An artificial neural network (ANN) based on particle swarm optimization (PSO) was developed for the time series prediction. This hybrid ANN+PSO algorithm was applied on Mackey-Glass series in the short-term prediction x(t+6) and the long-term prediction x(t+84), from the current value x(t) and the past values: x(t-6), x(t-12), x(t-18). Four cases were studied, alternating the timedelay parameter as 17 or 30. Also, the first four largest Lyapunov exponents were obtained for different time-delay. Simulation shows that this ANN+PSO method is a very powerful tool for making prediction of chaotic time series.
A convolution sum discrete process neural network (CSDPNN) is proposed. CSDPNN utilizes discrete samples as inputs directly and employs convolution sum to simulate the process inputs so as to deal with the time accumulation existing in many time series. Without the procedures of fitting the discrete samples into continuous functions to generate inputs and then to expand the input functions by basis functions, CSDPNN is better understandable and is with less precision reduction compared with process neural network (PNN) with function inputs. The approximation capacity of CSDPNN is analyzed in this paper, and it proved that CSDPNN can approximate PNN and has approximation capacity not worse than traditional artificial neural network (ANN). Finally, CSDPNN, PNN and ANN are utilized to predict the Logistic chaos time series and the iron concentration in the lubrication oil of aircraft engine, and the application test results indicate that CSDPNN performs better than PNN and ANN given the same conditions.