Scaling Region Research Articles

A novel method for feature extraction using crossover characteristics of nonlinear data and its application to fault diagnosis of rotary machinery

Defective rotary machinery typically exhibits a complex dynamical behavior, which is hard to analyze. Detrended Fluctuation Analysis (DFA) is a robust tool for uncovering long-range correlations hidden in nonstationary data. By DFA, an original series can be compressed into a fluctuation series, which can well preserve the dynamical characteristics of the original series. Lately, the fluctuation series has been separately analyzed by principal component analysis (PCA) and neural network (NN) for fault diagnosis of rotary machinery. However, the feature parameters extracted by PCA or NN normally lack clear physical meaning. In addition, the execution of PCA or NN usually consumes extra time. Interestingly, the scaling-law curve, by which the relation between the fluctuation function and the time scale can be illustrated graphically in a log–log plot, usually exhibits crossover properties. As a result, this study exploited the interesting crossover properties for fault diagnosis of rotary machinery and proposed a novel method for feature extraction of nonlinear data. The proposed method consists of three parts. Firstly, the vibration data from defective rotary machinery are analyzed by DFA and the resultant scaling-law curve is obtained. Secondly, the crossover points in the scaling-law curve are located and then employed to segment the entire scaling-law curve into several different scaling regions, in each of which a single Hurst exponent can be estimated. Thirdly, the whole or a part of the Hurst exponents are used as feature parameters for describing the conditions of defective rotary machinery. Next, the performance of the proposed method was measured using both real gearbox and rolling bearing vibration data with different fault types and severity. The results indicate that the proposed method can ease the problems mentioned previously and performs well in identifying fault types and severity of rotary machinery.

The synchronization club: classification of global economic groups by inequality

We find that, from 1970 to 2006, the GDPs of 181 countries are described by a log-normal with a power law tail before 1992, but by a kinked power law distribution after 1992. In the 15 years from 1992 to 2006, there are two obvious scale-free zones for annual GDPs, ranked from the largest to smallest. If the countries in each scaling region are regarded as a group, the world is divided into two groups, each with a roughly stable number of members. The power exponents of the two groups are different and hence lead to different inequalities. Therefore, the basis for classification is the macro-consistent inequality within each group. The wealth grows in a synchronous nonlinear manner within groups that have a stable wealth distribution and rank structure. If each group is considered as a club, we name it a ‘synchronization club’.

Morphological Dilation as Fractal Scaling in Roughness Measurement

In this work we propose, that the morphological dilation acts as fractal filters rebuilding white noise roughness surfaces into fractal 1/fm noise surfaces. The fractality indicates that the dilation does not have characteristic length scale, and the structuring element follows power-law distribution. Yashchuk's binary pseudo-random grating standard has been dilated with spherical and free form tips between 50 nm and 2000 nm and two scaling regions are referred to the tip diameter versus scaling exponent diagram. The first one in the smaller tip diameter region has a fast slope and the second one in the intermediate and larger tip diameter has a gradual slope. The results show that the dilated surfaces arise from the activity of at least two dynamical systems.

Comparative study of the upper and lower limb skin blood flow control mechanisms in patients with essential hypertension

To investigate limb specific differences in cutaneous vascular function in patients (n=33) with essential hypertension (EHT). In this observational cross-sectional study, baseline skin blood flow and the response to local heating were measured with a laser Doppler flowmeter (LDF) from the volar region of the forearm and the gaiter area of the foot at supine rest. The fractal analysis, detrended fluctuation analysis (DFA), was used to calculate the correlation properties of skin blood flow, LDF signal. The paired t-test and repeated measures ANOVA were used to determine the response to local heating and compare the scaling exponents of different anatomical locations respectively. We found three linear scaling regions that describe the fractal behavior of LDF signal with their slopes, scaling exponents. For cardiac (α(C)) and cardio-respiratory (α(CR)) scaling exponents, thermal hyperemia (T) induced greater change in the leg (α(C)=1.49 ± 0.26; α(CT)=1.62 ± 0.20 p<0.01 and α(CR)=0.84 ± 0.29 α(CRT)=0.42 ± 0.28 p<0.001) than in forearm (α(C)=1.28 ± 0.13; α(CT)=1.33 ± 0.13 p>0.05 and α(CR)=0.73 ± 0.15; α(CRT)=0.65 ± 0.018 p<0.05). Local scaling exponents (α(L) ≈ α(LT) ~ 1) were not significantly different (p>0.05) and, local lines did not shift in parallel with local heating in both extremities. The results of the present study suggest that skin microvascular function is impaired in both extremities in EHT patients. However, myogenic response is not uniform in both extremities and pronounced response to local thermal hyperemia has been observed in the gaiter area compared with the volar region. Further studies are needed to determine if these limb specific microvascular differences is the result of posture-induced structural and functional adaptation.

