Fractal Scaling Analysis Research Articles

The fractal turbulent–non-turbulent interface in the atmosphere

Abstract. With their constant increase in size, wind turbines are reaching unprecedented heights. Therefore, at these heights, they are influenced by wind conditions that have not yet been studied in detail. With increasing height, a transition to laminar conditions becomes more and more likely. In this paper, the presence of the turbulent–non-turbulent interface (TNTI) in the atmosphere is investigated. Three different on- and offshore locations are investigated. Our fractal scaling analysis leads to typical values known from ideal laboratory and numerical studies. The height distribution of the probability of the TNTI is determined and shows a frequent occurrence at the height of the rotor of future multi-megawatt turbines. The indicated universality of the fractality of the TNTI allows the use of simplified models in laboratory and numerical investigations.

Breast cancer: A fractal approach, scaling analysis and quantum cluster algorithm

Tumor growth in the breast is a complex process, understanding this type of phenomenon is important in order to establish adequate strategies at diagnostic and therapeutic levels, beginning with the evaluation morphology, through adequate descriptors. Scaling analysis is used to extract parameters such as exponents of local roughness and fractal dimension, which characterize tumor growth. The results show that invasive lobular and ductal tumors in situ behave as proposed by Family-Vicsek (Family and Vicsek, 1991), while benign tumors behave differently.

Multidimensional fractal scaling analysis using higher order moving average polynomials and its fast algorithm

The detrending moving average (DMA) analysis demonstrates excellent performance for the characterization of long-range correlations and fractal scaling and is performed in various research fields. The conventional DMA with a simple moving average can remove linear trends embedded in the observed time series. To improve the detrending ability of the DMA, higher-order DMA including a higher order polynomial detrending was also introduced using the Savitzky-Golay filter and its fast implementation algorithm was developed. However, the higher-order DMA applicable to higher dimensional data is yet to be well established. As the data dimension increases, an increase in the computational cost becomes a problem that needs to be resolved. Further, the implementation of the higher order DMA is a time-consuming procedure. To resolve this problem, we here proposed a fast algorithm for multidimensional DMA with higher order polynomial detrending. In the proposed algorithm, to reduce the computational complexity, parallel translation and recurrence techniques are introduced. Monte Carlo experiments for two-dimensional data show that the computational time of the proposed algorithm is approximately proportional to the cubic of the data length, whereas the computational time of the conventional implementation is approximately proportional to the quartic of the data length. Moreover, we evaluate the estimation accuracy of the Hurst exponent of the proposed method. Finally, we demonstrate the possible application of the proposed method by estimating the Hurst exponent of images.

Fractal correlations in the Covid-19 genome sequence via multivariate rescaled range analysis

An approach based on fractal scaling analysis to characterize the organization of the Covid-19 genome sequences is presented in this work. The method is based on a multivariate version of the fractal rescaled range analysis implemented on a sliding window scheme to detect variations of long-range correlations over the genome sequence domains. As a preliminary step, the nucleotide sequence is mapped in a numerical sequence by following a Voss rule, resulting in a multichannel sequence represented as a binary matrix. Fractal correlations, quantified in terms of the Hurst exponent, depending on the region of the sequence, where the Covid-19 genome sequences are predominantly random, with some patches of weak long-range correlations. The analysis shows that the regions of randomness are more abundant in the Covid-19 sequences than in the primitive SARS sequence, which suggests that the Covid-19 virus possesses a more diverse genomic structure for replication and infection. The analysis constrained to the surface glycoprotein region shows that the Covid-19 sequence is less random as compared to the SARS sequence, which indicates that the Covid-19 virus can undergo more ordered replications of the spike protein. The Omicron variation exhibits an interesting pattern with some randomness similarities with the other SARS and the Covid-19 genome sequences. Overall, the results show that the multivariate rescaled range analysis provides a suitable framework to assess long-term correlations hidden in the internal organization of the Covid-19 genome sequence.

