Randomness and Apparent Fractality

Critical Reviews in Biomedical Engineering, 2012

Complexity is maybe one of the less understood concepts, even within the scientific community. Recent theoretical and experimental advances, however, based on the close relationship between the complexity of the system and the presence of 1/f fluctuations in its macroscopic behavior, have opened new domains of investigation, which consider fundamental questions as well as more applied perspectives. These approaches allow a better understanding of how essential macroscopic functions could emerge from complex interactive networks. In this review we present the current state of the theoretical debate about the origins of 1/f fluctuations, with a special focus on recent hypotheses that establish a direct link between complexity and fractal fluctuations, and clarify some lines of opposition, especially between idiosyncratic vs nomothetic conceptions, and global vs componential approaches. Finally, we discuss the deep questioning that this approach can generate with regard to current theories of motor control and psychological processes, and some future developments which may be evoked, especially in the domain of physical medicine and rehabilitation.

Physical Review E, 1997

Fractal structures appear in a vast range of physical systems. A literature survey including all experimental papers on fractals which appeared in the six Physical Review journals (A-E and Letters) during the 1990's shows that experimental reports of fractal behavior are typically based on a scaling range ∆ which spans only 0.5 -2 decades. This range is limited by upper and lower cutoffs either because further data is not accessible or due to crossover bends. Focusing on spatial fractals, a classification is proposed into (a) aggregation; (b) porous media; (c) surfaces and fronts; (d) fracture and (e) critical phenomena. Most of these systems, [except for class (e)] involve processes far from thermal equilibrium. The fact that for self similar fractals [in contrast to the self affine fractals of class (c)] there are hardly any exceptions to the finding of ∆ ≤ 2 decades, raises the possibility that the cutoffs are due to intrinsic properties of the measured systems rather than the specific experimental conditions and apparatus. To examine the origin of the limited range we focus on a class of aggregation systems. In these systems a molecular beam is deposited on a surface, giving rise to nucleation and growth of diffusion-limited-aggregation-like clusters. Scaling arguments are used to show that the required duration of the deposition experiment increases exponentially with ∆. Furthermore, using realistic parameters for surfaces such as Al it is shown that these considerations limit the range of fractal behavior to less than two decades in agreement with the experimental findings. It is conjectured that related kinetic mechanisms that limit the scaling range are common in other nonequilibrium processes which generate spatial fractals. * URL: http://shum.cc.huji.ac.il∼malcai.

Monthly Notices of the Royal Astronomical Society, 2008

Homogeneity and isotropy of the universe at sufficiently large scales is a fundamental premise on which modern cosmology is based. Fractal dimensions of matter distribution are a parameter that can be used to test the hypothesis of homogeneity. In this method, galaxies are used as tracers of the distribution of matter and samples derived from various galaxy redshift surveys have been used to determine the scale of homogeneity in the Universe. Ideally, for homogeneity, the distribution should be a monofractal with the fractal dimension equal to the ambient dimension. While this ideal definition is true for infinitely large point sets, this may not be realized as in practice, we have only a finite point set. The correct benchmark for realistic data sets is a homogeneous distribution of a finite number of points and this should be used in place of the mathematically defined fractal dimension for infinite number of points (D) as a requirement for approach towards homogeneity. We derive the expected fractal dimension for a homogeneous distribution of a finite number of points. We show that for sufficiently large data sets the expected fractal dimension approaches D in absence of clustering. It is also important to take the weak, but non-zero amplitude of clustering at very large scales into account. In this paper we also compute the expected fractal dimension for a finite point set that is weakly clustered. Clustering introduces departures in the fractal dimensions from D and in most situations the departures are small if the amplitude of clustering is small. Features in the two-point correlation function, like those introduced by baryon acoustic oscillations can lead to non-trivial variations in the fractal dimensions where the amplitude of clustering and deviations from D are no longer related in a monotonic manner. We show that in the concordance model, the fractal dimension makes a rapid transition to values close to 3 at scales between 40 and 100 Mpc.

