# Local Coupled Extreme Learning Machine Based on Particle Swarm Optimization
## 1 Introduction  
The mathematical model of single-hidden layer feed-forward neural networks (SLFNs) has been widely used in many domains because of its ability to approximate strongly nonlinear input-output mappings. However, traditional learning methods are usually much slower than required while few faster learning algorithms for SLFNs are generated [1]. In 2006, a novel learning algorithm for SLFNs called extreme learning machine (ELM) [1, 2] was presented by [PERSON] et al. for decreasing the training time of SLFNs. Different from the existing learning algorithms of SLFNs, the weights and biases between the input layer and hidden layer of the ELM were chosen randomly, then the weights between the hidden layer and the output layer were determined based on the ordinary least squares. The ELM learning algorithm has fast learning speed and good generalization performance with little human intervention, which makes the algorithm applicable to many areas, such as stock prediction [3], image classification [4], fault diagnosis [5], etc. In the ELM, the number of hidden neurons is required to be greater than or equal to the number of the training samples so as to guarantee the convergence of the algorithm. Therefore, there will be quite a lot of input-hidden weights when the number of input neurons is large [6], which may reduce the generalization performance of SLFNs. The original ELM model has been equipped with various extensions to make it more suitable and efficient for specific applications [7]. For example, based on the structure of the local coupled feed-forward neural network (LCFNN) [8, 9] and the learning mechanism of the ELM algorithm, the local coupled extreme learning machine (LC-ELM) learning algorithm was proposed by [PERSON] in 2014 [10]. The algorithm could decrease the researching complexity of the weights between the input layer and the hidden layer by means of assigning the addresses tothe hidden neurons [10]. The advantage of the LC-ELM on image watermarking was examined by [PERSON] et al. [11]. In the LC-ELM learning algorithm, the addresses and radiuses were generally preset empirically or randomly. And, thus, those parameters might not be optimal for the LC-ELM, and the algorithm may yield an inappropriate underlying model. In 2015, [PERSON] et al. presented an evolutionary local coupled extreme learning machine (ELC-ELM). In the ELC-ELM, the differential evolutionary (DE) algorithm was used to optimize the addresses and the radiuses of the fuzzy membership functions in hidden neurons for improving the generalization performance [12]. However, it should be noted that the hidden biases and input weights in the ELC-ELM were also set randomly. The DE algorithm has good global converge property by means of utilizing the differential information of the population. However, the instability performance of DE can also be caused because of the above reason and the algorithm may be trapped in local optima [13; 14]. Moreover, three parameters of the DE algorithm should be controlled manually [15]. In 1995, the particle swarm optimization (PSO) algorithm was presented by [PERSON] et al. [16] and has been used in many optimization fields as it can converge to the global minima quickly. Compared with other stochastic optimization techniques, the advantages of the PSO algorithm are that it is easy to be implemented in practice and few parameters need to be adjusted [17; 18]. The PSO algorithm and its improved variants, such as APSO (Adaptive PSO) and PSOGSA (The hybrid PSO and gravitational search algorithm), were used to select the optimal parameters between the input layer and the hidden layer (input weights and biases) of the ELM [19; 20]. Therefore, in order to overcome the limitation of the DE, a new method combining the LC-ELM with an improved PSO called LC-PSO-ELM is proposed in this paper. In the proposed algorithm, the improved PSO algorithm is used to optimize the address and window radius of the local coupled parameters.