Growth of Point Distances in Embedded Space

Suppose you are sitting at the far end of the stadium (the end zone) at ground level during a football game and your depth perception deserts you. That means, looking down the field, you see all the players as if they were distributed along a single line that stretches across the width of the field. In your view, many of the players are very close together. If you measure the distances between them in the view that you have, those distances are small. To see the true spatial distribution of the players, you'd need a two-dimensional view of the field, as if from a helicopter. Most measured distances between players from that view are larger than from your end-on view. The phase space analogy of all this is as follows. Embedding a two-dimensional attractor in one dimension causes many points to appear closer together than they really are. Calculated distances between points in that case are short. Relaxing the embedding by embedding the data in two dimensions lets many points move to their true positions, thereby increasing their distance from one another. Points that seem to be close to a given point in a low dimension and are farther away in the correct embedding dimension are false nearest neighbors.
The same thing happens if we look at a three-dimensional attractor as if we see it only in two dimensions. Say the two dimensions are the length and width of this printed page you are reading and that the third dimension goes from the page to your eyes. The attractor really consists of a group of points between the page and your eyes. Forcing the images of all points of our three-dimensional attractor onto a two-dimensional plane (the page) increases their apparent closeness or density. Again, many points that seem to be close to another point can be false nearest neighbors; their true or desired distance can be much greater.
When the embedding dimension is low (say, two), there isn't enough space for a higher-dimensional attractor to express itself fully, so to speak. Our pseudo phase space plot crams the poor attractor into fewer dimensions than it wants to be in. In that case, computed distances between points tend to be small. Increasing the embedding dimension releases more and more of the attractor's points from the straight-jacket constraint we've imposed (i.e. the low number of embedding dimensions). In other words, they move closer to their real relative locations, even if still partly confined by the embedding dimension. So, distances between points increase with increase in embedding dimension. With further increases in embedding dimension, an attractor gradually assumes its true spatial configuration. Computed distances between points then stop increasing and become constant. That notion is basic to various proposed methods for estimating lag and true embedding dimension.

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