Algorithm to estimate the Hurst exponent of high-dimensional fractals

Quantifying multifractal anisotropy in two dimensional objects

An efficient method of exploring the effects of anisotropy in the fractal properties of 2D surfaces and images is proposed. It can be viewed as a direction-sensitive generalization of the multifractal detrended fluctuation analysis into 2D. It is tested on synthetic structures to ensure its effectiveness, with results indicating consistency. The interdisciplinary potential of this method in describing real surfaces and images is demonstrated, revealing previously unknown directional multifractality in data sets from the Martian surface and the Crab Nebula. The multifractal characteristics of Jackson Pollock's paintings are also analyzed. The results point to their evolution over the time of creation of these works.

Assessment of Fractal Synchronization during an Epileptic Seizure

In this paper, we define fractal synchronization (FS) based on the idea of stochastic synchronization and propose a mathematical apparatus for estimating FS. One major advantage of our proposed approach is that fractal synchronization makes it possible to estimate the aggregate strength of the connection on multiple time scales between two projections of the attractor, which are time series with a fractal structure. We believe that one of the promising uses of FS is the assessment of the interdependence of encephalograms. To demonstrate this approach in evaluating the cross-dependence between channels in a network of electroencephalograms, we evaluated the FS of encephalograms during an epileptic seizure. Fractal synchronization demonstrates the presence of desynchronization during an epileptic seizure.

Fractional Brownian motion with random Hurst exponent: Accelerating diffusion and persistence transitions

Fractional Brownian motion, a Gaussian non-Markovian self-similar process with stationary long-correlated increments, has been identified to give rise to the anomalous diffusion behavior in a great variety of physical systems. The correlation and diffusion properties of this random motion are fully characterized by its index of self-similarity or the Hurst exponent. However, recent single-particle tracking experiments in biological cells revealed highly complicated anomalous diffusion phenomena that cannot be attributed to a class of self-similar random processes. Inspired by these observations, we here study the process that preserves the properties of the fractional Brownian motion at a single trajectory level; however, the Hurst index randomly changes from trajectory to trajectory. We provide a general mathematical framework for analytical, numerical, and statistical analysis of the fractional Brownian motion with the random Hurst exponent. The explicit formulas for probability density function, mean-squared displacement, and autocovariance function of the increments are presented for three generic distributions of the Hurst exponent, namely, two-point, uniform, and beta distributions. The important features of the process studied here are accelerating diffusion and persistence transition, which we demonstrate analytically and numerically.

Corrosion resistance and surface microstructure of Mg3 N2 /SS thin films by plasma focus instrument

Utilizing a plasma focus (PF) instrument, magnesium nitride (Mg3 N2 ) thin films were synthesized on stainless steel substrates. Twenty five optimum focus shots at 8 cm distance from the anode tip were used to deposit the films at different angular positions regarded to the anode axis. Scanning electron microscopy (SEM), atomic force microscopy (AFM), and X-ray diffraction (XRD) analyses were performed to assess the surface morphology and structural characteristics of Mg3 N2 films. Based on AFM images, these films were studied to understand the effect of angular position variation on their surfaces through morphological and fractal parameters. By increasing the angle, we verify that the grain size decreased from 130(0) nm to 75(5) nm and also the mean quadratic surface roughness of the films reduced in its average values from (28.97 ± 3.24) nm to (23.10 ± 1.34) nm. Power spectrum density analysis indicated that films become more self-affine at larger angles. Furthermore, the corrosion behavior of the films was investigated through a potentiodynamic polarization test in H2 SO4 solution. It was found that the ion energy and flux, varying with the angular positions from the anode tip, directly affected the nanostructured roughness and surface morphology of the samples. The electrochemical studies of films show that the uncoated sample presented the lowest corrosion resistance. The highest corrosion resistance was obtained for the sample deposited with 25 optimum shots and at 0° angular position reaching a reduction in the corrosion current density of almost 800 times compared to the pure stainless steel-304 substrate. HIGHLIGHTS: Mg3 N2 /SS films have been deposited at different angles by plasma focus (PF) instruments. The effect of angular position on the surface microtexture, morphological parameters, and corrosion features of the films was studied. The RBS measurement and X-ray diffraction are utilized to identify the crystalline phases and thickness of films.

