Journal of Mathematical Sciences: Advances and ApplicationsAuthor's: Jiin-Po Yeh
Pages: [73] - [94]
Received Date: July 2, 2009
Submitted by:
With delay coordinates as inputs of neural networks, this paper presents a forecasting technique for chaotic time series. By using delay coordinates, the chaotic time series is first embedded in a reconstructed state space. Then the delay coordinates of each state in the reconstructed state space serve as the input vector of the neural network and the first delay coordinate of the next state as the target of the neural network. The neural networks used in this paper are two-layer feedforward networks with one hidden layer of tan-sigmoid neurons followed by an output layer of one linear neuron. Meanwhile, Bayesian regularization and early stopping of training are applied to improve the network generalization. Traffic flows of three different time scales are used as examples to show the effectiveness of the technique. Numerical results show that with the number of neurons in the hidden layer not more than the number of elements in the input vector and for only a few iterations, the neural network will have acceptable performance. Although, more neurons or iterations can enhance the network performance for the training set, it does not have the tendency to benefit the validation and the prediction sets. In addition, the prediction accuracy becomes higher, when the traffic volume time scale increases.
neural networks, chaotic time series, delay coordinates, reconstructed state space.
