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.