128.

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		2.5.15— New Methods: Surrogate Data Set

		The errors in evaluating the fractal dimension of the phase space set should be similar when similar data sets are analyzed. Instead of trying to determine the absolute value of the fractal dimension, we can determine how it changes when we change the data. For example, we can scramble the original data to remove any deterministic relationship that might be present. The scrambled version of the original time series is called the surrogate data set. If we find that the dimension of the phase space set of the surrogate data set is different from that of the original data, then that difference must be due to the fact that we removed a deterministic relationship that was present in the original data.

		The original data can be scrambled in different ways to generate the surrogate data set. For example, the surrogate data set can consist of the same values of the original time series in a random order. This scrambling may be too severe because consecutive values of the surrogate data set are uncorrelated. Theiler suggested that a better surrogate data set might be one that has the same first order correlations as those in the original time series but where all the higher order correlations have been removed. This can be generated by randomizing the phases of the Fourier components of the original time series.

		1— Random: Original and Its Surrogate Are the Same

		A surrogate time series with the same first order correlations but no higher order correlations is generated from the original time series. If the fractal dimension of the phase space set of the original time series is the same as that of its surrogate, then no deterministic relationship was present in the original time series.

		2— Deterministic: Original and Its Surrogate Are Different

		If the fractal dimension of the phase space set of the original time series is less than that of its surrogate, then the scrambling used to generate the surrogate removed a deterministic relationship that was present in the original time series.

 

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