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Introduction

As an opening to his lecture, Edward Lorenz, Emeritus professor of meteorology at MIT holds a piece of paper above the stage and gently lets it go, watching it leisurely float down to the ground. Lorenz repeats the experiment, and the paper falls again, landing in a different place on the ground. This serves an illustration of the theory of chaos.

Chaotic (highly sensitive to initial conditions) behaviour of many systems was observed by many researchers for a number of decades, but was first described as such by Lorenz (1963). In 1961 he discovered the manifestation of chaotic behaviour when he was working with computer models of weather prediction. It appeared that the model he was using was extremely sensitive to a small change in one of the parameters - a change from 0.506127 to 0.506 lead to gradual deviation of the original sequence of output to a very different one. This sensitivity dependence on initial conditions is common to chaos theory. Such a small amount of difference in a measurement might be considered experimental noise, background noise, or an inaccuracy of the equipment.

Basics of chaos

One of the important foundations behind the methods of non-linear signal processing and the chaos theory is an embedding theorem (Takens, 1981). It shows that the use of a single measured variable x(n) = x(t0 + nt) with t0 some starting time and t the sampling time, and its time delays provides N-dimensional space that is a proxy for the full multivariate state space for the observed system. The N-dimensional state vectors x(t) are then defined as

where x(t) is a value of the time-series at time t, t is a suitable time delay (sampling time) and N is the embedding dimension. This vector fully represents the non-linear dynamics when N is a large enough. The embedding theorem guarantees that a full knowledge of the behavior over a system is contained in the time series of any one measurement and that a proxy for the full multivariate phase space can be constructed from the single time series. To perform the state space recognition at time delay t and an embedding dimension N are needed.

There exist several methods for estimating time delay and embedded dimension, which are summarized as follows:

Analytical methods for estimating time delay t:

  • autocorrelation and power spectrum functions;
  • average mutual information (AMI) function;
  • degree of separation function;
  • Lyapunov exponents;

Analytical methods for estimating embedded dimension N:

  • false nearest neighbours;
  • bad prediction method;
  • fractal and correlation dimensions;

Empirical methods (for estimating both the time delay and dimension):

  • neural networks;
  • derivative-free global optimization methods, like genetic algorithms

For finding t we use the AMI function, and for finding N - method of false nearest neighbour. The time delay t must be a large enough that independent information about the system is in each component of the vector. However, t must not be so large that the components of the vectors x(t) are independent with respect to each other. Conversely, if the time delay is too short, the vector components will be independent enough and will not contain any new information. A possible rule for good time delay t is to use the first minimum of the average mutual information (Frazer and Swinney 1986). Average mutual information is derived from notions of entropy in communications systems (Shannon 1949). It determines how much information the measurements x(t) at some time have relative to measurements and some other time x(t+t).

The global embedding dimension N is the minimum number of time-delay coordinates needed so that the trajectories x(t) do not intersect in N dimensions. In dimensions less than N, trajectories can intersect because their projected down into too few dimensions. Subsequent calculations, such as predictions, may then be corrupted. If it is too large, noise and other contamination may corrupt other calculations because noise fills any dimension.

The method of finding a proper N can be described using geometrical considerations: as N increases, attractors "unfold" and the vectors that are close in dimension N move to a significant distance apart in N+1. They are "false" neighbours in dimension N. The method of false nearest neighbours (Kennel et al., 1992) measures the percentage of false neighbours as N increases. Points that are close in N are marked and the number of these points that become widely separated in N+1 is calculated.

Having parameters t and N identified and thus phase space reconstructed, one can build the prediction model in a form of multidimensional maps:

where the phase space x(t) is the current state of the system and x(t+T) is the state of the system after a time interval T and fT is a mapping function. The problem is then to find a good expression (local models) for the function fT.

The data is embedded and then divided into training and testing set. Based on the training set, the embedded data space is quantified (using K-NN algorithm at this stage). Local data sets are then constructed for each of the prototype vectors.

Finally, local data models (linear at this stage) are constructed based on the local data sets which are then used to predict the dynamics of the system (move the system from state x(t) into state x(t+T)).

Applications of chaos

Chaos theory is currewntly very widely used in various areas of engineering and even social sciences.

References

Abarbanel H.D.I. (1996). Analysis of observed chaotic data. Springer.

Tsonis, A., A. (1992). Chaos: From Theory to Applications. Plenium Press, New York

Chaos-related tools are in preparation.