Pernicious effect of physical cutoffs in fractal analysis

Scientific Reports, 2017

Fractal behavior is scale-invariant and widely characterized by fractal dimension. However, the correspondence between them is that fractal behavior uniquely determines a fractal dimension while a fractal dimension can be related to many possible fractal behaviors. Therefore, fractal behavior is independent of the fractal generator and its geometries, spatial pattern, and statistical properties in addition to scale. To mathematically describe fractal behavior, we propose a novel concept of fractal topography defined by two scale-invariant parameters, scaling lacunarity (P) and scaling coverage (F). The scaling lacunarity is defined as the scale ratio between two successive fractal generators, whereas the scaling coverage is defined as the number ratio between them. Consequently, a strictly scale-invariant definition for self-similar fractals can be derived as D = log F /log P. To reflect the direction-dependence of fractal behaviors, we introduce another parameter H xy , a general Hurst exponent, which is analytically expressed by H xy = log P x /log P y where P x and P y are the scaling lacunarities in the x and y directions, respectively. Thus, a unified definition of fractal dimension is proposed for arbitrary self-similar and self-affine fractals by averaging the fractal dimensions of all directions in a d-dimensional space, which D H d F P = ∑ (/)log /log i d xi x =1. Our definitions provide a theoretical, mechanistic basis for understanding the essentials of the scale-invariant property that reduces the complexity of modeling fractals.

Physica A: Statistical Mechanics and its Applications, 1998

The multifractal concept was introduced in the 1980s by Mandelbrot. This theory arose from the analysis of complex and=or discontinuous objects. In this study, we analyzed the data scatter obtained by a modiÿed box-counting method. Considering the curved shape of the data scatter, it is noticeable that there is more than one slope corresponding to di erent fractal behavior of an object. In this work, to discriminate di erent fractal dimensions from data scatter obtained by box counting, we suggest a rigorous selection of data points. The results show that large "'s usually characterize the embedding surface of the whole object and that small "'s approximate the dimension of the substructure for discontinuous objects. They also show that a dimension can be associated with a density distribution of singularities.

Symposium - International Astronomical Union, 2005

The debate about the possible smoothness of the Universe on large scales as opposed to an unbounded fractal hierarchy has been the subject of increasing interest in recent years. The controversy arises as a consequence of different statistical analyses performed on surveys of galaxy redshifts. I review the observational evidence supporting the idea that a gradual transition occurs in the galaxy distribution: from a fractal regime at small scales to large scale homogeneity.

Nonlinear Dynamics, 2019

In this work, we reported there are two kinds of independent complexities in mono-scale-invariance, namely to be behavioral complexity determined by fractal behavior and original one wrapped in scaling object. Quantitative characterization of the complexities is fundamentally important for mechanism exploration and essential understanding of nonlinear systems. Recently, a new concept of fractal topography (Ω) was emerged to define the fractal behavior in self-similarities by scaling lacunarity (P) and scaling coverage (F) as Ω(P, F). It is, however, a “special fractal topography” because fractals are rarely deterministic and always appear stochastic, heterogeneous, and even anisotropic. For that, we reviewed a novel concept of “general fractal topography”, Ω(P, F). In Ω, P is generalized to a set accounting for directiondependent scaling behaviors of the scaling object G, while F is extended to consider stochastic and heterogeneous effects. And then, we decomposed G into G(G+, G−) to ease type controlling and measurement quantification, with G+ wrapping the original complexity while G− enclosing behavioral complexity. Together with Ω and G, a mathematical model F3S (Ω,G) was then established to unify the definition of deterministic or statistical, self-similar or selfaffine, single- or multi-phase properties. For demonstration, algorithms are developed to model natural scale-invariances. Our investigations indicated that the general fractal topography is an open mathematic framework which can admit most complex scaling objects and fractal behaviors.

