ABSTRACT
Here, r = 90, If the embedding dimension d and the time delay r have been chosen for the phase-space reconstruction, then a reconstructed vector can be written as follows:
Bv Takens’ embedding theorem or its extensions, there exists a function F such that
Yi+i = F(Yi), i = 1 ,2 ,---
Let the reconstructed attractor be denoted by A, then F : A H> A. Thus we necessarily fit F only in the domain A by using our prediction algorithms. In this case, a good prediction technique should have a high enough resolving power so that it is able to resolve clearly the fine structure of the reconstructed attrac tors, A powerful analytical tool capable of doing this has been developed recently, which is the so-called “wavelet analysis,” also named multiresolution analysis. It has been widely applied to signal analysis due to some excellent properties, for example, in making local analysis. Because of such properties, wavelet
analysis is often regarded as a “microscope” in mathematics. The details of these could be found in any introductory mate rial on wavelets [220], When applying wavelet analysis, the hne structures of chaotic attractors ought to be able to be clearly re solved by adjusting two parameter values (i.e., a,b in Eq, 8,1) of the wavelets. This greatly motivates us to use the tool of wavelet analysis to approximate chaotic time series and then build a good predictive model. In practice, wavelet decompo sition has emerged as a new powerful tool for approximation. Furthermore, wavelet analysis is also very well suited for ana lyzing fractal signals since the wavelet basis is constructed only from one single function by means of dilations and translations. Based on the discussion above, we believe that wavelet analysis might be a powerful tool for predicting chaotic time series.
