Embedding
Those who have knowledge, don't predict. Those who predict,
don't have knowledge. --Lao Tzu, 6th-century B.C. Chinese poet
The pendulum we are looking at in Figure 1 obviously has a phase-space
portrait (a circle) that fits nicely onto a two-dimensional plane. The
reconstructed phase portrait in Figure 5 is also a round curve without
intersections. No doubt this is a valid and useful reconstruction.
But what if the system in the field has more than two variables, and
perhaps their interdependence is not so smooth? While our pendulum
simply goes around and around in phase space, there may be other
mechanisms with three independent variables, needing three dimensions
in their original phase space. Even worse, their portrait may look
more like a roller coaster than a flat circle. During the 1980s, scientists
worked on this more general problem and found methods and justifications
to deal with the situation:
- Can delayed phase space reconstruct the original phase space in
all cases? Yes, but only if some inevitable requirements are met.
There must be significant interaction among variables, otherwise the influence of
a (weak) variable may get lost. Furthermore, a weak interaction may get lost in
the presence of environmental noise.
- How many dimensions must the reconstructed phase space have?
Choosing too low a dimension for delayed phase space can lead to
``flattened'' curves with intersections. Fortunately, topologists
proved a theorem giving an upper bound: if the mechanism
has n independent variables, a
delayed phase space with 2n+1 variables can always avoid intersections.
In most cases, fewer are necessary. This is called the embedding
dimension de (always an integer).
- What are good values for tau and de, yielding a good
reconstruction?
There is no general way of telling in advance which values to use.
However, many algorithms have been developed to judge the quality of
given parameters tau and de. A popular one is the
False Nearest Neighbours algorithm. It looks at a given
delayed-phase portrait, and remembers which points are neighbours.
If these points are no longer close to each other after
increasing de, then the current de looks insufficient.
One of the most common mistakes among practitioners of the 1980s was
using a fixed tau without thinking about it. If the quality of your data
is questionable (because of a bad reconstruction or noise of any origin),
it makes no sense to chase for signs of chaos.
