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## Embedding parameters

A reasonable choice of the delay gains importance through the fact that we always have to deal with a finite amount of noisy data. Both noise and finiteness will prevent us from having access to infinitesimal length scales, so that the structure we want to exploit should persists up to the largest possible length scales. Depending on the type of structure we want to explore we have to choose a suitable time delay. Most obviously, delay unity for highly sampled flow data will yield delay vectors which are all concentrated around the diagonal in the embedding space and thus all structure perpendicular to the diagonal is almost invisible. In [24] the terms redundancy and irrelevance were used to characterize the problem: Small delays yield strongly correlated vector elements, large delays lead to vectors whose components are (almost) uncorrelated and the data are thus (seemingly) randomly distributed in the embedding space. Quite a number of papers have been published on the proper choice of the delay and embedding dimension. We have argued repeatedly [11, 2, 3] that an ``optimal'' embedding can - if at all - only be defined relative to a specific purpose for which the embedding is used. Nevertheless, some quantitative tools are available to guide the choice.

The usual autocorrelation function (autocor, corr) and the time delayed mutual information (mutual), as well as visual inspection of delay representations with various lags provide important information about reasonable delay times while the false neighbors statistic (false_nearest) can give guidance about the proper embedding dimension. Again, ``optimal'' parameters cannot be thus established except in the context of a specific application.

- Mutual information
- False nearest neighbors

Next: Mutual information Up: Phase space representation Previous: Delay coordinates