Multidimensional Fractal Scaling Analysis Using Higher Order Moving Average Polynomials and Its Fast Algorithm

Multidimensional Fractal Scaling Analysis Using Higher Order Moving Average Polynomials and Its Fast Algorithm

A fractal scaling analysis of the SARS-CoV-2 genome sequence

An approach based on fractal scaling analysis to characterize the organization of the SARS-CoV-2 genome sequence was used. The method is based on the detrended fluctuation analysis (DFA) implemented on a sliding window scheme to detect variations of long-range correlations over the genome sequence regions. The nucleotides sequence is mapped in a numerical sequence by using four different assignation rules: amino-keto, purine-pyrimidine, hydrogen-bond and hydrophobicity patterns. The originally reported sequence from Wuhan isolates (Wuhan Hu-1) was considered as a reference to contrast the structure of the 2002–2004 SARS-CoV-1 strain. Long-range correlations, quantified in terms of a scaling exponent, depended on both the mapping rule and the sequence region. Deviations from randomness were attributed to serial correlations or anti-correlations, which can be ascribed to ordered regions of the genome sequence. It was found that the Wuhan Hu-1 sequence was more random than the SARS-CoV-1 sequence, which suggests that the SARS-CoV-2 possesses a more efficient genomic structure for replication and infection. In general, the virus isolated in the early 2020 months showed slight correlation differences with the Wuhan Hu-1 sequence. However, early isolates from India and Italy presented visible differences that led to a more ordered sequence organization. It is apparent that the increased sequence order, particularly in the spike region, endowed some early variants with a more efficient mechanism to spreading, replicating and infecting. Overall, the results showed that the DFA provides a suitable framework to assess long-term correlations hidden in the internal organization of the SARS-CoV-2 genome sequence.

An Extended Correlation Dimension of Complex Networks.

Fractal and self-similarity are important characteristics of complex networks. The correlation dimension is one of the measures implemented to characterize the fractal nature of unweighted structures, but it has not been extended to weighted networks. In this paper, the correlation dimension is extended to the weighted networks. The proposed method uses edge-weights accumulation to obtain scale distances. It can be used not only for weighted networks but also for unweighted networks. We selected six weighted networks, including two synthetic fractal networks and four real-world networks, to validate it. The results show that the proposed method was effective for the fractal scaling analysis of weighted complex networks. Meanwhile, this method was used to analyze the fractal properties of the Newman–Watts (NW) unweighted small-world networks. Compared with other fractal dimensions, the correlation dimension is more suitable for the quantitative analysis of small-world effects.

Application of the Force Balance Model and Fractal Scaling Analysis for Size Estimation of the Complex-Agglomerates in a Conical Fluidized Bed

The size estimating of fluidized Titania agglomerates in a conical fluidized bed was studied by force balance model and fractal scaling analysis. The primary size of Titania Nano Particles (NPs) was 21 nm, while for complex agglomerates was in the size range of several hundred micrometers. The formation mechanism of simple-agglomerate and complex-agglomerate structures was studied experimentally. The size distribution and morphology of agglomerates were determined by advanced laser dynamic imaging and scanning electron microscopy. The AFM-nanoindentation test was used to determine the elastic modulus of agglomerates with porous structures. The size distribution of Titania NP agglomerates was estimated by the fractal analysis through the relationship between the number of particles and gyration diameter. The fractal exponent obtained from the power-law scaling of agglomerates and the complex agglomerate sizes were determined experimentally and theoretically. A simple theoretical model was applied to estimate the complex agglomerates' size based on the equilibrium of the separation and cohesion forces. The proposed model showed satisfactory results compared with the experimental data. The results of the present study can help to determine the critical gas velocity in achieving the desired agglomerate size of Titania NPs.

