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Generalizations

All theory of the previous sections is based on the probability of the distance between two points in phase space, which depends on the correlation dimension $D_{2}$ or $\nu$. By taking the maximum of distances between $q$-couples of points [Ellner, 1988,Grassberger and Procaccia, 1984], the $D_{q}$ for integer values of $q$ can be estimated, but this requires a much larger number of points. It could be that the method of generalizing the correlation integral as described in Chapter 4 can be used to derive maximum likelihood estimators of $D_q$. Furthermore, Göttsche and Huber [Göttsche and Huber, ] estimate the $f(\nu)$ spectrum of singularities [Halsey et al., 1986], from which the $D_{q}$ can be computed.



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