Dynamic universality class of Model C from the functional renormalization group

We establish new scaling properties for the universality class of Model C, which describes relaxational critical dynamics of a nonconserved order parameter coupled to a conserved scalar density. We find an anomalous diffusion phase, which satisfies weak dynamic scaling while the conserved density diffuses only asymptotically. The properties of the phase diagram for the dynamic critical behavior include a significantly extended weak scaling region, together with a strong and a decoupled scaling regime. These calculations are done directly in 2 < d < 4 space dimensions within the framework of the nonperturbative functional renormalization group. The scaling exponents characterizing the different phases are determined along with subleading indices featuring the stability properties.

A non-perturbative study of massive gauge theories

We consider a non-perturbative formulation of an SU(2) massive gauge theory on a space-time lattice, which is also a discretised gauged non-linear chiral model. The lattice model is shown to have an exactly conserved global SU(2) symmetry. If a scaling region for the lattice model exists and the lightest degrees of freedom are spin one vector particles with the same quantum numbers as the conserved current, we argue that the most general effective theory describing their low-energy dynamics must be a massive gauge theory. We present results of a exploratory numerical simulation of the model and find indications for the presence of a scaling region where both a triplet vector and a scalar remain light.

Turbulence in the two-dimensional Fourier-truncated Gross–Pitaevskii equation

We undertake a systematic, direct numerical simulation of the two-dimensional, Fourier-truncated, Gross–Pitaevskii equation to study the turbulent evolutions of its solutions for a variety of initial conditions and a wide range of parameters. We find that the time evolution of this system can be classified into four regimes with qualitatively different statistical properties. Firstly, there are transients that depend on the initial conditions. In the second regime, power-law scaling regions, in the energy and the occupation-number spectra, appear and start to develop; the exponents of these power laws and the extents of the scaling regions change with time and depend on the initial condition. In the third regime, the spectra drop rapidly for modes with wave numbers k > kc and partial thermalization takes place for modes with k < kc; the self-truncation wave number kc(t) depends on the initial conditions and it grows either as a power of t or as log t. Finally, in the fourth regime, complete thermalization is achieved and, if we account for finite-size effects carefully, correlation functions and spectra are consistent with their nontrivial Berezinskii–Kosterlitz–Thouless forms. Our work is a natural generalization of recent studies of thermalization in the Euler and other hydrodynamical equations; it combines ideas from fluid dynamics and turbulence, on the one hand, and equilibrium and nonequilibrium statistical mechanics on the other.

Multifractal analysis of neutral community spatial structure

The spatial structure of neutral communities has nontrivial properties, which are described traditionally by the Species–area relationship (SAR) and the Species Abundance Distribution, (SAD). Fractal analysis is an alternative way to describe community structure, the final product of which – a multifractal spectrum – combines information both on the scaling parameters of species richness (similar to SAR), and about species' relative abundances (similar to SAD). We conducted a multifractal analysis of community spatial structure in a neutral lattice-based model.In a realistic range of dispersal distances, moments of the species abundance distribution form a family of curves of the same shape, which are reduced to a single universal curve through a scaling collapse procedure. Trivial scaling is observed on small and large scales, which reflects homogeneity of species distribution at small scales and a limiting log-series distribution at large scales. Multifractal spectra for different speciation rates and dispersal kernels are obtained for the intermediate region of scaling. Analysis of spectra reveals that the key model parameters determine not only the species richness and its scaling, but also of species dominance and rarity. We discovered a phenomenon of negative dimensions in the multifractal spectrum. Negative dimensions have no direct interpretation from a purely physical point of view, but have biological meaning because they reflect the negative relationship between the number of singletons and the area.

SU(3) deconfining phase transition with finite volume corrections due to a confined exterior

Using the geometry of a double-layered torus we investigate the deconfining phase transition of pure SU(3) lattice gauge theory by Markov chain Monte Carlo simulations. In one layer, called "outside", the temperature is set below the deconfining temperature and in the other, called "inside", it is iterated to a pseudo-transition temperature. Lattice sizes are chosen in a range suggested by the physical volumes achieved in relativistic heavy ion collisions and both temperatures are kept close enough to stay in the SU(3) scaling region. Properties of the transition are studied as function of the volume for three outside temperatures. When compared with infinite volume extrapolations, small volume corrections of the deconfining temperature and width become competitive with those found by including quarks. Effective finite size scaling exponents of the specific and Polyakov loop susceptibilities are also calculated.