Fractal scaling laws for the dynamic evolution of sentiments in Never Let Me Go and their implications for writing, adaptation and reading of novels

A good novel can often elicit from a reader strong sentiments similar to the moods, feelings, and attitudes depicted in the novel. With the rapid progress in AI, sentiment-based arcs in novels can now be reliably extracted and used to summarize the novel’s plot in the story arc. Are there salient mathematical properties that underlie such story arcs and have far-reaching implications in the writing, adaptation, and reading of the novel? To gain insights into this question, we employ multifractal theory to characterize the narrative coherence and dynamic evolution of sentiments of the novel, Never Let Me Go, by Kazuo Ishiguro, the winner of the 2017 Nobel Prize for Literature as an example. Three methods are compared for fractal scaling analysis, the classic variance-time method, an improvement of the variance-time relation based on adaptive filtering, and adaptive fractal analysis. We find that while variance-time relation fails to accurately extract the fractal scaling exponent, adaptive fractal analysis succeeds in fully characterizing the fractal variations in the sentiment dynamics. The finding may be indicative of the potential that multifractal theory has for computational narratology and large-scale literary analysis, especially for inferring the degree of narrative coherence and variation of the plot of a novel.

Fractal and multifractal analysis of complex networks: Estonian network of payments

Complex networks have gained much attention from different areas of knowledge in recent years. Particularly, the structures and dynamics of such systems have attracted considerable interest. Complex networks may have characteristics of multifractality. In this study, we analyze fractal and multifractal properties of a novel network: the large scale economic network of payments of Estonia, where companies are represented by nodes and the payments done between companies are represented by links. We present a fractal scaling analysis and examine the multifractal behavior of this network by using a sandbox algorithm. Our results indicate the existence of multifractality in this network and consequently, the existence of multifractality in the Estonian economy. To the best of our knowledge, this is the first study that analyzes multifractality of a complex network of payments.

Fractal scaling analysis of groundwater dynamics in confined aquifers

Abstract. Groundwater closely interacts with surface water and even climate systems in most hydroclimatic settings. Fractal scaling analysis of groundwater dynamics is of significance in modeling hydrological processes by considering potential temporal long-range dependence and scaling crossovers in the groundwater level fluctuations. In this study, it is demonstrated that the groundwater level fluctuations in confined aquifer wells with long observations exhibit site-specific fractal scaling behavior. Detrended fluctuation analysis (DFA) was utilized to quantify the monofractality, and multifractal detrended fluctuation analysis (MF-DFA) and multiscale multifractal analysis (MMA) were employed to examine the multifractal behavior. The DFA results indicated that fractals exist in groundwater level time series, and it was shown that the estimated Hurst exponent is closely dependent on the length and specific time interval of the time series. The MF-DFA and MMA analyses showed that different levels of multifractality exist, which may be partially due to a broad probability density distribution with infinite moments. Furthermore, it is demonstrated that the underlying distribution of groundwater level fluctuations exhibits either non-Gaussian characteristics, which may be fitted by the Lévy stable distribution, or Gaussian characteristics depending on the site characteristics. However, fractional Brownian motion (fBm), which has been identified as an appropriate model to characterize groundwater level fluctuation, is Gaussian with finite moments. Therefore, fBm may be inadequate for the description of physical processes with infinite moments, such as the groundwater level fluctuations in this study. It is concluded that there is a need for generalized governing equations of groundwater flow processes that can model both the long-memory behavior and the Brownian finite-memory behavior.

Autoantibody recognition mechanisms of MUC1

The most cost-effective blood-based, noninvasive molecular cancer biomarkers are based on p53 epitopes and MUC1 tandem repeats. Here we use dimensionally compressed bioinformatic fractal scaling analysis to compare the two distinct and comparable probes, which examine different sections of the autoantibody population, achieving combined sensitivities of order 50%. We discover a promising MUC1 epitope in the SEA region outside the tandem repeats.

