H}\) of the hidden layer is computed based on Equation (4). (2) The weight \(\mathbf{\beta}\) is calculated based on Equation (5). ```  
**Algorithm 2**The algorithm flow of LC-PSO-ELM  
Similar to the LC-ELM, the combinational function \(F(S(\mathbf{x},\mathbf{d}_{i}))\) between the similarity relation \(S(x)\) and the fuzzy membership \(F(x)\) in the LC-PSO-ELM also has many selection strategies.