On the transition to hyperchaos and the structure of hyperchaotic attractors

Using a recently proposed algorithmic scheme for correlation dimension analysis of hyperchaotic attractors, we study two well-known hyperchaotic flows and two standard time delayed hyperchaotic systems in detail numerically. We show that at the transition to hyperchaos, the nature of the scaling region changes suddenly and the attractor displays two scaling regions for embedding dimension M ≥ 4. We argue that it is an indication of a strong clustering tendency of the underlying attractor in the hyperchaotic phase. Because of this sudden qualitative change in the scaling region, the transition to hyperchaos can be easily identified using the discontinuous changes in the dimension (D2) at the transition point. We show this explicitely for the two time delayed systems. Further support for our results is provided by computing the spectrum of Lyapunov Exponents (LE) of the hyperchaotic attractor in all cases. Our numerical results imply that the structure of a hyperchaotic attractor is topologically different from that of a chaotic attractor with inherent dual scales, at least for the two general classes of hyperchaotic systems we have analysed here.

A new assessment of the second-order moment of Lagrangian velocity increments in turbulence

The behaviour of the second-order Lagrangian structure functions on state-of-the-art numerical data both in two and three dimensions is studied. On the basis of a phenomenological connection between Eulerian space-fluctuations and the Lagrangian time-fluctuations, it is possible to rephrase the Kolmogorov 4/5-law into a relation predicting the linear (in time) scaling for the second-order Lagrangian structure function. When such a function is directly observed on current experimental or numerical data, it does not clearly display a scaling regime. A parameterisation of the Lagrangian structure functions based on Batchelor model is introduced and tested on data for 3d turbulence, and for 2d turbulence in the inverse cascade regime. Such parameterisation supports the idea, previously suggested, that both Eulerian and Lagrangian data are consistent with a linear scaling plus finite-Reynolds number effects affecting the small- and large timescales. When large-time saturation effects are properly accounted for, compensated plots show a detectable plateau already at the available Reynolds number. Furthermore, this parameterisation allows us to make quantitative predictions on the Reynolds number value for which Lagrangian structure functions are expected to display a scaling region. Finally, we show that this is also sufficient to predict the anomalous dependency of the normalised root mean squared acceleration as a function of the Reynolds number, without fitting parameters.

The fractal dimension of software networks as a global quality metric

We analyzed the source code of various releases of two large Object Oriented Open Source Java software systems, Eclipse and Netbeans, investigating the complexity of the whole release and of its subprojects. We show that when the classes in the source code and the dependencies between them are considered, such systems can be viewed as complex software networks, and emerging structures, characteristic of fractals, appear at different length scales – on the entire systems and on subprojects of any size.We were able to find in all examined cases a scaling region where it is possible to compute a self-similar coefficient, the fractal dimension, using “the box counting method”. Such a coefficient is a single metric related to the system’s complexity.More importantly, we were able to show that this measure looks fairly related to software quality, acting as a global quality software metric. In particular, we computed the defects of each software system, and we found a clear correlation among the number of defects in the system, or in a subproject, and its fractal dimension. This correlation exists across all the subprojects and also along the time evolution of the software systems, as new releases are delivered.

An Inverse Laplace Transform on Lattice Spacing

An inverse Laplace transform on lattice spacing is introduced as a computational framework of the extrapolation of the strong coupling expansion to the scaling region. We apply the transform to the two-dimensional nonlinear O(N) model at N ≥ 3 and show that the approximation of the continuum limit of the susceptibility agrees with the existing theoretical and Monte Carlo data.

Investigation on very large scale motions (VLSMs) and their influence in a dust storm

Based on the real-time synchronous measurements of the wind velocity, temperature, the PM10 concentration at 16 m and 47 m during a dust storm event, in which Reynolds number Re τ exceeds 6×106, this study reveals the existence of the very large scale motions (VLSMs) during the stable stage both in the stream velocity and the temperature field at the two heights, whose streamwise scales reach up to 10 times the thickness of the boundary layer. The streamwise velocity and the PM10 concentration display a similar frequency corresponding to the peaks of their energy spectra, which implies that the VLSMs of streamwise flow have a significant role in dust transportation. In contrast, the salient deviations of the PM10 concentration at 47 m from the Gaussian distribution are revealed, which means that 47 m is not in the dust transportation layer, but is a region where the dust transportation layer and the outer flow intersect each other. Analysis demonstrates that the energy spectra of the PM10 concentrations at 16 m and 47 m display the “−1” scaling law feature, which has the same frequency range (0.001–0.1 Hz) as that of the wind velocity. This provides a new paradigm for the existence of the self-similarity scaling region in turbulent flow.