Autoantibody recognition mechanisms of p53 epitopes

There is an urgent need for economical blood based, noninvasive molecular biomarkers to assist in the detection and diagnosis of cancers in a cost-effective manner at an early stage, when curative interventions are still possible. Serum autoantibodies are attractive biomarkers for early cancer detection, but their development has been hindered by the punctuated genetic nature of the ten million known cancer mutations. A landmark study of 50,000 patients (Pedersen et al., 2013) showed that a few p53 15-mer epitopes are much more sensitive colon cancer biomarkers than p53, which in turn is a more sensitive cancer biomarker than any other protein. The function of p53 as a nearly universal “tumor suppressor” is well established, because of its strong immunogenicity in terms of not only antibody recruitment, but also stimulation of autoantibodies. Here we examine dimensionally compressed bioinformatic fractal scaling analysis for identifying the few sensitive epitopes from the p53 amino acid sequence, and show how it could be used for early cancer detection (ECD). We trim 15-mers to 7-mers, and identify specific 7-mers from other species that could be more sensitive to aggressive human cancers, such as liver cancer. Our results could provide a roadmap for ECD.

New Perspectives in Human Movement Variability

Movement variability is defined as the normal variations that occur in motor performance across multiple repetitions of a task (Stergiou & Decker, 2011). Bernstein described movement variability quite eloquently as “repetition without repetition”. Traditionally, movement variability has been linked to noise and error, being considered to be random and independent. This theoretical approach blends well with traditional statistical and assessment methods of movement variability that assume randomness and independence of observations. However, numerous studies have indicated that when movement is observed over time variations are closely related with each other neither being random nor independent. Practically, traditional methods can mask the temporal structure of movement variability and contain little information about how movement changes over time. Importantly, a growing body of literature has shown that the variations seen in a wide variety of physiological systems may offer insight into the control of these motor systems. For example, they can enhance our understanding of the effect of various pathologies on movement allowing us to develop better prognostic and diagnostic biomedical engineering tools. Thus, many new methods have been presented recently to assess the temporal structure of movement variability. These nonlinear methods can also allow us to develop alternative theoretical models to guide experimental designs and scientific inquiry. Therefore, in this special issue we provide a thorough view of the state-of-the-art methods currently used in the mainstream biomechanics for evaluating and quantifying human movement variability. These new perspectives in human movement variability are further elaborated in the light of the temporal structure of movement variability. It is our hope that this special issue will provide a thorough background in understanding of the fundamental concepts of human movement variability and their utility in solving particular problems in motor control, and related peripheral topics and fields in the biomedical engineering. Our collection of 14 papers from leading laboratories highlights these advances in distinct human movement arenas. The articles featured in this special issue address several different motor control systems utilizing various nonlinear dynamical systems tools. The first article discusses the theory of Optimal Movement Variability (Kaipust, McGrath, Mukherjee, and Stergiou) investigating the complexity and predictability of time series data of gait characteristics by assessing fractal indexes in a dual task paradigm (detrended fluctuation analysis). Similarly, the second and third articles explore the fluctuation dynamics of cycling (Warlop, Bollens, Crevecoeur, Detrembleur, and Lejeune) and eye saccades using a regime-switching approach (Wong and Shelhamer). Fractal fluctuations of center-of-pressure data were determined in the fourth paper to better understand the cascade dynamics (Plantinus, Dixon, and Kelty-Stephen). The fifth (Ihlen and Vereijken) and sixth (Kuznetsov, bonnette, Gao, and Riley) articles propel us into the new frontier of the multifractal scaling analyses perhaps more fitting for postural stability modeling than the monofractal scaling methods. The seventh article investigates the utility of fractal scaling analysis in biofeedback training during walking (Tirosh, Cambell, Begg, and Sparrow). The next three articles employed varied linear and nonlinear methods (maximal Lyapunov exponents using Wolf’s and Rosenstein’s methods, approximate entropy, and detrended fluctuation analysis) to explore the notions of complexity and stability. The eighth article (Huisinga, Mancini, George, and Horak) probed the movement variability of walking using a trunk accelerometer data of patients with Multiple Sclerosis and healthy controls. The ninth article (Nessler, McMillan, Schoulten, Shallow, Stewart, and De Leone) elaborates on the nonlinear dynamics of desynchronized gait with an interesting walking protocol – side-by-side treadmill walking. The tenth article (Myers, Johanning, Pipinos, Schmid, and Stergiou) evaluates gait variability changes post vascular occlusion using complementary linear and nonlinear analyses. The next set of articles describes independent linear and nonlinear methods to assess characteristics related to movement variability. The eleventh article (Kurz, Arpin, Davies, and Harbourne) elaborates on the stochastic and deterministic components within the COP movement variability of children with balance impairments using a Langevin equation methodology. The next article (Ramdani, Tallon, Bernard, and Blain) discusses an interesting approach of differentiating older fallers and non-fallers utilizing the recurrence quantification analysis. The thirteenth article (Roerdink, Ridderikhoff, Peper, and Beek) exploits informational timing and neuromuscular properties in the control of rhythmic movements. In the last article of our special issue, footfall placement variability is investigated using a novel Fourier based variability concept (Socie, Sandroff, Pula, Hsiao-Wecksler, Motl, and Sosnoff). We hope that you will find these articles stimulating and informative, and timely in your research in understanding human movement variability.

