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		--> -->2.5.11— Problems: The Lag Dt

		When the phase space set is constructed from the time series of one measured variable X(t), then the coordinates of each point in the N-dimensional phase space are X(t), X(t+Dt), X(t+2Dt), . . . . X(t+(N-1) Dt). The properties of the phase space set depend on the value used for the lag Dt. This can be seen by constructing the 2-dimensional phase space from the variable X(t) of the Lorenz system.

		When the lag Dt is too small, then the value of X(t+Dt) is almost the same as the value of X(t). Thus the phase space set lies along the diagonal X(t+Dt) = X(t). The phase space set is l-dimensional. Since the fractal dimension of the phase space set is low, you would conclude that the time series X(t) was generated by a deterministic mechanism.

		When the lag Dt is too large, because of the sensitivity to initial conditions, the value of X(t+Dt) is uncorrelated with the value of X(t). Thus the phase space set fills the phase space and is therefore high dimensional. Since, the fractal dimension of the phase space set is high, you would conclude that the time series X(t) was generated by a random mechanism.

		It is only when the lag Dt is just right, that the phase space set has the real form of the attractor of the Lorenz system and the real fractal dimension of approximately 2.03.

		There are a number of excellent methods to determine Dt in principle. These methods are based on finding the time scale Dt of the correlations between X(t) and X(t+Dt) using the autocorrelation or mutual information functions. These methods were developed and tested on data generated by differential equations where there is a unique correlation time scale. However, in practice, experimental data often have more than one correlation time scale. The fractal dimension of the phase space set will depend on which time scale is used for the lag Dt. Different values of the lag will produce phase space sets with different fractal dimensions. There is no way to choose which is the "real" time scale and its associated lag Dt, that gives the "real" value of the fractal dimension of the phase space set.

 

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