New signal scaling strategies for return phase training in motion-base spaceflight simulator

Signal scaling is an essential step in spaceflight simulation. Thus far, the third-order polynomial scaling method has been widely used for signal scaling; however, in this method, parameter tuning is complicated and may induce perceptible distortion during large-range monotonic signal scaling. In the simulation of spacecraft return, specifically, that of re-entry, acceleration and angular velocity signals may vary considerably over short time periods. Motion perception is important for training astronauts in this phase. In this study, two strategies are proposed to solve these problems using the ‘scaling scope’ parameter. The first strategy is based on the Hermite interpolation polynomial, and the other is based on third-order polynomial scaling. Two methods were developed which make use of the stable region of third-order polynomial scaling. The first method maximizes the stable region to prevent signal distortion, and the other restricts the scaling scope in the stable region. Based on the dynamic characteristics of spacecraft in the return phase, the signal scaling strategies proposed in this study are simulated for trainees’ perception in a motion-base simulator. Simulations were implemented by utilizing the full curves of spacecraft return phase for the first time, and results show that these methods are more advantageous for parameter tuning and can eliminate signal distortion for all input signals. While these methods have a shortcoming in that the trigger velocity (onset cue) is slowed down, this shortcoming is eliminated by employing the moving cueing algorithm. Both the strategies proposed in this article show good performance and can be applied potentially to the motion simulation of the spacecraft return phase.

Detecting Long-range Correlations in NDVI over Greater Khingan Mountains

The spatial-temporal power-law distributions are found in many natural systems.In this paper,multi-temporal series of satellite GIMMS Normalized Difference of Vegetation Index(NDVI) data from 1982 to 2006 were exploited for studying long-range correlations in the Greater Khingan Mountains.We used Detrended Fluctuation Analysis(DFA),which permits the detection of persistent properties in nonstationary signal fluctuations.Firstly,anomaly NDVI fluctuations are the nonstationary signal and a complex driving system.Secondly,the first-order detrended fluctuation analysis is suitable for GIMMS NDVI datasets,and in NDVI fluctuations little time-scale won't be ignored.NDVI fluctuations are found to be persistent long-range power-law correlations,with exponent 0.85(significance 99%),for time scales longer than 2 months and shorter than 15 months in the Greater Khingan Mountains.The DFA1 was applied to randomly shuffled versions of each of the original series.The exponent for the shuffled version of NDVI is about 0.5(significance 99%).The result suggests that NDVI have self-similarity characteristics.Finally,the four fluctuation curves,calculated for the north,northwest,middle and south of the Greater Khingan Mountains present two scaling regions with crossover timescales at about 6.5 to 8.5 months.Break points of northwest and south came earlier than north and middle in the Greater Khingan Mountains.The first timescale regions are characterized by scaling exponents α1 varying from 0.93 to 1.17.It is shown that the interval sequences of NDVI closely resemble that of 1/f noise at a small timescale.The second scaling regions are featured by exponents α2 ranging between 0.61 and 0.84 and exhibit persistent long-range power-law correlations.North and middle correlations are stronger than northwest and south.Vegetation type and climate may affect the long-range correlation in some way.These findings could be proved useful in testing the results of climate-vegetation models.

Chaotic analysis of daily rainfall series in Koyna reservoir catchment area, India

