The application of Lyapunov exponents to the prediction of time series

Monika Miśkiewicz
Prace naukowe Akademii Ekonomicznej we Wrocławiu Nr 1189 / 2007
Summary (according to author)
“The paper presents the new method of chaotic series prediction. This method is based on a basic characteristic of chaotic systems which is sensitive dependence upon initial conditions (SDUIC). This figures refers to as Lyapunov exponents are a measure of the SDUIC. They measure the rate of the divergence of trajectories in state space. The method involves, first the reconstruction of the phase space using a time series and then the prediction of unknown phase space point making use of Lyapunov exponents as a qualitative parameter. As a result of the transformation of this phase space point the predicted time series data can be obtained. Numerical examples have proved that the method is effective.”
Commentary.
Studying the numerical examples it is hard to agree that the method is really effective. Without even understanding the mathematical equations behind the method it is visible how few numbers are predicted with a tolerable deviation. It would be surprising if it were to the contrary, so that the prediction were accurate. Then we would have an effective method of predicting the currency rates. Unfortunately we haven’t worked out such a method yet.
It is not easy to judge the applied method without referring to detailed mathematical and strictly formal description. It also requires very comprehensive knowledge on Lyapunov exponents and various trials of its application. It looks that author possess such a knowledge and can use it with impressive skill. It looks however that the problems lies in the assumptions. Lyapunov exponents can be applied to chaotic systems which are deterministic. In other words, there are equations which describes the movements of elements within the phase space, even if we don’t know them yet. This is a big advantage of this method of instability measuring comparing to e.g. metric entropy of Kolmogorov. Simply studying the series of data we can firstly find an attractor (assuming that one exists) and then estimate the Lyapunov exponent and predict the subsequent series. However if there is no equation behind the series of data, and if there is no attractor the results of predictions will be highly accidental.

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