Statistical Self-similarity Research Articles

Symmetric Proliferation: An Examination of the Fractal Geometry of Tumors

S ymmetric P roliferation : A n E xamination of the F ractal G eometry of T umors B S J Rachel Lew Sierpinski carpet model of plane fractals From flowers to faces, nature is abound with symmetry. Most natural objects tend to form according to patterns, a tendency which mathematicians readily exploit in order to create theoretical models of the world around us. The height of a cliff, for instance, is modeled by a one-dimensional line; a snake’s path through the sand is modeled across a two- dimensional surface; a block of ice is modeled as a mass extending into three dimensions. But what about objects in between dimensions? In fact, between the 1-D and the 2-D there exist objects known as fractals. These mathematical objects are infinitely self-similar, which means that upon magnification of a certain part of a fractal, one sees the figure of the overall fractal, and so on unto infinity. Self-symmetry “Picture, for example, water trickling through only the most loosely packed areas of soil in a pot; in the same way, the blood vessels of a tumor grow into the weakest areas of the tissue around it.” allows a fractal to have fractional dimensions because it is not purely linear--the border of a fractal cannot be traced--but this lack of boundedness also means that the fractal never encircles a defined area. Imagine a tree whose branches branch out infinitely, or a snowflake with six tips, each of which looks like the original snowflake, and so on and so forth. Clearly, in fractal models, as in all models of nature, there is a difference between the mathematical and the natural. Natural fractal objects are composed of discrete units and are not infinitely divisible--the fractal pattern must end somewhere, or else, for instance, one might find tiny branches at the cellular level of a tree branch. To correct for this quality, scientists define natural fractal objects as having statistical self-similarity, or when “the statistical properties of the pieces [of an object] are proportional to the statistical properties of the whole” (Grizzi et al., 2008). In the human body, statistically self-similar models are most commonly applied to branching structures in the lung and in networks of blood vessels. The latter application has had particular importance in medical studies of cancer, as there is evidence that understanding the fractal geometry of tumor vasculature may aid in identification and targeted treatment of cancers. Tumor vasculature can in fact be described by a fractal model, and is often distinguished from normal vasculature by either an abnormally high or abnormally low fractal dimension (Zook and Iftekharuddin, 2005). An object’s fractal dimension is a constant between the integers 1 and 2, and might be described as how ‘proliferative’ the object looks; i.e., an object with a higher fractal dimension looks more like an object with true area than like a curve. Baish and Jain observed that blood vessels in the tumors of mice had higher fractal dimensions than the mice’s normal arteries and veins, claiming that “the fractal dimension quantified the degree of randomness to the vascular distribution, a characteristic not easily captured by the vascular density” (Baish and Jain, 2000). Moreover, the researchers noted that tumor vessels tended to be more twisted than normal vessels, having “many smaller bends upon each larger bend” (Baish and Jain, 2000). They also found that the way tumor vessels grew and branched closely matched a type of statistical growth called invasive percolation. In invasive percolation, a substance moves through a medium which has varying degrees of strength, penetrating the weakest areas of the medium and thus branching out to form a network. Picture, for example, water trickling through only the most loosely packed areas of a pot of soil; in the same way, the blood vessels of a tumor grow into the weakest areas of the tissue around it. On the other hand, normal capillaries are traditionally modeled by the Krogh cylinder model, which assumes that the capillaries are straight, relatively spaced, and only reach a cylindrical volume of tissue immediately 18 • B erkeley S cientific J ournal • S ymmetry • F all 2015 • V olume 20 • I ssue 1

Fractal structure enables temporal prediction in music.

1/f serial correlations and statistical self-similarity (fractal structure) have been measured in various dimensions of musical compositions. Musical performances also display 1/f properties in expressive tempo fluctuations, and listeners predict tempo changes when synchronizing. Here the authors show that the 1/f structure is sufficient for listeners to predict the onset times of upcoming musical events. These results reveal what information listeners use to anticipate events in complex, non-isochronous acoustic rhythms, and this will entail innovative models of temporal synchronization. This finding could improve therapies for Parkinson's and related disorders and inform deeper understanding of how endogenous neural rhythms anticipate events in complex, temporally structured communication signals.