Universal Markers Unveil Metastatic Cancerous Cross-Sections at Nanoscale

Simple Summary

We propose the use of two universal morphometric indices whose synergetic potency leads to the classification of a cancerous tissue of a few nanometers in size as metastatic or non-metastatic. The method is label-free, operates on conventional histological cross-sections, recording surface height–height roughness by AFM, and detects nanoscale changes associated with the progress of carcinogenesis which are the output of combined statistical approaches, namely multifractal analysis and the generalized moments method. The benefit of this approach is at least two-fold. On the one hand, its application in the context of early diagnosis can increase the life expectancy of patients, and on the other hand, differentiation between metastatic and non-metastatic tissues at the singular cell level can lead to new methodologies to treat cancer biology and therapies.

Abstract

The characterization of cancer histological sections as metastatic, M, or not-metastatic, NM, at the cellular size level is important for early diagnosis and treatment. We present timely warning markers of metastasis, not identified by existing protocols and used methods. Digitized atomic force microscopy images of human histological cross-sections of M and NM colorectal cancer cells were analyzed by multifractal detrended fluctuation analysis and the generalized moments method analysis. Findings emphasize the multifractal character of all samples and accentuate room for the differentiation of M from NM cross-sections. Two universal markers emphatically achieve this goal performing very well: (a) the ratio of the singularity parameters (left/right), which are defined relative to weak/strong fluctuations in the multifractal spectrum, is always greater than 0.8 for NM tissues; and (b) the index of multifractality, used to classify universal multifractals, points to log-normal distribution for NM and to log-Cauchy for M tissues. An immediate large-scale screening of cancerous sections is doable based on these findings.

Hurst analysis of dynamic networks

The sequence of network snapshots with time stamps is an effective tool for describing system dynamics. First, this article constructs a multifractal analysis of a snapshot network, in which the Hurst integral is used to describe the fractal structure hidden in structural dynamics. Second, we adjusted the network model and conducted comparative analysis to clarify the meaning of the Hurst exponent and found that the snapshot network usually includes multiple fractal structures, such as local and global fractal structures. Finally, we discussed the fractal structure of two real network datasets. We found that the real snapshot network also includes rich dynamics, which can be distinguished by the Hurst exponent. In particular, the dynamics of financial networks includes multifractal structures. This article provides a perspective to study the dynamic networks, thereby indirectly describing the fractal characteristics of complex system dynamics.

Discriminating chaotic and stochastic time series using permutation entropy and artificial neural networks

Extracting relevant properties of empirical signals generated by nonlinear, stochastic, and high-dimensional systems is a challenge of complex systems research. Open questions are how to differentiate chaotic signals from stochastic ones, and how to quantify nonlinear and/or high-order temporal correlations. Here we propose a new technique to reliably address both problems. Our approach follows two steps: first, we train an artificial neural network (ANN) with flicker (colored) noise to predict the value of the parameter, [Formula: see text], that determines the strength of the correlation of the noise. To predict [Formula: see text] the ANN input features are a set of probabilities that are extracted from the time series by using symbolic ordinal analysis. Then, we input to the trained ANN the probabilities extracted from the time series of interest, and analyze the ANN output. We find that the [Formula: see text] value returned by the ANN is informative of the temporal correlations present in the time series. To distinguish between stochastic and chaotic signals, we exploit the fact that the difference between the permutation entropy (PE) of a given time series and the PE of flicker noise with the same [Formula: see text] parameter is small when the time series is stochastic, but it is large when the time series is chaotic. We validate our technique by analysing synthetic and empirical time series whose nature is well established. We also demonstrate the robustness of our approach with respect to the length of the time series and to the level of noise. We expect that our algorithm, which is freely available, will be very useful to the community.

