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# The correlation integral

We start with a brief description of the correlation integral method, since it is the basis of our maximum likelihood approach. To characterize a dynamical system from a time series , first the phase space of the system is reconstructed with the method of time delayed coordinates [Takens, 1981,Packard et al., 1980], i.e. with vectors
 (1)

where is the sample time, is the embedding dimension and is an appropriate lag. The correlation integral is defined as
 (2)

where denotes a norm. It is assumed that distances between two randomly chosen points in phase space obey the cumulative probability distribution function [Grassberger and Procaccia, 1983b]
 (3)

and that for sufficiently small and large . We also assume that and do not depend on (see, however, Theiler, 1988) and that does not depend on the embedding dimension. The correlation dimension is now given by [Eckmann and Ruelle, 1985]
 (4)

if is large enough [Takens, 1981]. The correlation entropy is given by
 (5)

With experimental data, a scaling region'' must be identified, for which the distribution (3) is valid. The dimension can be estimated by the slope of a straight line through a number of points in the scaling region. The entropy can be estimated by observing the behaviour of as a function of but using equation (5) directly may result in slow convergence. Therefore one usually studies the quotient of two correlation integrals, at embedding dimensions and [Grassberger and Procaccia, 1983a] :

 (6)

We will now show how to obtain maximum likelihood estimators for both and , together with their asymptotic variances, under the assumption that the finite number of distances is the only source of error.

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