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GLOSSARY OF TERMS USED IN TIME SERIES ANALYSIS
OF CARDIOVASCULAR DATA
FRACTAL  DIMENSION

Definition of the geometric dimension of an object which includes fractal objects.

For a smooth (i.e., nonfractal) line, an approximate length L(r) is given by the minimum number N of segments of length r needed to cover the line: L(r)=Nr. As r goes to zero, L(r) approaches a finite limit, the length L of the curve. Similarly one can define the area A or the volume V of nonfractal objects as the limit of an integer power law of r:

where the integer exponent is the Euclidean dimension E of the object.

This definition can not be used for fractal objects: as r tends to 0, we enter finer and finer details of the fractal and the product NrE may diverge to infinity. However, a real number D exists so that the limit of NrD stays finite. This exponent is called Hausdorff Dimension DH:

This is not the only definition of dimension proposed for fractal objects. Very popular is the Correlation Dimension D2 :

where C(r) is the number of points which have a smaller (euclidean) distance than a given distance r. This measure is widely used because it is easy to evaluate for experimental data, when the fractal comes from a "dust" of isolated points. A method for measuring D2of strange attractors can be found in (Grassberger and Procaccia, 1983).
D2 may also be used to determine whether a time-series derives from a random process or from a deterministic chaotic system. M-dimensional data vectors are constructed from m measurements spaced equidistant in time, and D2is evaluated for this m-dimensional set of points. If the time-series is a random process, D2 increases with m; if the time-series is a deterministic signal, D2 does not increase further when the embedding dimension m exceeds D2. Thus a plot of the correlation dimension as a function of the embedding dimension may easily shows whether a signal is random noise of deterministic chaos. Note that D2DH

Another way to calculate the fractal dimension is through the estimation of the Hurst's exponent.

References:
Schroeder M. (1991) Fractals, Chaos, Power Laws-Minutes from an Infinite Paradise. Freeman, NY.
Glenny RW et al  (1991) Applications of fractal analysis to physiology  J. Appl. Physiol.
Grassberger P., Procaccia I. (1983) Measuring the strangeness of a strange attractor. Physica D, 9, 189-208.
Schepers HE, van Beek JHGM, Bassingthwaighte JB (1992) Four methods to estimate the fractal dimension from self-affine signals IEEE Engineering in Medicine and Biology, 11, 57-64
Butler GC et al. (1994) Fractal nature of short-term systolic BP and HR variability during lower body negative pressure. Am J Physiol.

(PC, 14 Dec 99)

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