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### Choice of a Proper Embedding Dimension for a Noise-Free Time Series of Finite Length

As a first step we choose some . We want to embed in . In general will be unknown so that it is not obvious how to meet Takens' condition . But if we choose large enough to ensure that then we can embed in , according to Takens' theorem 2, because it is a well known fact that every Euclidean space can again be embedded in any higher-dimensional Euclidean space without any problems. So we start by making an ansatz for and our concern for the next few paragraphs will be to find a more appropriate such that is a space containing Takens' embedding space. Thus will be a better embedding dimension than the we started with, since it gives a lower-dimensional embedding space. We want to use the method of delays to construct vectors in from the of the time series. Doing this one realizes that there is even one more quantity which is not yet specified: One could, for example, take the series of -vectors
 (27)

which we are obviously allowed to use in accordance with Takens' statements. So we also have to choose the lag time'' , where . We will see in section 3.4.2 that we do not have to spend much effort on choosing : when we use the singular systems technique the influence of the lag time becomes insignificant, hence we will choose it from now on to equal (i.e. ). Consider a sequence of vectors , (i.e. we take a time series containing data points)12. There seems to be no analytical way to compute the proper (i.e. minimal) embedding dimension from the time series. However, it is possible to determine a reasonable estimate for it: For some given the -vectors usually do not explore the whole space . Rather than that they are restricted to some subspace of ; T contains the embedded manifold which contains the picture of the attractor:
 (28)

When we assume that the really visit the whole attractor in the embedding space (more or less) uniformly and we bear in mind that usually is much larger than then is a sensible upper bound for the minimal embedding dimension. In order to determine we compute the maximum number of linearly independent vectors that can be constructed as linear combinations of the . To do this, we define the -trajectory matrix :
 (29)

which is built out of all vectors we want to use to reconstruct the attractor. Notice that when operating with on some -vector we get an -vector:
 (30)

Since we are interested in linearly independent -vectors, we choose a set of vectors such that the -vectors
 (31)

are orthonormal. We introduce some real constants into this equation, in order to normalize the :
 (32)

The important point about this equation is that, after transposing, it can be rewritten as
 (33)

i.e. as a linear combination of the reconstructed trajectory vectors. This tells us, when we keep in mind the definition of , that we can get linearly independent vectors , using eq. (34); so we have vectors and numbers , too. The are elements of an orthonormal basis of . Thus we are left to determine as the number of those which are non-zero. Define the structure matrix ; then it follows from eq. (34) that the are the eigenvalues of this matrix:
 (34)

We could determine as the number of the non-zero eigenvalues of . But is a huge -matrix, and singular, and its diagonalization is in practice impossible. Instead, we notice that the covariance matrix has the same non-zero eigenvalues as , and is much easier to diagonalize, because it is only an -matrix. So all one has to do in order to calculate which, cum grano salis, estimates the minimal embedding dimension , is to determine the number of the non-zero eigenvalues of
 (35)

Now we know that the trajectory is confined to an -dimensional subspace of , and we can use as the embedding space. However, this treatment only makes sense in the case that we have noise- free data.

#### Footnotes

... points)12
We will generally assume , which is obviously true in most cases.

Next: Singular Value Decomposition as Up: Singular System Analysis Previous: Singular System Analysis   Contents
Martin_Engel 2000-05-25