research, “random” (if it doesn’t mean measurement error) indicates unknown
degrees of freedom, this Do> 5 rule is also without relevance for brain research.
D, is called the “information dimension” and is computed by counting the number of e-cubes, M(g), it takes to cover the points constituting some fixed fraction of all of the points of the set of orbital points on the attractor and can be regarded as the “core dimension” (without the outliers) of the set. The counterintuitive finding is that D, is nearly constant across a range of fixed fractions that are less than the
whole measure (Farmer et al, 1983). The invariance of D; can even be taken to the
extreme by computing the D, = lim in (4)
around (typical, not all) single points. In
this context, D; is called the “pointwise dimension” or “singularity exponent” and, as
might be anticipated, its value is usually less than that of Do.
The scaling exponent that is both sensitive to point densities and easiest to compute from real data is the “correlation dimension,” D2 Here, analogous to the relationship between the amplitudes of the variance and the correlation function in
conventional statistics, the measure squared is of interest for the computation of Dz,
M(e) e.g. /(2,¢€)= SLAC, )F (see below for this use of measure u on sum = of cubes C)). i=l
The selection of Dz as the fractal measure dominates the studies that invoke scaling exponents to quantify the distributions of points on the attractor as reconstructed from time series in the neurosciences (Grassberger and Procaccia, 1983; Mayer- Kress, 1986; Ott et al, 1994). Several sets of programs are available for its computation (for example, Sprott and Rowlands, 1991). Generally, a correlation sum (“integral’, R(e) ) is computed from a starting point by counting all subsequent point pairs with distances between them less than ¢ as e—>0 and plotting __ lim In(R(e) 逗>0 Ine)
, . D2 is computed for increasing embedding (and therefore
hypercube) dimension, d, until Dz achieves an asymptotic plateau, it “saturates” (Ding et al, 1993). It is generally the case that Do > D;> Dz (Farmer et al, 1983).
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