PRODUCTION RUNS

The same two time series used by Sugihara and May were used in this study. The two time series were:

1.  a tent map with parameter value 2, which is known to generate a chaotic time series [x _t+1 =2x _t for 0<x<.5, x _t+1 =2-2x _t for .5<x<1].

2.  a modified sine map with 50% noise [x_t = sin(.5t) + N(-.5,.5)].

For each of the above maps, about 1000 time series values were generated, representing 500 pattern and 500 test points. Vectors for both of these were created using the time delay methodology described in step 2 of the section entitled: Solution Formulation. Twelve runs were made for each map, each run representing one of the twelve time steps under review. Each run produced about 500 pairs of numbers: one for the result of the fitness function (projecting the test vector into the future) and one for the true value (obtained directly from the time series). The correlation coefficient was then calculated for each run (of about 500 pairs). The results are plotted in Chart 1, with similar results from the original study by Sugihara and May plotted in Chart 2. (Note: the horizontal axis of each chart represents an increasing number of time steps, while the vertical axis represents the correlation coefficient.)

RESULTS

Comparison of the two charts demonstrates that the application of genetic operators and the fitness function were successful in distinguishing chaotic behavior from noise. Both charts contain two graphs which behave in similar ways:

a.  they both contain a graph (Series 1) that is fairly constant in the horizontal direction. This graph represents the results of using noisy data (i.e., the sine map with 50% noise). The reason for the fairly constant graph is that the noise is independent of the time horizon. That is, it basically has the same value across the time different time steps.

b.  they both contain a graph (Series 2) that initially has a relatively higher correlation coefficient that decreases as the time horizon increases. This graph represents the results of using chaotic data (i.e., the tent map). The fall off of accuracy (in this case the correlation coefficient) is characteristic of chaotic systems. The technical term for this drop in accuracy is called “sensitive dependence on initial conditions” and means that two trajectories that are initially close become very distant over time. In the runs made, the two trajectories were represented by the projected test reults (from the fitness functions) and the actual results (obtained directly from the time series). When the two trajectories were close (at time step 1), the results showed a relatively higher correlation. As the time steps grew larger, the two trajectories separated more, resulting in a lower correlation between them.

Thus, the implementation of genetic operations was successful in obtaining the same characteristic results as obtained by Sugihara and May.

For comparison, the results obtained by Sugihara and May are plotted in Chart 2.

There is, however, one important difference between the two charts. The results obtained by Sugihara and May show significantly higher levels of correlation coefficients for both maps than do the results using genetic operators. Several attempts to improve the correlation coefficients using genetic operators were made, but no appreciable improvements were obtained. These attempts included changing the parameter values of the probabilities of crossover and mutation, varying the crossover points, experimenting with numerous new operators, and increasing the embedding dimension values. In particular, experiments using a shorter coding scheme did not appear to improve the level of the correlation coefficient.

CONCLUSION

Although numerous attempts were made to improve the correlation, the results may suggest that the benefits in using genetic operators to distinguish chaotic behavior from noise (i.e., significantly reduced search time) may come at a cost - accuracy. In some instances, this may not be acceptable. For example, when deciding to use filtering techniques, fuzzy expert systems, genetic algorithms, or some other approach to the analysis, the analyst might feel more comfortable with a higher level of accuracy. However, in other instances, it may be quite acceptable to work with a lower level of correlation. This could happen if the analyst wanted to obtain a quick understanding of the behavior of the underlying system.

REFERENCES

1  Sugihara, G. & May, R. (1990). Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series. Nature, 344, April 19, 734-741.

2  Sauer, T. (1992). Time series prediction by using delay coordinate embedding, Non Linear Modeling and Forecasting, edited by Casdagli and Ewbank, Sante Fe Institute, 175-195.

3  Pineda, F. & Sommerer, J. (1992). Estimating generalized dimensions and choosing time delays: A fast algorithm, Non Linear Modeling and Forecasting, edited by Casdagli and Ewbank, Sante Fe Institute, 367-387.

4  Kantz, H. (1992). Noise reduction by local reconstruction of the dynamics, Non Linear Modeling and Forecasting, edited by Casdagli and Ewbank, Sante Fe Institute, 475-491.

5  Fraser, A. & Dimitriadis, A. (1992). Forecasting probability densities by using hidden Markov models, Non Linear Modeling and Forecasting, edited by Casdagli and Ewbank, Sante Fe Institute, 265-283.

6  Gershenfeld, N. & Weigend, A. (1992). The future of time series: Learning and understanding, Non Linear Modeling and Forecasting, edited by Casdagli and Ewbank, Sante Fe Institute, 1-71.

7  Weigend, D. & Rumelhart, G. (1992). Predicting sunspots and exchange rates with connectionist networks. Non Linear Modeling and Forecasting, edited by Casdagli and Ewbank, Sante Fe Institute, 395-432.¹

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¹This article felt that the simplex projection method relied too heavily on the embedding dimension. Their approach to prediction is one that is not as sensitive to the embedding dimension. Theirs has “an advantage over other prediction methods such as the simplex algorithm employed by Sugihara and May,” (page 416).

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8  Grassberger, P. and Proccacia, I. (1983). Physical review of letters. 50, 346-369