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Next: Discussion Up: Dynamical analyses of postural Previous: Dynamic Stability of

Problems in Reconstructing a Postural Attractor

In order to reconstruct the attractor of a dynamic system, two problems will need to be solved. The first concerns how to select the time delay, , for reconstructing the trajectory in phase space. With very small delays, the components of the vectors will be nearly the same, and so the trajectories in the embedding space will all be compressed into a long thin volume equivalent to a diagonal in the state space. The goal is to have the reconstructed trajectory be as uniformly distributed as possible so that its properties can be accurately measured, i.e., select time delay which gives least compression of trajectories.

One method for resolving this problem is to analyze the Average Mutual Information in the time series. Mutual information is a general measure, based on information theory, of the extent to which the values in a time series can be predicted by earlier values. It is not limited to linear dependence as is the autocorrelation function. Fraser & Swinney (1986) [3] argue that when a phase portrait is reconstructed from a time series using a time delay that minimizes mutual information, the trajectories in the reconstructed state space will be maximally separated, which in turn will facilitate their analysis.

[FIGURE 12] The next figure shows how mutual information decreases as the time delay increases. Based on these analyses, we selected a time delay of 5 samples or 0.10 sec. Thus, each vector was constructed by selecting components that were separated by a tenth of a second.

Figure 12: Mutual information is plotted against increasing values of time delay for free sitting infants age seven months.  

The second problem concerns how to determine the embedding dimension of the system. Our approach to this problem was to use an analytic method known as False Nearest Neighbors [5]. In contrast to many other methods, it has the advantage of not requiring very large samples of data. In essence, this method determines when points in dimension d are neighbors of one another by virtue of their projection onto too low a dimension. [FIGURE 13] Notice how the red and green dots appear as nearest neighbors when projected onto a line in one-dimensional space, but are shown to be much further apart when projected onto a disk in two- dimensional space. By contrast, the red and blue dots are close together in two-dimensional space and continue to appear close together in three- dimensional space. By examining this question in dimension one, then dimension two, etc. until there are no incorrect or false neighbors remaining, one should be able to establish from geometrical considerations alone, a value for the necessary embedding dimension.

Figure: A simplified explanation of False Nearest Neighbors. When the red, green and blue points are projected in one dimension, they appear to be close to one another. When projected in two dimensions, the green point is no longer near the red and blue points. When projected in three dimensions, there is no further change in the relative distances between the points.  

When the percent of false nearest neighbors are plotted as a function of the embedding dimension, two important results emerge. [FIGURE 14] The minimum of the function represents an upper bound on the structure or the dimensionality of the control process. When the COP data from our research are evaluated, the minimum of the function corresponds to an embedding dimension of 3 which remains constant across age. This finding suggests that the dimensionality of the system does not change from one age to the next. The positive slope of the function calculated from that minimum represents the percent of noise in the signal. As a frame of reference for evaluating this effect, We show what happens when measurements from a known dynamical system are corrupted with noise. [FIGURE 15] This figure shows the effect of adding uniform random numbers to a signal from a chaotic system known as the Lorenz attractor. As can be seen, the slope of the function increases as more noise is added.

Figure 14: A prototypical False Nearest Neighbors curve. The percent of false nearest neighbors is plotted against the number of embedding dimensions.  

Figure 15: The effect on False Nearest Neighbors when one adds uniform random noise to a known low dimensional system (the Lorenz attractor).  

Infants' postural sway responses (COP) were analyzed with this False Nearest Neighbor test. [TABLE 3] Recall that the embedding dimension was approximately 3.0 and did not vary with age. [TABLE 4] By contrast, the slope of the function indexing the magnitude of noise in the signal showed a steady decline between 5 and 9 months of age. Similar to previous analyses, the function appears to asymptote beyond 9 months. Interestingly, this measure did not vary as a function of the stimulus condition, suggesting that the stability of the system is the same across different stimulus velocities.

Table 3: Mean false nearest neighbor noise detection slopes by condition.  

Table 4: Mean false nearest neighbor noise detection slopes pooled over all conditions.  

next up previous
Next: Discussion Up: Dynamical analyses of postural Previous: Dynamic Stability of

Steven M. Boker
Thu Aug 17 10:10:02 EDT 1995