Study On Long-term Prediction Technology Of Chaotic Time Series

This paper discusses the mechanism of modeling and prediction of chaotic time series. On the base of analyzing chaotic characteristics, the aim is constructing a methodology to describe univariate or multivariate series of complex natural system. Within the range of predictability, long term prediction is made by two ways. One is tracking the strange attractor's trajectory in real time and the other is tracking the multiple neighboring trajectories at the same time. Corresponding theory is also analyzed. The main research contents and research conclusions are listed as follows.(1) Method study on chaotic noise reduction combining with the attractor trajectories features in phase space. The chaotic time series behave unknown chaotic character and have a big data set, which requires the improvement of the traditional local average nonlinear noise reduction method. A nonlinear local approximation noise reduction method is presented. This method uses the simple °°-norm to compute the distance between two points and builds the weighted model in the neighborhood, which can better correct the position of data points in phase space to approximate the real chaotic attractor trajectories more closely. Simulation results show that the improved method can effectively reduce the noise, keep the chaotic characteristics of the nonlinear system and better distinguish the closed trajectories in the phase space.(2) Self-adaptive prediction study using reconstructing system equations. A new concept named Generalized State Vector is introduced, which consists of two parts, the unlcnown system parameters and the original system state variables. Comparison of attractor characteristics between typical chaotic equations and observed time series is made to better define the initial values of the Generalized State Vector. Then the real-time capability of expanded Kalman filter is used to synchronously predict chaotic time series and adjust the parameters of chaotic governing system equations. Simulations to sunspot chaotic system show that this method realizes precise on-line prediction and effectively constructs the system equations.(3) Prediction study using multi-branches time delay neural network. Owing to the strong sensitivity of chaotic system to initial conditions, it is highly difficult to make long term prediction. In this paper, the probability to make long term prediction using time delay neural networks is first discussed. Based on it, a new recurrent predictor neural network (RPNN) is built. The network is one-layer and consists of a number of locally inter-connected nodes. Theconnections may contain multiple branches with time delays. Considering that the nodes represent the processing elements or time instants in series, and the branches between the nodes describe the relation among the processes, RPNN is more accordant with the practical complex chaotic systems. Based on the learning of a great deal of input-output samples, the network tracks all the paths of the nearest neighbors. Consequently, the influence of prediction lead time is weakened and a more accurate prediction can be made. Five performance measures are introduced to quantitatively measure the performance of prediction models. The advantages and disadvantages are also analyzed. On the other hand, the stability analysis of multi-branches time delay neural network is discussed. Consisting of nonlinearly operated nodes connected by multi-branches with time delays, the dynamics of this network is highly complex. This paper introduces the branch weights matrix to denote all weights. When deviated from the equilibrium under the disturbances brought by inputs or weights, the network can keep stable, that is to say the internal structure of networks could limit the outputs vibration to a finite neighborhood. The corresponding structure conditions are analyzed.(4) Multivariate time series prediction study using the combination of principal component analysis and time-delay neural network. For a practical complex system, because the internal dynamics is often containe...