Time-averaged mean squared displacement ratio test for Gaussian processes with unknown diffusion coefficient

The time-averaged mean squared displacement (TAMSD) is one of the most common statistics used for the analysis of anomalous diffusion processes. Anomalous diffusion is manifested by non-linear (mostly power-law) characteristics of the process in contrast to normal diffusion where linear characteristics are expected. One can distinguish between sub- and super-diffusive processes. We consider Gaussian anomalous diffusion models and propose a new approach used for their testing. This approach is based on the TAMSD ratio statistic for different time lags. Similar to the TAMSD, this statistic exhibits a specific behavior in the anomalous diffusion regime. Through its structure, it is independent of the diffusion coefficient, which, in general, does not influence anomalous diffusion behavior. Thus, the TAMSD ratio-based approach does not require preliminary knowledge of the diffusion coefficient's value, in contrast to the TAMSD-approach, where this value is crucial in the testing procedure. Based on the quadratic form representation of the TAMSD ratio, we calculate its main characteristics and propose a step-by-step testing procedure that can be applied for any Gaussian process. For the anomalous diffusion model used here, namely, the fractional Brownian motion, we demonstrate the effectiveness of the proposed methodology. We show that the new approach outperforms the TAMSD-based one, especially for small sample sizes. Finally, the methodology is applied to the real data from the financial market.

Microscopy-based infrared spectroscopy as a tool to evaluate the influence of essential oil on the surface of loaded bilayered-nanoparticles

Increasing interest in nanoparticles of technological application has been improving their fabrication processes. The encapsulation of essential oils as bioactive compounds has proved to be an excellent alternative to the use of less environment friendly compounds. However, the difficulty of identifying their constitution and interaction with carrier agents have aroused scientific interest and a problem to overcome. Bilayer-based nanoparticles were developed using gelatin and poly-ε-caprolactone (PCL) aiming the encapsulation ofPiper nigrumessential oil. based on atomic force microscopy images and dynamic light scattering analysis, the size of the unloaded and loaded nanoparticles was found around (194 ± 40) and (296 ± 54) nm, respectively. The spatial patterns revealed that the surface of nanoparticles presented different surface roughness, similar shapes and height distribution asymmetry, lower dominant spatial frequencies, and different spatial complexity. Traditional infrared spectroscopy allowed the identification of the nanoparticle outermost layer formed by the gelatin carrier, but microscopy-based infrared spectroscopy revealed a band at 1742 cm^-1related to the carbonyl stretching mode of PCL, as well as a band at 1557 cm^-1due to the amide II group from gelatin. The combination of microscopy and spectroscopy techniques proved to be an efficient alternative to quickly identify differences in chemical composition by evaluating different functional groups in bilayer PLC/gelatin nanoparticles of technological application.

Probabilistic properties of detrended fluctuation analysis for Gaussian processes

Detrended fluctuation analysis (DFA) is one of the most widely used tools for the detection of long-range dependence in time series. Although DFA has found many interesting applications and has been shown to be one of the best performing detrending methods, its probabilistic foundations are still unclear. In this paper, we study probabilistic properties of DFA for Gaussian processes. Our main attention is paid to the distribution of the squared error sum of the detrended process. We use a probabilistic approach to derive general formulas for the expected value and the variance of the squared fluctuation function of DFA for Gaussian processes. We also get analytical results for the expected value of the squared fluctuation function for particular examples of Gaussian processes, such as Gaussian white noise, fractional Gaussian noise, ordinary Brownian motion, and fractional Brownian motion. Our analytical formulas are supported by numerical simulations. The results obtained can serve as a starting point for analyzing the statistical properties of DFA-based estimators for the fluctuation function and long-memory parameter.

Detrended fluctuation analysis using orthogonal polynomials

An alternative analysis approach, namely, orthogonal detrended fluctuation analysis (ODFA), is proposed to quantify the long-range correlation exponent. This method uses an orthogonal polynomial to attenuate any trends and quantify the (auto-) correlations in the data. The method is tested using numerically simulated data with long-range correlation. A matrix formalism of this approach is also proposed. Furthermore, the extension to high-order polynomial detrending is discussed. The proposed approach quantifies the long-range exponent with an error rate of about 8% for short datasets (3000 samples) and an error rate of about 1% for long datasets (100 000 samples). ODFA can find applications that involve processing long datasets as well as in real-time processing.

