Fractal dimension computation from equal mass partitions

Abstract

While the numerical methods which utilizes partitions of equal-size, including the box-counting method, remain the most popular choice for computing the generalized dimension of multifractal sets, two massoriented methods are investigated by applying them to the one-dimensional generalized Cantor set. We show that both mass-oriented methods generate relatively good results for generalized dimensions for important cases where the box-counting method is known to fail. Both the strengths and limitations of the methods are also discussed.Fractal sets are characterized by self-similarity, and power laws can be associated with them. Examples of fractals in nature are ubiquitous. Their discovery led to the extension of the notion of dimension. For monofractals, the scaling pattern is homogeneous while it varies over the set for multifractals. By introducing the generalized dimension D q , not only a non-integer dimension can be assigned to a set, but also a spectrum of dimensions can be attributed to a single set if the set is a multifractal. In finding the generalized dimensions, the box-counting method has been by far the most popular choice among researchers across various fields. However, it is known that the class of methods which deal with partitions of equal size, including the box-counting method, is illsuited for computing the generalized dimensions on some domain of q. In this paper, two promising methods which utilize mass-oriented partitions, rather than partitions of equal-size, are investigated.

Cosmology in one dimension: Vlasov dynamics

“…Nevertheless, these preliminary results, which are potentially sensitive to the numerical resolution of the code, would require more studies to be fully confirmed. Other approaches based on equal mass partitions [36] may provide further information on the low-density regions. to ω −1 J (standard scaling with β = 1) for an N-body simulation with initial condition similar to that used for the Vlasov case.…”

Cosmology in one dimension: A two-component model

“…Instead, we applied mass-oriented methods, all of which may be considered variants of the method originally proposed by van de Water and Schram [18]. In general, mass-oriented methods have advantages over size-oriented methods on low-density regions and are therefore more suitable for studying void structure formation [19]. Mass-oriented methods are based on the statistics of the k th nearest distances from reference points sampled from a given set.…”

“…When the value of k is fixed with some small integer, the method is often referred to as the near neighbor method in the literature with a special case, called the nearest neighbor method, for k = 1 [20]. The near-neighbor method is also known to be problematic when estimating the generalized dimensions in the positive range of q, or equivalently, the Dimension Function in the negative range of γ [19]. The theoretical range where the singularities exist is for k < γ D(γ) .…”

“…Here, following Broggi, we also consider values of k up to 1000, i.e., a hundredth of the number of test particles in the system. If a set is a fractal on all scales, the associated dimensions should be independent of the choice of the value of k. In dynamics, however, a simulated set often exhibits scale-dependent properties, meaning that the associated dimensions vary with the value of k. Instead of fixing the value of k, we can also fix the number of the reference points and study how scaling arise when increasing the value of k. From the earlier studies on mass oriented methods, this k-neighbor approach is known to work on the entire range of the spectrum but the results are highly sensitive to the choice of a scaling range [19]. If a given set has scale-dependent properties, multiple scaling ranges may exist.…”

Cosmology in one dimension: Symmetry role in dynamics, mass oriented approaches to fractal analysis

“…The second approach is realized in the near-neighbor [22] and k-neighbor [23] methods and has the advantage of only including occupied cells in the partition. Recently, to gain insight concerning their useful regimes, we have applied these methods to standard sets with well-characterized fractal properties [24].…”