Auditory-motor synchronization with temporally fluctuating sequences is dependent on fractal structure but not musical expertise.

Fractal structure is a ubiquitous property found in nature and biology, and has been observed in processes at different levels of organization, including rhythmic behavior and musical structure. A temporal process is characterized as fractal when serial long-term correlations and statistical self-similarity (scaling) are present. Previous studies of sensorimotor synchronization using isochronous (non-fractal) stimuli show that participants' errors exhibit persistent structure (positive long-term correlations), while their inter-tap intervals (ITIs) exhibit anti-persistent structure (negative long-term correlations). Auditory-motor synchronization has not been investigated with anti-persistent stimuli. In the current study, we systematically investigated whether the fractal structure of auditory rhythms was reflected in the responses of participants who were asked to coordinate their taps with each event. We asked musicians and non-musicians to tap with 12 different rhythms that ranged from anti-persistent to persistent. The scaling exponents of the ITIs were strongly correlated with the scaling exponents of the stimuli, showing that the long-term structure of the participants' taps scaled with the long-term structure of the stimuli. Surprisingly, the performance of the musicians was not significantly better than that of the non-musicians. Our results imply that humans are able to readily adapt (rather than simply react) to the overall statistical structure of temporally fluctuating stimuli, regardless of musical skill.

Operating system efficiency evaluation on the base of measurements analysis with the use of non-extensive statistics elements

AbstractThe major goal of this article was to evaluate the efficiency of Linux operating system using statistical self-similarity and multifractal analysis. In order to collect the necessary data, the tools available in Linux such as vmstat, top and iostat were used. The measurement data collected with those tools had to be converted into a format acceptable by applications which analyze statistical selfsimilarity and multifractal spectra. Measurements collected while using the MySQL database system in a host operating system were therefore analyzed with the use of statistical self-similarity and allowed to determine the occurrence of long-range dependencies. Those dependencies were analyzed with the use of adequately graduated diagrams. Multifractal analysis was conducted with the help of FracLab application. Two methods were applied to determine the multifractal spectra. The obtained spectra were analyzed in order to establish the multifractal dependencies.

A universal mechanism for long-range cross-correlations

Cross-correlations are thought to emerge through interaction between particles. Here we present a universal dynamical mechanism capable of generating power-law cross-correlations between non-interacting particles exposed to an external potential. This phenomenon can occur as an ensemble property when the external potential induces intermittent dynamics of Pomeau-Manneville type, providing laminar and stochastic phases of motion in a system with a large number of particles. In this case, the ensemble of particle-trajectories forms a random fractal in time. The underlying statistical self-similarity is the origin of the observed power-law cross-correlations. Furthermore, we have strong indications that a sufficient condition for the emergence of these long-range cross-correlations is the divergence of the mean residence time in the laminar phase of the single particle motion (sporadic dynamics). We argue that the proposed mechanism may be relevant for the occurrence of collective behaviour in critical systems.

Network anomaly detection based on the statistical self-similarity factor for HTTP protocol

Self-similarity analysis and anomaly detection in networks are interesting field of research and scientific work of scientists around the world. Simulation studies have demonstrated that the Hurst parameter estimation can be used to detect traffic anomaly – the Hurst values are compared with confidence intervals of normal values to detect anomaly in few kinds of traffic: HTTP protocol, email, SSL. Slowa kluczowe: wspolczynnik Hursta, wykrywanie anomalii, samopodobienstwo, zalezności dlugozasiegowe.

Traffic prediction in 3G mobile networks based on multifractal exploration

Traffic prediction plays an integral role in telecommunication network planning and network optimization. In this paper, we investigate the traffic forecasting for data services in 3G mobile networks. Although the Box-Jenkins model has been proven to be appropriate for voice traffic (since the arrival of calls follows a Poisson distribution), it has been demonstrated that the Internet traffic exhibits statistical self-similarity and has to be modeled using the Fractional AutoRegressive Integrated Moving Average (FARIMA) process. However, a few studies have concluded that the FARIMA process may fail in modeling the Internet traffic. To this end, we conducted experiments on the modeling of benchmark Internet traffic and found that the FARIMA process fails because of the significant multifractal characteristic inherent in the traffic series. Thereafter, we investigate the traffic series of data services in a 3G mobile network from a province in China. Rich multifractal spectra are found in this series. Based on this observation, an integrated method combining the AutoRegressive Moving Average (ARMA) and FARIMA processes is applied. The obtained experimental results verify the effectiveness of the integrated prediction method.