Theoretical foundation of detrending methods for fluctuation analysis such as detrended fluctuation analysis and detrending moving average

We present a bottom-up derivation of fluctuation analysis with detrending for the detection of long-range correlations in the presence of additive trends or intrinsic nonstationarities. While the well-known detrended fluctuation analysis (DFA) and detrending moving average (DMA) were introduced ad hoc, we claim basic principles for such methods where DFA and DMA are then shown to be specific realizations. The mean-squared displacement of the summed time series contains the same information about long-range correlations as the autocorrelation function but has much better statistical properties for large time lags. However, the scaling exponent of its estimator on a single time series is affected not only by trends on the data but also by intrinsic nonstationarities. We therefore define the fluctuation function as mean-squared displacement with weighting kernel. We require that its estimator be unbiased and exhibit the correct scaling behavior for the random component of a signal, which is only achieved if the weighting kernel implies detrending. We show how DFA and DMA satisfy these requirements and we extract their kernel weights.

Dirac fermions duality in graphene: Ripples and fractional dimensions as function of temperature

Graphene consists of coupled direct/dual fermionic sub-systems and, consequently, the thermal properties of both are intrinsically correlated. The dual is characterized by negative temperatures, and its free energy keeps opposite sign concerning the direct. The growth of ripples in graphene becomes related to temperature rises with fractional spatial dimension ~2.19 at 300 °K. An analytical, and suitable, expression for ripples dimension as a function of temperature is presented. Further, internal energy, entropy, specific heat and free energy are evaluated as a function of temperature and dimension for both sub-systems. Free energy supports a simple, functional expression inversely proportional to ripples dimension.

Direct determination approach for the multifractal detrending moving average analysis

In the canonical framework, we propose an alternative approach for the multifractal analysis based on the detrending moving average method (MF-DMA). We define a canonical measure such that the multifractal mass exponent τ(q) is related to the partition function and the multifractal spectrum f(α) can be directly determined. The performances of the direct determination approach and the traditional approach of the MF-DMA are compared based on three synthetic multifractal and monofractal measures generated from the one-dimensional p-model, the two-dimensional p-model, and the fractional Brownian motions. We find that both approaches have comparable performances to unveil the fractal and multifractal nature. In other words, without loss of accuracy, the multifractal spectrum f(α) can be directly determined using the new approach with less computation cost. We also apply the new MF-DMA approach to the volatility time series of stock prices and confirm the presence of multifractality.

Multifractality and Network Analysis of Phase Transition

Many models and real complex systems possess critical thresholds at which the systems shift dramatically from one sate to another. The discovery of early-warnings in the vicinity of critical points are of great importance to estimate how far the systems are away from the critical states. Multifractal Detrended Fluctuation analysis (MF-DFA) and visibility graph method have been employed to investigate the multifractal and geometrical properties of the magnetization time series of the two-dimensional Ising model. Multifractality of the time series near the critical point has been uncovered from the generalized Hurst exponents and singularity spectrum. Both long-term correlation and broad probability density function are identified to be the sources of multifractality. Heterogeneous nature of the networks constructed from magnetization time series have validated the fractal properties. Evolution of the topological quantities of the visibility graph, along with the variation of multifractality, serve as new early-warnings of phase transition. Those methods and results may provide new insights about the analysis of phase transition problems and can be used as early-warnings for a variety of complex systems.