Annals of the New York Academy of Sciences, 1985

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.

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.

Modern Physics Letters A, 2009

A stochastic model relating the parameters of astrophysical structures to the parameters of their granular components is applied to the formation of hierarchical, large-scale structures from galaxies assumed as point-like objects. If the density profile of galaxies on a given scale is described by a power law then the stochastic model leads naturally to a mass function that is proportional to the square of the distance from an occupied point, which corresponds to a two-point correlation function that is inversely proportional to the distance. This result is consistent with observations indicating that galaxies are, on the largest scales, characterized by a fractal distribution with a dimension of order 2 and well-fit with transition to homogeneity at cosmological scales.

Physica D: Nonlinear Phenomena, 1989

Department t)f Physws. Emorv Umt'¢rsto'. .4tlanta, (7.4 30322, USA Four deterministic models are presented in order to get more insight into the geometry and multifi'actal behavior associated ~,th fractal growth phenomena. These models allow for exact treatment and are used to demonstrate such properties as directed self-affinity and self-similarit)' and multifractal growth probability and mass distribution, it ns pointed out that in some cases the varim~s fractal dimension defimtions may lead to inconsistent results. 0167-2~89,,89/$03.50 ,t', Elsevier Science PuOhshers B V. l North-Holland IPh.~slcs Publishing Oi~ isiL)n )

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.

The development from fractal geometry to fractal statistics was established in this paper. Interesting features such as self similarity, scale invariance, and the space-filling property of objects (fractal dimension) of fractal geometry provided an enormous groundwork to build a link towards the statistical paradigm since most data sets are endowed with its non-normal and irregular characteristic. A new probability distribution called fractal distribution was modeled to accommodate this non-normal conforming characteristic of most data sets and sample investigations were presented at the end of the paper. 1.0 Introduction: Fractal is a general term used to describe both the geometry and the processes which exhibit self-similarity, scale invariance, and fractional dimension (Mandelbrot, 1982). Deterministic fractals had been extensively studied and consist essentially of establishing an initiator structure and a generator (Mandelbrot, 1967), Palma (1992), Orbach (1992). The initial structure is transformed by the generator repeatedly at various scales. Common examples of fractals geometric structures include the fractal disks or Cantor sets (dimension d=0.63), the Von Koch curve (dimension d=1.26), Julia sets and others. The applications of the fractal geometry have been enormous; in astronomy, biology and chemistry, data compression, economy, art and music, and weather to name a few. As knowledge seeking progresses, fractals not only gained its grandeur in geometrical sphere, but also in the area of statistics. The study of Padua, et al. convincingly provided the idea on the strong connection between fractals and probability theory called statistical fractal (fractal statistics). This is because most real-life situations involve a non-existent mean like dealing with stock market crashes, earthquakes, test scores of students, and others (Selvam (2008)). The non-existence of the mean guarantees a non-existence of the standard deviation and variance hence performing an inferential test (which commonly assumes normality) is impossible. They argued that most of the phenomena that had been modeled using the normal distribution can be more accurately analyzed using statistical fractals instead since most of the data sets exhibit self-similar stochastic patterns (Padua, et.al, 2013). There still have been few formal studies exemplifying the evolution from fractal geometry to statistical fractal. This study continues to uncover that link from fractal geometry to statistical fractal by showing that the properties of fractal objects (self-similar, scale invariant, heterogeneous, and has fractional dimension) can be linked to the properties exhibited by a random variable in the statistical fractal. 2.0 Fractal Geometry: Self-Similarity, Scale Invariance, Fractional Dimension Two of the core properties of fractals are self similarity and scale invariance. Formally, self-similarity is dened as a property where a subset, when magnied to the size of the whole, is indistinguishable from the whole (Mandelbrot, 1967). The structure of the shape is made of smaller or bigger copies of itself: same shape but different size. For example, if the Barnsley fern leaf (Figure 1) is closely examined, it is noticed that the every little leaf, part of the bigger one, has a similar shape with the whole fern leaf. So we say that the fern leaf is exhibiting self-Figure 1