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## On the Required Number of Data Points

The question of how many reconstructed phase space vectors () we need to secure that the reconstructed picture of the attractor is accurate'' is seldom addressed in the literature. Often it is simply implicitely assumed that is large enough to make the error small. It is clear (when keeping Takens' results in mind) that for our picture converges to an object which is (diffeomorphically) equivalent to the attractor. We will take a brief look on what is known about the requirements on to come as near to this ideal as possible. The first thing to mention is that the needed depends on the fractality'' of the attractor, that is the more complex the attractor is (i.e. the larger the modulus of the difference between the attractor's dimension and the nearest integer is) the more points we need to reproduce it correctly. Secondly the attractor-picture is not only constructed as an end in itself but to derive characteristic properties from it. From this it follows that one should choose according to the quantity to be computed. For example, calculations of Lyapunov exponents usually require more data points than calculations of the correlation dimension. Thirdly, an inhomogeneous distribution of the on the reconstructed attractor obviously results in the need for more data points, since we want to resolve all parts of the attractor accurately enough. Considering the above hints which seem to suggest that usually a large is required it is good news that one can conclude from systematic studies surveyed by Albano et al. [8] that often a surprisingly small number of data points is sufficient, especially when one computes numerically robust quantities such as the correlation dimension. Unfortunately this observation seems to be correct only if the required accuracy is small. If one wants to compute for instance dimensions up to several decimal places then it can become necessary to use millions of data points. This observation is illustrated by the numerical experiment with the Rössler system, mentioned above: Although the geometrical structures of the objects in Figs. 5.a/5.b and e.g. in Fig. 5.f/5.g are very similar, one cannot expect to derive very accurate numerical invariants from the latter ones, since they are quite coarse-grained. Much more effort must be spent on slightly more exact computations.

Next: Conclusion Up: Analysis of Real-World Data Previous: Choice of First Embedding   Contents
Martin_Engel 2000-05-25