Short-Term Prediction of Chaotic Time Series by Local Fuzzy Reconstruction Method

Abstract

In recent years, the study of chaos has been drawing attention. As chaos application, we propose a short-term prediction of the time series under chaotic behavior. Short-term prediction is to find some deterministic regularity in a phenomenon that was thought of as noise or an irregularity and thereby predict its state in the near future. Short-term prediction employs the following technique: the observed time series is reconstructed in a multidimensional state space according to the Takens' theorem of embedding. Local reconstruction is the mode that uses the vector neighboring the data vector, including the latest observed data. Proposed as the practical methods of this prediction are Gram-Schmidt's orthogonal system method, the tessellation method, etc. However, the drawback to these methods is that prediction becomes impossible if the selected neighboring vector is not linearly independent. 1n the tessellation method, it also has the shortcoming that the time period of calculation increases abruptly as the dimension of reconstructed state space increases. To overcome such disadvantages, this article proposes the local fuzzy reconstruction method. First we explain deterministic chaos. Next we show the approach to short-term prediction of a chaotic time series in detail. Finally, examination is made of the result of the preceding local fuzzy reconstruction method applied to the short-term prediction of the logistic map and the Lorenz attractor, which are known as typical chaotic time series, and tap water demand data, traffic density, and power demand data as social phenomena.