Conventionally the process of rainfall is considered as a stochastic process and the last century witnessed many developments, however recently many rainfall series are proved to be a chaotic series. Out of various chaotic methods available to analyse time series data, the most commonly employed method, correlation dimension method is used in the present study to carry out the behavioral study of catchment rainfall process. The daily average rainfall observed over a period of 49 years (1961–2009) from the catchment area of Koyna reservoir in Maharashtra, India, has been taken up to study its behaviour. Daily rainfall from nine rain gage stations is averaged in order to obtain the above mentioned daily average rainfall. From the detailed non-linear dynamic analysis using correlation dimension method, it is found that the average rainfall of Koyna region is showing a chaotic behaviour. The effect of radius ‘r’ and its steps in the Heaviside function for plotting log r versus log Cr graph, the effect of scaling region in determining correlation exponent from the above plot and the effect of longer length of zeros in time series are analyzed in the present study. In addition to this, the chaotic behaviour of rainfall for various time steps (daily, weekly, monthly and seasonal) is also analyzed. It is found that the selection of radius ‘r’ and its steps is important in Heaviside function as there will be initial disturbances for smaller ‘r’ which is difficult to account while determining the correlation exponent whereas higher ‘r’ leads to the log Cr values close to zero making it impossible to estimate the slope of log r versus log Cr graph for determining correlation exponent. Nevertheless, the selection of scaling region is very crucial in determining the correlation exponent which subsequently determines the behaviour of the system as chaotic or not. Wrong selection of scaling region results in misinterpretation of results. In the present study, the correlation exponent value is saturating in case of daily full year rainfall, weekly rainfall and 10 daily rainfall indicating its chaotic behaviour whereas the correlation exponent value go on increasing with embedding dimension but not as in case of a stochastic series for daily monsoon rainfall, monthly rainfall and seasonal rainfall. The saturated value of correlation exponent is in between zero and one indicating correlation dimension of the three series (daily full year, weekly and 10 daily) as one.

The derivation of the generalized functional equations describing self-similar processes

Abstract The generalized functional equations describing a wide class of different self-similar processes are derived. These equations follow from the observation that microscopic function describing an initial self-similar process increases monotonically or even cannot have a certain value. The last case implies the behavior of trigonometric functions cos(zζ n), sin(zζ n) at ζ &gt; 1 and n ≫ 1 that can enter to the microscopic function and when the limits of the initial scaling region are increasing and becoming large. The idea to obtain the desired functional equations is based on the approximate decoupling procedure reducing the increasing microscopic function to the linear combination of the same microscopic functions but having smaller scales. Based on this idea the new solutions for the well-known Weierstrass-Mandelbrot function were obtained. The generalized functional equations derived in this paper will help to increase the limits of applicability in description of a wide class of self-similar processes that exist in nature. The procedure that is presented in this paper allows to understand deeper the relationship between the procedure of the averaging of the smoothed functions on discrete self-similar structures and continuous fractional integrals.

Fractal-based exponential distribution of urban density and self-affine fractal forms of cities

Urban population density always follows the exponential distribution and can be described with Clark’s model. Because of this, the spatial distribution of urban population used to be regarded as non-fractal pattern. However, Clark’s model differs from the exponential function in mathematics because that urban population is distributed on the fractal support of landform and land-use form. By using mathematical transform and empirical evidence, we argue that there are self-affine scaling relations and local power laws behind the exponential distribution of urban density. The scale parameter of Clark’s model indicating the characteristic radius of cities is not a real constant, but depends on the urban field we defined. So the exponential model suggests local fractal structure with two kinds of fractal parameters. The parameters can be used to characterize urban space filling, spatial correlation, self-affine properties, and self-organized evolution. The case study of the city of Hangzhou, China, is employed to verify the theoretical inference. Based on the empirical analysis, a three-ring model of cities is presented and a city is conceptually divided into three layers from core to periphery. The scaling region and non-scaling region appear alternately in the city. This model may be helpful for future urban studies and city planning.

Adaptive Fractal Analysis Reveals Limits to Fractal Scaling in Center of Pressure Trajectories

Fractal time series analysis methods are commonly used for analyzing center of pressure (COP) signals with the goal of revealing the underlying neuromuscular processes for upright stance control. The use of fractal methods is often coupled with the assumption that the COP is an instance of fractional Gaussian noise (fGn) or fractional Brownian motion (fBm). Our purpose was to evaluate the applicability of the fGn-fBm framework to the COP in light of several characteristics of COP signals revealed by a new method, adaptive fractal analysis (AFA). AFA quantifies how the variance of the residuals to fits of a globally smooth trend signal scales with the time scale at which the fits are performed. Application of AFA to COP signals revealed that there are potentially three fractal scaling regions in the COP as opposed to one as expected from a pure fGn or fBm process. The scaling region at the fastest scale was anti-persistent and spanned ~30-90 ms, the intermediate was persistent and spanned ~200 ms-1.9 s, and the slowest was anti-persistent and spanned ~5-40 s. The intermediate fractal scaling region was the most clearly defined, but it only contributed around 11% of the total spectral energy of the COP signal, indicating that other features of the COP signal contribute more importantly to the overall dynamics. Also, more than half of the Hurst exponents estimated for the intermediate region were greater than the theoretically expected range [0,1] for fGn-fBm processes. These results suggest the fGn-fBm framework is not appropriate for modeling COP signals. ON-OFF intermittency might provide a better modeling framework for the COP, and multiscale approaches may be more appropriate for analyzing COP data.