Automatic Detection of Pathological Voices Using Complexity Measures, Noise Parameters, and Mel-Cepstral Coefficients

This paper proposes a new approach to improve the amount of information extracted from the speech aiming to increase the accuracy of a system developed for the automatic detection of pathological voices. The paper addresses the discrimination capabilities of 11 features extracted using nonlinear analysis of time series. Two of these features are based on conventional nonlinear statistics (largest Lyapunov exponent and correlation dimension), two are based on recurrence and fractal-scaling analysis, and the remaining are based on different estimations of the entropy. Moreover, this paper uses a strategy based on combining classifiers for fusing the nonlinear analysis with the information provided by classic parameterization approaches found in the literature (noise parameters and mel-frequency cepstral coefficients). The classification was carried out in two steps using, first, a generative and, later, a discriminative approach. Combining both classifiers, the best accuracy obtained is 98.23% ± 0.001.

Characterizing heart rate variability by scale-dependent Lyapunov exponent

Previous studies on heart rate variability (HRV) using chaos theory, fractal scaling analysis, and many other methods, while fruitful in many aspects, have produced much confusion in the literature. Especially the issue of whether normal HRV is chaotic or stochastic remains highly controversial. Here, we employ a new multiscale complexity measure, the scale-dependent Lyapunov exponent (SDLE), to characterize HRV. SDLE has been shown to readily characterize major models of complex time series including deterministic chaos, noisy chaos, stochastic oscillations, random 1/f processes, random Levy processes, and complex time series with multiple scaling behaviors. Here we use SDLE to characterize the relative importance of nonlinear, chaotic, and stochastic dynamics in HRV of healthy, congestive heart failure, and atrial fibrillation subjects. We show that while HRV data of all these three types are mostly stochastic, the stochasticity is different among the three groups.

Discriminating brain activity from task-related artifacts in functional MRI: Fractal scaling analysis simulation and application

Functional magnetic resonance imaging (fMRI) signal changes can be separated from background noise by various processing algorithms, including the well-known deconvolution method. However, discriminating signal changes due to task-related brain activities from those due to task-related head motion or other artifacts correlated in time to the task has been little addressed. We examine whether three exploratory fractal scaling analyses correctly classify these possibilities by capturing temporal self-similarity; namely, fluctuation analysis, wavelet multi-resolution analysis, and detrended fluctuation analysis (DFA). We specifically evaluate whether these fractal analytic methods can be effective and reliable in discriminating activations from artifacts. DFA is indeed robust for such classification. Brain activation maps derived by DFA are similar, but not identical, to maps derived by deconvolution. Deconvolution explicitly utilizes task timing to extract the signals whereas DFA does not, so these methods reveal somewhat different information from the data. DFA is better than deconvolution for distinguishing fMRI activations from task-related artifacts, although a combination of these approaches is superior to either one taken alone. We also present a method for estimating noise levels in fMRI data, validated with numerical simulations suggesting that Birn's model is effective for simulating fMRI signals. Simulations further corroborate that DFA is excellent at discriminating signal changes due to task-related brain activities from those due to task-related artifacts, under a range of conditions.

