LJ 75: Detecting Chaos in the Field

Determinism, Prediction and Filtering

Prediction is very difficult, especially of the future. --Niels Bohr, 1885-1962

How do we know whether our recorded signal is a deterministic signal or just noise? In the case of our recorded sine wave, the orderly behaviour is obvious from the graph over time (Figure 3). In the general case, it is easier to discover determinism in phase space (Figure 1). Look at a specific point in phase space, for example in Figure 2. Determinism means if the swinging system ever returns to the vicinity of this point, we can tell the next step in advance. Given the same circumstances again, the pendulum is determined to advance in the same way. A nondeterministic system may hop around like mad.

This understanding of deterministic behaviour rests on some assumptions:

Precise measurement: but, our measurements are always noisy. Taking the average of the three motions in Figure 2 will improve the quality of prediction significantly.
Adequate phase space: if the next step of the system depends on an additional variable (like total energy), the direction of progress might not be uniquely determined. This corresponds to a curve which could not be unfolded due to an insufficient reconstruction of phase space.
Locality: this simple scheme works only if our measurement uses a fine resolution in time and space. Small steps in time and space tend to be locally bounded, allowing for easier detection of deterministic behaviour.

Prediction is always error-prone, and it makes sense to calculate the extent of failure. A comparison of deviation to the signal's range is appropriate. Averaging all the local prediction errors produces a robust quantity for the estimation of the quality called determinism.

Building upon this reasoning, there is a wide range of applications. Adaptive linear predictive filtering is a very important paradigm in contemporary digital signal processing. Noise removal, speech and image compression and feature extraction are examples. Nonlinear generalizations of these applications are now feasible.