Cascade of Kinetic Energy in Three-Dimensional Compressible Turbulence

The conservative cascade of kinetic energy is established using both Fourier analysis and a new exact physical-space flux relation in a simulated compressible turbulence. The subgrid scale (SGS) kinetic energy flux of the compressive mode is found to be significantly larger than that of the solenoidal mode in the inertial range, which is the main physical origin for the occurrence of Kolmogorov's -5/3 scaling of the energy spectrum in compressible turbulence. The perfect antiparallel alignment between the large-scale strain and the SGS stress leads to highly efficient kinetic energy transfer in shock regions, which is a distinctive feature of shock structures in comparison with vortex structures. The rescaled probability distribution functions of SGS kinetic energy flux collapse in the inertial range, indicating a statistical self-similarity of kinetic energy cascades.

Blowup as a driving mechanism of turbulence in shell models

Since Kolmogorov proposed his phenomenological theory of hydrodynamic turbulence in 1941, the description of the mechanism leading to the energy cascade and anomalous scaling remains an open problem in fluid mechanics. Soon after, in 1949, Onsager noticed that the scaling properties in the inertial range imply nondifferentiability of the velocity field in the limit of vanishing viscosity. This observation suggests that the turbulence mechanism may be related to a finite-time singularity (blowup) of incompressible Euler equations. However, the existence of such blowup is still an open problem too. In this paper, we show that the blowup indeed represents the driving mechanism of the inertial range for a simplified (shell) model of turbulence. Here, blowups generate coherent structures (instantons), which travel through the inertial range in finite time and are described by universal self-similar statistics. The anomaly (deviation of scaling exponents of velocity moments from the Kolmogorov theory) is related analytically to the process of instanton creation using the large deviation principle. The results are confirmed by numerical simulations.

Vibration Signal Fault Diagnosis Based on Generalized Fractal Dimension

To process the vibration signal fault diagnosis automatically and effectively, a method to calculate the generalized fractal dimension (Dq) based on the traditional G-P correlation dimension algorithm is presented. First, the vibration signal, a kind of random fractal that meets the statistical self-similarity, is reconstructed according to the phase space reconstruction theory. Second, for various weight factor values (q), calculation of the qth-order correlation integral, followed by the Dq, is processed. Range of the weight factor is discussed. With the generalized fractal dimension, the dynamic characteristics of the system are then reproduced more entirely and accurately. The effectiveness of the proposed method is verified by calculating the generalized fractal dimension for various measured vibration signals. The generalized fractal dimension distinguishes the state of vibration signal unambiguously. The analysis also indicates that the fluctuation amplitude of Dq is related to the frequency distribution of vibration signal. This would be helpful for the vibration signal fault diagnosis.

Computer network traffic analysis with the use of statistical self-similarity factor

The optimal computer network performance models require accurate traffic models, which can capture the statistical characteristic of actual traffic. If the traffic models do not represent traffic accurately, one may overestimate or underestimate the network performance. The paper presents confirmation of the self-similar nature of the selected protocols in the computer network communication layer. It shows that the good measure of self-similarity is a Hurst factor.

The Two-parameter Analyzing Model and Statistical Distribution Law of the Random Signal in Optical Sensor

In this paper, on the basis of giving a reference amplitude value, the two-parameter analyzing model of the random signal is established by introducing two basic conceptions---the amplitude and the width. Based on which, the statistical distribution law of the random signal group’s characteristic parameters is studied. The experimental results show that the amplitude and the width counting statistical distributions of the random pulse signal group and subsets are obviously asymmetric. Besides, the distribution functions of the whole signal group and its characteristic subsets match well with the lognormal distribution while the natural number as the independent variable. That is to say, the statistical self-similarity character of the random signals’ internal structure is definitely shown by applying the two-parameter analyzing model to deal with them.