Recoletos Multidisciplinary Research Journal, 2013

The natural world is recognized to be fractal: from the growth of a leaf to how the trees propagate to form a forest, the neural networks, the DNA and the galaxies in space. The patterns and geometry that nature creates seem familiar and predetermined; actually it is random and unpredictable. It is this characteristic that made fractals a convenient method for such studies. Insights into the unpredictability could be a key in understanding the natural random events, like earthquakes, typhoons and other more subtle, natural occurrence like growths and cell developments. Data in different studies have always been thought of as normally distributed; however, recent investigations found that most of these statistical data are non-normal, and in a lot of cases are found to be fractal. According to Padua et al 2013, statistical fractal observations are random observations that possess stochastically self-similar patterns at various scale. In this light, this paper aims to illustrate the pervasiveness of statistical fractal observations in real-life by examining old data sets that used to be modeled in the framework of normal distribution theory. Samples of such observations will be presented in this paper.

Journal of Statistical Physics, 1995

These nine essays provide succinct yet mostly accessible summaries of research at the frontier in fractal mathematics as applied to medicine, biology, ecology, polymers, condensed matter physics, and surface science. In these fields scaling concepts are particularly effective in quantifying and describing inherently complex phenomena. This second volume of a series highlighting applications of fractals in science continues the practice introduced in the first volume, Fractals and Disordered Systems (A. Bunde and S. Havlin, Springer-Verlag, Berlin, 1991) of providing an introductory chapter by the editors, some background material and literature citations within each chapter, plus relevant cross-references to other chapters in this and the first volume of the series. Notation for the various fractal dimensions is consistent from chapter to chapter and also with that found in the first volume.

1996

The authors first introduce the basic concepts of fractal geometry and the methods of correlation analysis. The main body of the paper discusses extensively the application of fractal analysis to galaxy redshift surveys and the derived three-dimensional distribution of galaxies in the universe. The authors consider at length the important issue that the scale of the largest inhomogeneities is comparable with the extent of the surveys in which they are detected. These surveys probe scales from 100 h-1-200 h-1 Mpc for wide angle surveys ...

Geophysical Journal International, 2000

The box-counting algorithm is the most commonly used method for evaluating the fractal dimension D of natural images. However, its application may easily lead to erroneous results. In a previous paper (Gonzato et al. 1998) we demonstrated that a crucial bias is introduced by insu¤cient sampling and/or by uncritical application of the regression technique. This bias turns out to be common in many practical applications. Here it is shown that an equally important additional bias is introduced by the orientation, placement and length of the digitized object relative to that of the initial box. Some additional problems are introduced by objects containing unconnected parts, since the discontinuities may or may not be indicative of a fractal pattern. Last, but certainly not least in magnitude, the thickness of the digitized pro¢le, which is implicitly controlled by the scanner resolution versus the image line thickness, plays a fundamental role. All of these factors combined introduce systematic errors in determining D, the magnitudes of which are found to exceed 50 per cent in some cases, crucially a¡ecting classi¢cation. To study these errors and minimize them, a program that accounts for image digitization, zooming and automatic box counting has been developed and tested on images of known dimension. The code automatically extracts the unconnected parts from a digitized shape given as input, zooms each part as optimally as possible, and performs the box-counting algorithm on a virtual screen. The size of the screen can be set to meet the sampling requirement needed to produce stable and reliable results. However, this code does not provide image vectorization, which must be performed prior to running this program. A number of image vectorizing codes are available that successfully reduce the thickness of the image parts to one pixel. Image vectorization applied prior to the application of our code reduces the sampling bias for objects with known fractal dimension to around 10^20 per cent. Since this bias is always positive, this e¡ect can be readily compensated by a multiplying factor, and estimates of the fractal dimension accurate to about 10 per cent are e¡ectively possible.

Physica A: Statistical Mechanics and its Applications, 2002

We present a generalized stochastic Cantor set by means of a simple cut and delete process and discuss the self-similar properties of the arising geometric structure. To increase the flexibility of the model, two free parameters, m and b, are introduced which tune the relative strength of the two processes and the degree of randomness respectively. In doing so, we have identified a new set with a wide spectrum of subsets produced by tuning either m or b. Measuring the size of the resulting set in terms of fractal dimension, we show that the fractal dimension increases with increasing order and reaches its maximum value when the randomness is completely ceased.