Chaotic Time Series Prediction Based on Improved MOPSO-ELM

Extreme learning machines have attracted increasing attention for setting the input weights and hidden biases randomly. However, excessive hidden neurons in ELMs may cause ill-posed problem when Moore-Penrose generalized inverse method is used to train the neural network. It generates large amplitude of output weights, which leads to bad generalization. To solve this problem, we propose an improved multi-objective particle swarm optimization algorithm to optimize the output weights of ELMs. It consists of two objective functions. One is the sum of the errors between predicted values and target values. The other one is the $L_{2}$ norm of output weights. We improve general MOPSO algorithm by adding chaotic mapping to promote the ergodicity and randomness of new solutions. In addition, a new strategy is proposed to select $pbest$ and $gbest$ to deal with the unbalance of multiple objectives. Experimental results on predicting two benchmark chaotic time series sustain the efficiency and effectiveness of our proposed algorithm. It has better generalization performance and smaller output weights than original ELM and other evolutionary ELMs.