Fractal dimension

In fractal geometry, the fractal dimension, D, is a statistical quantity that gives an indication of how completely a fractal appears to fill space, as one zooms down to finer and finer scales. There are many specific definitions of fractal dimension. The most important theoretical fractal dimensions are the Rényi dimension, the Hausdorff dimension and packing dimension. Practically, the box-counting dimension and correlation dimension are widely used, partly due to their ease of implementation. In a box counting algorithm the number of boxes covering the point set is a power law function of the box size. Fractal dimension is estimated as the exponent of such power law. Although for some classical fractals all these dimensions do coincide, in general they are not equivalent.

One non-trivial example is the fractal dimension of a Koch snowflake. It has a topological dimension, but it is by no means a rectifiable curve: the length of the curve between any two points on the Koch Snowflake is infinite. No small piece of it is line-like, but rather is composed of an infinite number of segments joined at different angles. The fractal dimension of a curve can be explained intuitively thinking of a fractal line as an object too big to be a one-dimensional object, but too thin to be a two-dimensional object. Therefore its dimension might best be described in a sense by the fractal dimension, which is a number between one and two.

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Specific definitions

There are two main approaches to generate a fractal structure. One is growing from a unit object (Fig. 1), and the other is to construct the subsequent divisions of an original structure, like the Sierpinski triangle (Fig.2).[2] Here we follow the second approach to define the dimension of fractal structures (See figure 1).

If we take an object with linear size equal to 1 residing in Euclidean dimension D, and reduce its linear size by the factor 1/l in each spatial direction, it takes N=l^D number of self similar objects to cover the original object(Fig.(1)). However, the dimension defined by

D = \frac{\log N(l)}{\log l}

(where the logarithm can be of any base) is still equal to its topological or Euclidean dimension.[1] By applying the above equation to fractal structure, we can get the dimension of fractal structure (which Hausdorff proved being identical to the Hausdorff dimension[3]) as a non-whole number as expected.

D = \lim_{\epsilon \rightarrow 0} \frac{\log N(\epsilon)}{\log\frac{1}{\epsilon}}

where N(ε) is the number of self-similar structures of linear size ε needed to cover the whole structure.

For instance, the fractal dimension of the Sierpinski triangle (Fig.(2)) is given by,

D = \lim_{\epsilon \rightarrow 0} \frac{\log N(\epsilon)}{\log\left(\frac{1}{\epsilon}\right)} =\lim_{k \rightarrow \infty} \frac{\log3^k}{\log2^k} = \frac{\log 3}{\log 2}\approx 1.585.

Closely related to this is the box-counting dimension, which considers, if the space were divided up into a grid of boxes of size ε, how many boxes of that scale would contain part of the attractor? Again,

D_0 = \lim_{\epsilon \rightarrow 0} \frac{\log N(\epsilon)}{\log\frac{1}{\epsilon}}.

Other dimension quantities include the information dimension, which considers how the average information needed to identify an occupied box scales, as the scale of boxes gets smaller:

D_1 = \lim_{\epsilon \rightarrow 0} \frac{-\langle \log p_\epsilon \rangle}{\log\frac{1}{\epsilon}}

and the correlation dimension, which is perhaps easiest to calculate,

D_2 = \lim_{\epsilon \rightarrow 0, M \rightarrow \infty} \frac{\log (g_\epsilon / M^2)}{\log \epsilon}

where M is the number of points used to generate a representation of the fractal or attractor, and gε is the number of pairs of points closer than ε to each other.

Rényi dimensions

The box-counting, information, and correlation dimensions, can be seen as special cases of a continuous spectrum of generalised or Rényi dimensions of order α, defined by

D_\alpha = \lim_{\epsilon \rightarrow 0} \frac{\frac{1}{1-\alpha}\log(\sum_{i} p_i^\alpha)}{\log\frac{1}{\epsilon}}

where the numerator in the limit is the Rényi entropy of order α. The Rényi dimension with α=0 treats all parts of the support of the attractor equally; but for larger values of α increasing weight in the calculation is given to the parts of the attractor which are visited most frequently.

