Embedding dimension In practice, there are two problems with delay coordinates as a method of phase space reconstruction. The first problem, which is common to all reconstruction methods, is that the necessary dimension of the reconstructed phase space, i. e. the minimum embedding dimension, is not known since the dimension of the original phase space is also unknown.

This problem may, however, be solved indirectly by utilizing a generic property of a faithful reconstruction, i. e. an embedding, namely, that the attractor in the original phase space is completely "unfolded", or contains no self-intersections, in the reconstructed phase space. In other words, if the embedding dimension is too low, the attractor is not completely unfolded which means that distant points on the attractor in the original phase space are close points in the reconstructed phase space.

Moreover, if the attractor is completely unfolded it is not necessary to view the attractor in a phase space with a higher embedding dimension. Thus, from a theoretical point of view, it does not matter whether the attractor is unfolded in an embedding dimension which is too high.

From a practical point of view, however, there are several reasons to determine the minimum embedding dimension mm . First, the number of points on the reconstructed attractor can be too few to obtain reliable estimates of, for example, the Lyapunov exponents. Second, the computational "cost" rises exponentially with the embedding dimension because all mathematical computations take place in R".

Third, in the presence of noise, the unnecessary dimensions of the phase space, i. e. M - mm,, are not populated by new information about the attractor because this information has already been captured in a smaller embedding dimension. In summary, it is desirable to determine the minimum embedding dimension.

Examples of methods developed for this purpose are false nearest neighbors, saturation of invariants on the attractor and true vector fields. The method of false nearest neighbors is based on the aforementioned property of a faithful reconstruction, namely, that when choosing an embedding dimension which is too low, distant points on the attractor in the original phase space are close points in the reconstructed phase space, i. e. false nearest neighbors exist. By increasing the embedding dimension, the attractor is completely unfolded when there are no false nearest neighbors (Kennel et al., 1992).

The second method, saturation of invariants on the attractor, is based on the observation that when the attractor is completely unfolded, any invariant on the attractor, e.g. the dimension of the reconstructed attractor, is independent of the embedding dimension. If, however, the attractor is not completely unfolded in the reconstructed phase space, these invariants depend on the embedding dimension. Thus, by increasing the embedding dimension, the attractor is completely unfolded when the value of the specific invariant on the attractor stops changing (Grassberger
and Procaccia, 1983).

The third method, true vector fields, is based on another property of a faithful reconstruction, namely, that the vector field associated with the vector function ft is unambiguous when the attractor is completely unfolded. In other words, the tangents to the evolution of the vector function are smoothly and uniquely given throughout the phase space.

Thus, if the embedding dimension is too low, the vector field in some neighborhoods of the attractor is not unique because the directional vectors, i. e. the tangents, in that neighborhood point in different directions. Thus, by increasing the embedding dimension, the attractor is completely unfolded when the directional vectors in each neighborhood point in the same direction. Kaplan and Glass (1992) specifically test whether the distribution of directions in a neighborhood of the attractor is consistent with being generated by a deterministic system.

When observational noise is present in the observed scalar time series, i. e. 'Y 0 in eq. (2.2), it is not an easy task to determine the minimum embedding dimension, especially when the noise to signal ratio is high. For example, if the method of saturation of invariants on the attractor is utilized, the specific invariant may never stop changing.

Reconstruction delay

A second problem with delay coordinates, which is specific to this reconstruction method, is that the reconstruction delay, i. e. J in eq. (2.4), must be chosen. If the reconstruction delay is too small, each coordinate is almost the same and this results in a reconstructed attractor that is compressed along the identity line, or the main diagonal, of the phase space. If, however, the reconstruction delay is too large, successive delay coordinates may become causally unrelated and the reconstructed attractor no longer represents the true dynamics.

These problems are the problems of redundance and irrelevance, respectively. It is thus desirable to determine the proper reconstruction delay. It should be noted that Takens' (1981) embedding theorems did not address problems about the proper reconstruction delay because if the observed scalar time series is noise-free, i. e. 'Y = 0 in eq. (2.2), almost every reconstruction delay is suitable. Examples of methods to determine the proper reconstruction delay are average mutual information and reconstruction expansion from the identity line.

The latter method is a geometrical method that focuses on the redundance error in a proposed reconstruction of the attractor. This method, proposed by Rosenstein et al. (1994), is based on the observation that the expansion of the reconstructed attractor from the identity line is best quantified by measuring the average displacement of the delay vectors, i. e. xt in eq. (2.4), from their original locations on the identity line.

These authors suggest that the proper reconstruction delay has been determined when the first maximum of the average displacement examined as a function of the reconstruction delay, occurs. This method thus neglects the irrelevance error in a proposed reconstruction of the attractor, because one typically cannot measure the irrelevance error.

The former method, average mutual information, focuses on the chaotic dynamical system as a producer of information. Because the resolution in the reconstructed phase space is, in practice, finite, points which are too close together cannot be distinguished. However, because close points separate exponentially when the system is chaotic, they can be distinguished if enough time has elapsed. The fact that these
points were close but not equal is revealed as time evolves. This means that if the reconstruction delay is too small, the chaotic dynamical system has not yet explored enough of its phase space to produce new information about that phase space.

Fraser and Swinney (1986) make use of the so-called average mutual information function when they determine the proper reconstruction delay. The average mutual information function can be considered as a generalization of the autocorrelation function where the latter provides a measure of the linear dependence, on the average over all observations, between measurements at a certain lag. Specifically, the average mutual information function gives the average amount of information about xt+J given xt . Fraser and Swinney (1986) suggest that when the first minimum of the average mutual information function occurs, the proper reconstruction delay is determined.

Prof. Mikael Bask

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