Time and frequency domain characteristics of detrending-operation-based scaling analysis: Exact DFA and DMA frequency responses

We develop a general framework to study the time and frequency domain characteristics of detrending-operation-based scaling analysis methods, such as detrended fluctuation analysis (DFA) and detrending moving average (DMA) analysis. In this framework, using either the time or frequency domain approach, the frequency responses of detrending operations are calculated analytically. Although the frequency domain approach based on conventional linear analysis techniques is only applicable to linear detrending operations, the time domain approach presented here is applicable to both linear and nonlinear detrending operations. Furthermore, using the relationship between the time and frequency domain representations of the frequency responses, the frequency domain characteristics of nonlinear detrending operations can be obtained. Based on the calculated frequency responses, it is possible to establish a direct connection between the root-mean-square deviation of the detrending-operation-based scaling analysis and the power spectrum for linear stochastic processes. Here, by applying our methods to DFA and DMA, including higher-order cases, exact frequency responses are calculated. In addition, we analytically investigate the cutoff frequencies of DFA and DMA detrending operations and show that these frequencies are not optimally adjusted to coincide with the corresponding time scale.

Detrending moving average algorithm: Frequency response and scaling performances

The Detrending Moving Average (DMA) algorithm has been widely used in its several variants for characterizing long-range correlations of random signals and sets (one-dimensional sequences or high-dimensional arrays) over either time or space. In this paper, mainly based on analytical arguments, the scaling performances of the centered DMA, including higher-order ones, are investigated by means of a continuous time approximation and a frequency response approach. Our results are also confirmed by numerical tests. The study is carried out for higher-order DMA operating with moving average polynomials of different degree. In particular, detrending power degree, frequency response, asymptotic scaling, upper limit of the detectable scaling exponent, and finite scale range behavior will be discussed.

Fast algorithm for scaling analysis with higher-order detrending moving average method

Among scaling analysis methods based on the root-mean-square deviation from the estimated trend, it has been demonstrated that centered detrending moving average (DMA) analysis with a simple moving average has good performance when characterizing long-range correlation or fractal scaling behavior. Furthermore, higher-order DMA has also been proposed; it is shown to have better detrending capabilities, removing higher-order polynomial trends than original DMA. However, a straightforward implementation of higher-order DMA requires a very high computational cost, which would prevent practical use of this method. To solve this issue, in this study, we introduce a fast algorithm for higher-order DMA, which consists of two techniques: (1) parallel translation of moving averaging windows by a fixed interval; (2) recurrence formulas for the calculation of summations. Our algorithm can significantly reduce computational cost. Monte Carlo experiments show that the computational time of our algorithm is approximately proportional to the data length, although that of the conventional algorithm is proportional to the square of the data length. The efficiency of our algorithm is also shown by a systematic study of the performance of higher-order DMA, such as the range of detectable scaling exponents and detrending capability for removing polynomial trends. In addition, through the analysis of heart-rate variability time series, we discuss possible applications of higher-order DMA.

Fractal and multifractal characteristics of swift heavy ion induced self-affine nanostructured BaF2 thin film surfaces

Fractal and multifractal characteristics of self-affine surfaces of BaF2 thin films, deposited on crystalline Si ⟨1 1 1⟩ substrate at room temperature, were studied. Self-affine surfaces were prepared by irradiation of 120 MeV Ag(9+) ions which modified the surface morphology at nanometer scale. The surface morphology of virgin thin film and those irradiated with different ion fluences are characterized by atomic force microscopy technique. The surface roughness (interface width) shows monotonic decrease with ion fluences, while the other parameters, such as lateral correlation length, roughness exponent, and fractal dimension, did not show either monotonic decrease or increase in nature. The self-affine nature of the films is further confirmed by autocorrelation function. The power spectral density of thin films surfaces exhibits inverse power law variation with spatial frequency, suggesting the existence of fractal component in surface morphology. The multifractal detrended fluctuation analysis based on the partition function approach is also performed on virgin and irradiated thin films. It is found that the partition function exhibits the power law behavior with the segment size. Moreover, it is also seen that the scaling exponents vary nonlinearly with the moment, thereby exhibiting the multifractal nature.

