In order to answer these questions, it is necessary to separate the signal from the noise, and then determine the number of parameters or dimensions contributing to the signal. One approach is to attempt to reconstruct the phase space of the system from the scalar values of the time series. The sequence of allowed states is plotted as a trajectory parameterized over time. If the dynamics governing this process is low-dimensional, then the trajectory will converge to an attractor, such as a point or limit cycle attractor.
Let me illustrate with some examples. A spring that is stretched will always return to the same state after a brief period of oscillation. [FIGURE 5] This example of a point attractor is represented by the following vector field. As can be seen, the position of the spring will return to the same state or equilibrium point no matter the direction or distance that it is stretched. [FIGURE 6] It is equivalent to dropping a ball onto the depicted convex surface and observing that it will eventually come to rest in the deepest location of the surface.
Figure 5: Vector field representation of a point attractor.
Figure 6: Surface representation of a point attractor.
Other physical systems, such as a forced pendulum, do not converge to a single state, but instead converge to a limit cycle of allowable states. [FIGURE 7] The next figure shows a vector field of initial states that all converge toward the limit cycle depicted as an elliptical orbit. [FIGURE 8] When a ball is dropped onto a surface conforming to this vector field, it will eventually enter the elliptical depression and continue to orbit along the path constrained to that basin.
Figure 7: Vector field representation of a limit cycle attractor.
Figure 8: Surface representation of a limit cycle attractor.
When the time series involving COP is plotted in phase space, it is apparent that the trajectory does not converge to a stable orbit. [FIGURE 9] Most notably, the trajectory continues to cross over itself. These crossings arise by virtue of projecting the attractor onto a too low dimensional space. [FIGURE 10, 11] What happens when the attractor is plotted in three-dimensions? As you can see, it becomes more difficult to determine by eye whether or not the structure of the orbit is stable. -- The answer depends on finding the correct perspective for viewing the attractor. Of course, it becomes even more difficult to evaluate the dimensionality of the system once you move beyond three dimensions.
Figure: Trajectory of COP plotted in 2-D phase space for one subject.
Figure: Trajectory of COP plotted in 3-D phase space for the same subject.
Figure: Trajectory of COP plotted in 3-D phase space for the same subject as seen from a different angle.
Takens proved that it is possible to reconstruct an attractor in state space by substituting time lagged observations for higher-order derivatives [8,7]. As such, each vector is constructed from a series of time-lagged observations. [TABLE 2] For example, a four dimensional embedding of a time series is constructed from the following data. Each row represents a single n- dimensional observation. Each column represents the next component of the vector.
Table 2: A four dimensional embedding of the time series