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Delay coordinates

The most important phase space reconstruction technique is the method of delays. Vectors in a new space, the embedding space, are formed from time delayed values of the scalar measurements:
The number m of elements is called the embedding dimension, the time tex2html_wrap_inline6553 is generally referred to as the delay or lag. Celebrated embedding theorems by Takens [21] and by Sauer et al. [22] state that if the sequence tex2html_wrap_inline6549 does indeed consist of scalar measurements of the state of a dynamical system, then under certain genericity assumptions, the time delay embedding provides a one-to-one image of the original set tex2html_wrap_inline6557, provided m is large enough.

Time delay embeddings are used in almost all methods described in this paper. The implementation is straightforward and does not require further explanation. If N scalar measurements are available, the number of embedding vectors is only tex2html_wrap_inline6563. This has to be kept in mind for the correct normalization of averaged quantities. There is a large literature on the ``optimal'' choice of the embedding parameters m and tex2html_wrap_inline6553. It turns out, however, that what constitutes the optimal choice largely depends on the application. We will therefore discuss the choice of embedding parameters occasionally together with other algorithms below.

Figure:   Time delay representation of a human magneto-cardiogram. In the upper panel, a short delay time of 10 ms is used to resolve the fast waveform corresponding to the contraction of the ventricle. In the lower panel, the slower recovery phase of the ventricle (small loop) is better resolved due to the use of a slightly longer delay of 40 ms. Such a plot can be conveniently be produced by a graphic tool such as gnuplot without generating extra data files.

A stand-alone version of the delay procedure (delay) is an important tool for the visual inspection of data, even though visualization is restricted to two dimensions, or at most two-dimensional projections of three-dimensional renderings. A good unfolding already in two dimensions may give some guidance about a good choice of the delay time for higher dimensional embeddings. As an example let us show two different two-dimensional delay coordinate representations of a human magneto-cardiogram (Fig. gif). Note that we do neither assume nor claim that the magneto- (or electro-) cardiogram is deterministic or even chaotic. Although in the particular case of cardiac recordings the use of time delay embeddings can be motivated theoretically [23], we here only want to use the embedding technique as a visualization tool.

next up previous
Next: Embedding parameters Up: Phase space representation Previous: Phase space representation

Thomas Schreiber
Wed Jan 6 15:38:27 CET 1999