Detrended partial cross-correlation analysis of two nonstationary time series influenced by common external forces

When common factors strongly influence two power-law cross-correlated time series recorded in complex natural or social systems, using detrended cross-correlation analysis (DCCA) without considering these common factors will bias the results. We use detrended partial cross-correlation analysis (DPXA) to uncover the intrinsic power-law cross correlations between two simultaneously recorded time series in the presence of nonstationarity after removing the effects of other time series acting as common forces. The DPXA method is a generalization of the detrended cross-correlation analysis that takes into account partial correlation analysis. We demonstrate the method by using bivariate fractional Brownian motions contaminated with a fractional Brownian motion. We find that the DPXA is able to recover the analytical cross Hurst indices, and thus the multiscale DPXA coefficients are a viable alternative to the conventional cross-correlation coefficient. We demonstrate the advantage of the DPXA coefficients over the DCCA coefficients by analyzing contaminated bivariate fractional Brownian motions. We calculate the DPXA coefficients and use them to extract the intrinsic cross correlation between crude oil and gold futures by taking into consideration the impact of the U.S. dollar index. We develop the multifractal DPXA (MF-DPXA) method in order to generalize the DPXA method and investigate multifractal time series. We analyze multifractal binomial measures masked with strong white noises and find that the MF-DPXA method quantifies the hidden multifractal nature while the multifractal DCCA method fails.

Information measure for long-range correlated sequences: the case of the 24 human chromosomes

A new approach to estimate the Shannon entropy of a long-range correlated sequence is proposed. The entropy is written as the sum of two terms corresponding respectively to power-law (ordered) and exponentially (disordered) distributed blocks (clusters). The approach is illustrated on the 24 human chromosome sequences by taking the nucleotide composition as the relevant information to be encoded/decoded. Interestingly, the nucleotide composition of the ordered clusters is found, on the average, comparable to the one of the whole analyzed sequence, while that of the disordered clusters fluctuates. From the information theory standpoint, this means that the power-law correlated clusters carry the same information of the whole analysed sequence. Furthermore, the fluctuations of the nucleotide composition of the disordered clusters are linked to relevant biological properties, such as segmental duplications and gene density.

Two-dimensional matrix algorithm using detrended fluctuation analysis to distinguish Burkitt and diffuse large B-cell lymphoma

A detrended fluctuation analysis (DFA) method is applied to image analysis. The 2-dimensional (2D) DFA algorithms is proposed for recharacterizing images of lymph sections. Due to Burkitt lymphoma (BL) and diffuse large B-cell lymphoma (DLBCL), there is a significant different 5-year survival rates after multiagent chemotherapy. Therefore, distinguishing the difference between BL and DLBCL is very important. In this study, eighteen BL images were classified as group A, which have one to five cytogenetic changes. Ten BL images were classified as group B, which have more than five cytogenetic changes. Both groups A and B BLs are aggressive lymphomas, which grow very fast and require more intensive chemotherapy. Finally, ten DLBCL images were classified as group C. The short-term correlation exponent α1 values of DFA of groups A, B, and C were 0.370 ± 0.033, 0.382 ± 0.022, and 0.435 ± 0.053, respectively. It was found that α1 value of BL image was significantly lower (P < 0.05) than DLBCL. However, there is no difference between the groups A and B BLs. Hence, it can be concluded that α1 value based on DFA statistics concept can clearly distinguish BL and DLBCL image.

Comparing the performance of FA, DFA and DMA using different synthetic long-range correlated time series

Notwithstanding the significant efforts to develop estimators of long-range correlations (LRC) and to compare their performance, no clear consensus exists on what is the best method and under which conditions. In addition, synthetic tests suggest that the performance of LRC estimators varies when using different generators of LRC time series. Here, we compare the performances of four estimators [Fluctuation Analysis (FA), Detrended Fluctuation Analysis (DFA), Backward Detrending Moving Average (BDMA), and Centred Detrending Moving Average (CDMA)]. We use three different generators [Fractional Gaussian Noises, and two ways of generating Fractional Brownian Motions]. We find that CDMA has the best performance and DFA is only slightly worse in some situations, while FA performs the worst. In addition, CDMA and DFA are less sensitive to the scaling range than FA. Hence, CDMA and DFA remain "The Methods of Choice" in determining the Hurst index of time series.