Computer network traffic analysis with the use of statistical self-similarity factor

The optimal computer network performance models require accurate traffic models, which can capture the statistical characteristic of actual traffic. If the traffic models do not represent traffic accurately, one may overestimate or underestimate the network performance. The paper presents confirmation of the self-similar nature of the selected protocols in the computer network communication layer. It shows that the good measure of self-similarity is a Hurst factor.

Computation of anomalous scaling exponents of turbulence from self-similar instanton dynamics

We show that multiscaling properties of developed turbulence in shell models, which lead to anomalous scaling exponents in the inertial range, are determined exclusively by instanton dynamics. Instantons represent correlated extreme events localized in space-time, whose structure is described by self-similar statistics with a single universal scaling exponent. We show that anomalous scaling exponents appear due to the process of instanton creation. A simplified model of instanton creation is suggested, which adequately describes this anomaly.

Estimating the Hausdorff–Besicovitch dimension of boundary of basin of attraction in helicopter trim

Helicopter trim involves solution of nonlinear force equilibrium equations. As in many nonlinear dynamic systems, helicopter trim problem can show chaotic behavior. This chaotic behavior is found in the basin of attraction of the nonlinear trim equations which have to be solved to determine the main rotor control inputs given by the pilot. This study focuses on the boundary of the basin of attraction obtained for a set of control inputs. We analyze the boundary by considering it at different magnification levels. The magnified views reveal intricate geometries. It is also found that the basin boundary exhibits the characteristic of statistical self-similarity, which is an essential property of fractal geometries. These results led the authors to investigate the fractal dimension of the basin boundary. It is found that this dimension is indeed greater than the topological dimension. From all the observations, it is evident that the boundary of the basin of attraction for helicopter trim problem is fractal in nature.

Fractal Model for Thermal Conductivity of Wetting, Fibrous Porous Media

The current paper deals with the fractal effective thermal conductivity model for fibrous porous media containing unsaturated water moisture. The model is based on the thermal-electrical analogy and statistical self-similarity of porous media. The fractal effective thermal conductivity model can be expressed as a function of the pore structure (fractal dimension) and architectural parameters of porous media. It is expected that the model will be helpful in the evaluation of thermal comfort for textiles in the whole range of porosity.

Long-range correlations in the structure of fractal films

Hierarchically granulated films formed on a material subjected to the action of a high-temperature plasma have been microscopically studied. The statistical self-similarity of the structure of the surface from nanoscale to macroscale, as well as long-range correlations in the relief that are due to the statistically inhomogeneous structure of granularity, has been revealed.

Multifractal Analysis and Modeling of Chaotic Channels

This paper introduces multifractal theory into the research of chaotic channels, and a novel model of chaotic channel is established from nonlinear science standpoint. By studying the relationship between multifractal dimensions and work environments, increments probability distribution, and statistical self-similarity of low power line communication (L-PLC) channel which is a typical chaotic channel, we find that multifractal dimensions are key parameters for describing the chaotic channel, and reveal the inherent relationship between the chaotic channel and Fractional Brownian Motion (FBM) model. Finally, the multifractal model of chaotic channel based on FBM theory is established and applied to reconstructing received signals accurately.

On the Fractal Nature of Nervous Cell System

On the Fractal Nature of Nervous Cell System

Universal self-similarity of propagating populations

This paper explores the universal self-similarity of propagating populations. The following general propagation model is considered: particles are randomly emitted from the origin of a d-dimensional Euclidean space and propagate randomly and independently of each other in space; all particles share a statistically common--yet arbitrary--motion pattern; each particle has its own random propagation parameters--emission epoch, motion frequency, and motion amplitude. The universally self-similar statistics of the particles' displacements and first passage times (FPTs) are analyzed: statistics which are invariant with respect to the details of the displacement and FPT measurements and with respect to the particles' underlying motion pattern. Analysis concludes that the universally self-similar statistics are governed by Poisson processes with power-law intensities and by the Fréchet and Weibull extreme-value laws.