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Reconstruction of phase space

The reconstruction method has two parameters: the embedding dimension and the time lag (see eq. (2.6)). In theory, (with an infinite amount of noise-free data,) taking
\begin{displaymath}
d \geq 2D + 1
\end{displaymath} (6.2)

should be enough, where $D$ is the dimension of a compact manifold containing the attractor and $d$ is the embedding dimension. In practice however, $D$ is unknown. This implies that we have to estimate the dimension for increasing embedding dimensions, until it ``saturates''. There may be lack of saturation due to noise or entropy. It might be that a reasonable estimate (e.g. for comparisons) is obtained as soon as eq. (6.2) is satisfied.

Almost every value for the time lag should work. However, it should not be too small because the attractor would be restricted to the diagonal of the reconstructed phase space and it should not be too large since the components would become uncorrelated. Furthermore, it should not be close to an integer multiple of a periodicity of the system. Many suggestions have been done for the choice of the time lag. For example, a criterion would be a minimum amount of fluctuations of the correlation integrals around the scaling region [Liebert and Schuster, 1989]. However, it could be that these fluctuations become smaller with decreasing lag. Furthermore, finding the minimum requires to repeat the analysis for many values of the lag. Alternatives are to let the lag be a quarter of the period of the characteristic frequency (greatest amplitude), the autocorrelation time (when the autocorrelation approaches 1/exp(1)), or the time when the mutual information function reaches its first minimum [Fraser and Swinney, 1986,Olofsen, 1987]). Another option is to use a ``fixed window'' [van Neerven, 1987,Albano et al., 1988]. Though this procedure does not allow the estimation of the entropy, there may be less artefacts due to noise and entropy. However, a spurious saturation of the dimension may occur (may be due to lack of additional information as the embedding dimension increases [Broomhead and King, 1986]) at a value which depends on the window length.


next up previous contents
Next: Singular value decomposition Up: PRACTICAL ASPECTS Previous: Theiler correction   Contents
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