Ridge network detection in crumpled paper via graph density maximization

Crumpled sheets of paper tend to exhibit a specific and complex structure, which is described by physicists as ridge networks. Existing literature shows that the automation of ridge network detection in crumpled paper is very challenging because of its complex structure and measuring distortion. In this paper, we propose to model the ridge network as a weighted graph and formulate the ridge network detection as an optimization problem in terms of the graph density. First, we detect a set of graph nodes and then determine the edge weight between each pair of nodes to construct a complete graph. Next, we define a graph density criterion and formulate the detection problem to determine a subgraph with maximal graph density. Further, we also propose to refine the graph density by including a pairwise connectivity into the criterion to improve the connectivity of the detected ridge network. Our experimental results show that, with the density criterion, our proposed method effectively automates the ridge network detection.

Self-similarity of higher-order moving averages

In this work, higher-order moving average polynomials are defined by straightforward generalization of the standard moving average. The self-similarity of the polynomials is analyzed for fractional Brownian series and quantified in terms of the Hurst exponent H by using the detrending moving average method. We prove that the exponent H of the fractional Brownian series and of the detrending moving average variance asymptotically agree for the first-order polynomial. Such asymptotic values are compared with the results obtained by the simulations. The higher-order polynomials correspond to trend estimates at shorter time scales as the degree of the polynomial increases. Importantly, the increase of polynomial degree does not require to change the moving average window. Thus trends at different time scales can be obtained on data sets with the same size. These polynomials could be interesting for those applications relying on trend estimates over different time horizons (financial markets) or on filtering at different frequencies (image analysis).

Multifractal detrending moving-average cross-correlation analysis

There are a number of situations in which several signals are simultaneously recorded in complex systems, which exhibit long-term power-law cross correlations. The multifractal detrended cross-correlation analysis (MFDCCA) approaches can be used to quantify such cross correlations, such as the MFDCCA based on the detrended fluctuation analysis (MFXDFA) method. We develop in this work a class of MFDCCA algorithms based on the detrending moving-average analysis, called MFXDMA. The performances of the proposed MFXDMA algorithms are compared with the MFXDFA method by extensive numerical experiments on pairs of time series generated from bivariate fractional Brownian motions, two-component autoregressive fractionally integrated moving-average processes, and binomial measures, which have theoretical expressions of the multifractal nature. In all cases, the scaling exponents h(xy) extracted from the MFXDMA and MFXDFA algorithms are very close to the theoretical values. For bivariate fractional Brownian motions, the scaling exponent of the cross correlation is independent of the cross-correlation coefficient between two time series, and the MFXDFA and centered MFXDMA algorithms have comparative performances, which outperform the forward and backward MFXDMA algorithms. For two-component autoregressive fractionally integrated moving-average processes, we also find that the MFXDFA and centered MFXDMA algorithms have comparative performances, while the forward and backward MFXDMA algorithms perform slightly worse. For binomial measures, the forward MFXDMA algorithm exhibits the best performance, the centered MFXDMA algorithms performs worst, and the backward MFXDMA algorithm outperforms the MFXDFA algorithm when the moment order q<0 and underperforms when q>0. We apply these algorithms to the return time series of two stock market indexes and to their volatilities. For the returns, the centered MFXDMA algorithm gives the best estimates of h(xy)(q) since its h(xy)(2) is closest to 0.5, as expected, and the MFXDFA algorithm has the second best performance. For the volatilities, the forward and backward MFXDMA algorithms give similar results, while the centered MFXDMA and the MFXDFA algorithms fail to extract rational multifractal nature.

