Discussion of the Two Methods
According to the analysis of Dubuc et al. (Dubuc et al. 1989), the Box-counting dimension and the Minkowski-Bouligand dimension are mathematically equivalent
in limit thus, Db = Dm. However, they are completely different in practice because of the way that limits are taken, and the manner in which they approach zero.
Experimental results published in the literature (Dubuc et al. 1989; Maragos and Sun 1993; Zeng et al. 2001) showed that these two methods suffer from inaccuracy and uncertainty. Indeed, according to Zeng et al. (2001) the precision of these estimators are mainly related to the following aspects:
- Real Value of the Fractal Dimension D: With big values of D, the estimation error is always very high. This can be explained by the effect of resolution (Huang et al. 1994). When the value of D increases, its estimates can not reflect the roughness of the object and higher resolution is then needed.
- Resolution: In the case of the temporal curves, the resolution consists of observation size of the curve (minute, hour, day...). According to Tricot et al. (1988) estimated fractal dimension decreases with the step of observation. This is due to the fact that a curve tends to become a horizontal line segment and appears more regular.
- Effect of Theoretical Approximations: Imprecision of the Box-counting and the Minkowski-Bouligand methods is also related to constraints occurring in theoretical approximations of these estimators. For example, the Box-counting dimension causes jumps on the log-log plots (Dubuc et al. 1989) which generate dispersion of the points of the log-log plots with respect to the straight line obtained by linear regression. Moreover, the value of N (є) must be integer in this method. The inaccuracy of the method of Minkowski-Bouligand is due to the fact that the Minkowski covering is too thick.
- Choice of the Interval [є0, єтax]: The precision of the estimators is influenced much by the choice of the interval [є0, єтах] through which the line of the log-log plots is adjusted. є0 is the minimum value that can be assigned to the step. When є0 is too large, the curve is covered per few elements (limp or balls). Conversely, when the value єтах is too small, the number of elements which cover the curve is too large and each element covers few points or pixels. Some researchers tried to choose this “optimal” interval in order to minimize the error in estimation (Dubuc et al. 1989; Huang et al. 1994). For example, Liebovitch and Toth (1989) proposed a method for determining this interval, Maragos and Sun (1993) used an empirical rule to determine єтах for temporal signals. In practice, these optimal intervals improve considerably the precision of the fractal dimension estimate for special cases but not in all cases.