Industrial Applications of Genetic Algorithms:Using Genetic Operators to Distinguish Chaotic Behavior from Noise in a Time Series

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## Chapter 13
Using Genetic Operators to Distinguish Chaotic Behavior from Noise in a Time Series

John Nilson

AT&T Corporation
600 Mountain Avenue
Murray Hill, NJ 07974
email: jvn@insight.att.com

ABSTRACT

In this study, genetic operators are used in reproducing an important study that distinguishes chaotic behavior from noise in a time series. Distinguishing chaotic behavior from noise is accomplished by predicting future time values from current time values. With chaotic behavior, the prediction accuracy declines as the time horizon increases while with noise, the accuracy essentially remains the same. The results are important for analysts in order to apply appropriate methodologies to nonlinear dynamics. It is especially useful to forecasters who use time series as part of their methodology.

INTRODUCTION

According to G. Sugihara and R. May, authors of an important article entitled: Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series (Nature, vol. 344, April 19, 1990, pages 734-741), there are two sources of uncertainty in forecasting the behavior of dynamical systems: 1) the error associated with measurement (e.g., sample size error, unpredictable environments), which is often called noise, and 2) the complexity of the system itself, which often exhibits chaotic behavior. Both of these sources of uncertainty appear as randomness in time series.

If an analyst could understand which of the two sources of uncertainty he/she was dealing with, he/she could take appropriate steps to improve the analysis. For example, if the source were identified as noise, then the analyst might choose a smoothing or a filtering technique as part of the methodology. If, however, the source were due to the inherent nature of the system, then the analyst might approach the analysis in a different manner. He/she might use nonlinear techniques such as neural networks, fuzzy expert systems, or genetic algorithms as the basis of their methodology. Thus the ability to distinguish between the two sources can be important in the approach and in the methodology of the analysis.

Some attempts have been made in the past to distinguish the two sources (8) but these have had mixed results and require large sample sizes (10,000 to 20,000 sample points). The study by Sugihara and May uses a non-parametric approach in order to reduce the required number of required sample points. A standard technique, called Time Delay Embedding, is used to create a geometric space using the time series values as components. The “smallest” simplex is then formed by using the nearest neighbors (of the current point in the series) as vertices. The simplex is projected into the future by keeping track of where the vertices end up after n number of time steps. By applying the weights established in the original simplex (calculated using the current point in relation to the vertices) to the projected simplex, they obtain a forecast of the current point.

The study is important because, as a non-parametric technique, there are no underlying assumptions about distributions or population parameters. It allows for the central thesis of the study: that forecasts of time series with measurement error exhibit a high level of noise even as the forecast time horizon increases. The accuracy of forecasts then tend to be constant for all time horizons. Forecasts of time series with chaotic behavior, on the other hand, get worse as the time horizon increases. The accuracy of the forecasts then tends to fall off as the time horizon increases.

PROBLEM

In the study, an extensive search is made in determining the “smallest” simplex surrounding the current point. This involves a lengthy search for the exact set of points surrounding the current point comprising the “smallest” simplex. A key parameter for this is the embedding dimension. A simplex in an embedding dimension of three requires four points (four vertices), while a simplex in an embedding dimension of five requires six points (six vertices). Determining the exact set of four points (in an embedding dimension of three) out of say 1000 points requires an exhaustive number of searches using combinations of 1000 points, taken four at a time. For larger series, say, 10,000 points the cost increases at an exponential rate. (The combination of 10,000 points taken four at a time is more than ten times the combination of 1000 points taken four at a time.)

The approach presented uses genetic reproduction operators to replace the determination of the “smallest” simplex. Crossover and mutation operators are used after the projected simplex has been determined. It is anticipated that the application of genetic operators will duplicate the results of the original study, while eliminating the extensive search costs.

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