An attractor for which the Rényi dimensions are not all equal is said to be a multifractal, or to exhibit multifractal structure. This is a signature that different scaling behaviour is occurring in different parts of the attractor.

Estimating the fractal dimension of real-world data

The fractal dimension measures, described above, are derived from fractals which are formally-defined. However, organisms and real-world phenomena exhibit fractal properties (see Fractals in nature), so it can often be useful to characterise the fractal dimension of a set of sampled data. The fractal dimension measures cannot be derived exactly but must be estimated. This is used in a variety of research areas including physics,[4] image analysis,[5][6] acoustics,[7] Riemann zeta zeros[8] and even (electro)chemical processes.[9]

Practical dimension estimates are very sensitive to numerical or experimental noise, and particularly sensitive to limitations on the amount of data. One should be cautious about claims based on fractal dimension estimates, particularly claims of low-dimensional dynamical behaviour — there is an inevitable ceiling, unless very large numbers of data points are presented. However, modelling as a multiplicative cascade leads to estimation of multifractal properties for relatively small datasets [10]: a maximum likelihood fit of a multiplicative cascade to the dataset not only estimates the complete Renyi dimensions, but also gives reasonable estimates of the errors (see the web service [1]).

See also

Notes

  1. ^ a b Fractals & the Fractal Dimension
  2. ^ Vicsek, Tamás (2001). Fluctuations and scaling in biology. Oxford [Oxfordshire]: Oxford University Press. ISBN 0-19-850790-9. 
  3. ^ Mandelbrot, Benoit B. (1982). The fractal geometry of nature. W. H. Freeman Press. pp. 44. ISBN 0-71-671186-9. 
  4. ^ B. Dubuc, J. F. Quiniou, C. Roques-Carmes, C. Tricot, and S. W. Zucker (1989). "Evaluating the fractal dimension of profiles". Phys. Rev. A 39 (3): 1500–12. Bibcode 1989PhRvA..39.1500D. doi:10.1103/PhysRevA.39.1500. 
  5. ^ P. Soille and J.-F. Rivest (1996). "On the validity of fractal dimension measurements in image analysis". Journal of Visual Communication and Image Representation 7 (3): 217–229. doi:10.1006/jvci.1996.0020. http://mdigest.jrc.ec.europa.eu/soille/soille-rivest96.pdf. 
  6. ^ Tolle, C. R., McJunkin, T. R., and Gorisch, D. J. (January 2003). "Suboptimal Minimum Cluster Volume Cover-Based Method for Measuring Fractal Dimension". IEEE Trans. Pattern Anal. Mach. Intell. 25 (1): 32–41. doi:10.1109/TPAMI.2003.1159944. 
  7. ^ P. Maragos and A. Potamianos (1999). "Fractal dimensions of speech sounds: Computation and application to automatic speech recognition". Journal of the Acoustical Society of America 105 (3): 1925–32. Bibcode 1999ASAJ..105.1925M. doi:10.1121/1.426738. PMID 10089613. 
  8. ^ O. Shanker (2006). "Random matrices, generalized zeta functions and self-similarity of zero distributions". J. Phys. A: Math. Gen. 39 (45): 13983–97. Bibcode 2006JPhA...3913983S. doi:10.1088/0305-4470/39/45/008. 
  9. ^ Ali Eftekhari (2004). "Fractal Dimension of Electrochemical Reactions". Journal of the Electrochemical Society 151 (9): E291–6. doi:10.1149/1.1773583. 
  10. ^ A.J. Roberts and A. Cronin (1996). "Unbiased estimation of multi-fractal dimensions of finite data sets". Physica A 233: 867–878. doi:10.1016/S0378-4371(96)00165-3. 

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