Snow metamorphism: A fractal approach

Snow is a porous disordered medium consisting of air and three water phases: ice, vapor, and liquid. The ice phase consists of an assemblage of grains, ice matrix, initially arranged over a random load bearing skeleton. The quantitative relationship between density and morphological characteristics of different snow microstructures is still an open issue. In this work, a three-dimensional fractal description of density corresponding to different snow microstructure is put forward. First, snow density is simulated in terms of a generalized Menger sponge model. Then, a fully three-dimensional compact stochastic fractal model is adopted. The latter approach yields a quantitative map of the randomness of the snow texture, which is described as a three-dimensional fractional Brownian field with the Hurst exponent H varying as continuous parameters. The Hurst exponent is found to be strongly dependent on snow morphology and density. The approach might be applied to all those cases where the morphological evolution of snow cover or ice sheets should be conveniently described at a quantitative level.

Detrending moving average algorithm for multifractals

The detrending moving average (DMA) algorithm is a widely used technique to quantify the long-term correlations of nonstationary time series and the long-range correlations of fractal surfaces, which contains a parameter θ determining the position of the detrending window. We develop multifractal detrending moving average (MFDMA) algorithms for the analysis of one-dimensional multifractal measures and higher-dimensional multifractals, which is a generalization of the DMA method. The performance of the one-dimensional and two-dimensional MFDMA methods is investigated using synthetic multifractal measures with analytical solutions for backward (θ=0), centered (θ=0.5), and forward (θ=1) detrending windows. We find that the estimated multifractal scaling exponent τ(q) and the singularity spectrum f(α) are in good agreement with the theoretical values. In addition, the backward MFDMA method has the best performance, which provides the most accurate estimates of the scaling exponents with lowest error bars, while the centered MFDMA method has the worse performance. It is found that the backward MFDMA algorithm also outperforms the multifractal detrended fluctuation analysis. The one-dimensional backward MFDMA method is applied to analyzing the time series of Shanghai Stock Exchange Composite Index and its multifractal nature is confirmed.

Fractal heterogeneous media

A method is presented for generating compact fractal disordered media by generalizing the random midpoint displacement algorithm. The obtained structures are invasive stochastic fractals, with the Hurst exponent varying as a continuous parameter, as opposed to lacunar deterministic fractals, such as the Menger sponge. By employing the detrending moving average algorithm [A. Carbone, Phys. Rev. E 76, 056703 (2007)], the Hurst exponent of the generated structure can be subsequently checked. The fractality of such a structure is referred to a property defined over a three-dimensional topology rather than to the topology itself. Consequently, in this framework, the Hurst exponent should be intended as an estimator of compactness rather than of roughness. Applications can be envisaged for simulating and quantifying complex systems characterized by self-similar heterogeneity across space. For example, exploitation areas range from the design and control of multifunctional self-assembled artificial nanostructures and microstructures to the analysis and modeling of complex pattern formation in biology, environmental sciences, geomorphological sciences, etc.

The compass rose pattern in electricity prices

The "compass rose pattern" is known to appear in the phase portraits, or scatter diagrams, of the high-frequency returns of financial series. We first show that this pattern is also present in the returns of spot electricity prices. Early researchers investigating these phenomena hoped that these patterns signaled the presence of rich dynamics, possibly chaotic or fractal in nature. Although there is a definite autoregressive and conditional heteroscedasticity structure in electricity returns, we find that after simple filtering no pattern remains. While the series is non-normal in terms of their distribution and statistical tests fail to identify significant chaos, there is evidence of fractal structures in periodic price returns when measured over the trading day. The phase diagram of the filtered returns provides a useful visual check on independence, a property necessary for pricing and trading derivatives and portfolio construction, as well as providing useful insights into the market dynamics.

Efficient scheme for parametric fitting of data in arbitrary dimensions

We propose an efficient scheme for parametric fitting expressed in terms of the Legendre polynomials. For continuous systems, our scheme is exact and the derived explicit expression is very helpful for further analytical studies. For discrete systems, our scheme is almost as accurate as the method of singular value decomposition. Through a few numerical examples, we show that our algorithm costs much less CPU time and memory space than the method of singular value decomposition. Thus, our algorithm is very suitable for a large amount of data fitting. In addition, the proposed scheme can also be used to extract the global structure of fluctuating systems. We then derive the exact relation between the correlation function and the detrended variance function of fluctuating systems in arbitrary dimensions and give a general scaling analysis.