TARGET DETECTION WITHIN SEA CLUTTER: A COMPARATIVE STUDY BY FRACTAL SCALING ANALYSES

Sea clutter refers to the radar returns from a patch of ocean surface. Accurate modeling of sea clutter and robust detection of low observable targets within sea clutter are important problems in remote sensing and radar signal processing applications. Due to lack of fundamental understanding of the nature of sea clutter, however, no simple and effective methods for detecting targets within sea clutter have been proposed. To help solve this important problem, we apply three types of fractal scaling analyses, fluctuation analysis (FA), detrended fluctuation analysis (DFA), and the wavelet-based fractal scaling analysis to study sea clutter. Our analyses show that sea clutter data exhibit fractal behaviors in the time scale range of about 0.01 seconds to a few seconds. The physical significance of these time scales is discussed. We emphasize that time scales characterizing fractal scaling break are among the most important features for detecting patterns using fractal theory. By systematically studying 392 sea clutter time series measured under various sea and weather conditions, we find very effective methods for detecting targets within sea clutter. Based on the data available to us, the accuracy of these methods is close to 100%.

Fractal analysis of heart rate variability and mortality after an acute myocardial infarction

The recently developed fractal analysis of heart rate (HR) variability has been suggested to provide prognostic information about patients with heart failure. This prospective multicenter study was designed to assess the prognostic significance of fractal and traditional HR variability parameters in a large, consecutive series of survivors of an acute myocardial infarction (AMI). A consecutive series of 697 patients were recruited to participate 2 to 7 days after an AMI in 3 Nordic university hospitals. The conventional time-domain and spectral parameters and the newer fractal scaling indexes of HR variability were analyzed from 24-hour RR interval recordings. During the mean follow-up of 18.4 ± 6.5 months, 49 patients (7.0%) died. Of all the risk variables, a reduced short-term fractal scaling exponent (α 1 <0.65), measured by detrended fluctuation analysis, was the most powerful predictor of mortality (univariate relative risk 5.05, 95% confidence intervals [CI] 2.87 to 8.89, p <0.001). A low scaling exponent α 1 predicted death in the patients with and without depressed left ventricular function (p <0.001 and p <0.01, respectively). Several other HR variability parameters also predicted mortality in univariate analyses, but in a multivariate analysis after adjustments for clinical variables and left ventricular ejection fraction, α 1 was the most significant independent HR variability index that predicted subsequent mortality (relative risk 3.90, 95% CI 2.03 to 7.49, p <0.001). Short-term fractal scaling analysis of HR variability is a powerful predictor of mortality among patients surviving an acute myocardial infarction.

Relation of heart rate dynamics to the occurrence of myocardial ischemia after coronary artery bypass grafting

Postoperative myocardial ischemia is a common finding after coronary artery bypass grafting (CABG) and is associated with an adverse short-term clinical outcome. The reasons and pathophysiologic background for the occurrence of ischemia after CABG are not well established. We tested the hypothesis that altered heart rate (HR) behavior precedes the onset of myocardial ischemic episodes in patients after CABG. Time-domain HR variability measurements, along with analysis of Poincaré plots and fractal scaling analysis were assessed in 40 CABG patients from 48-hour postoperative Holter recordings. Twenty patients experienced 195 ischemic episodes during the postoperative course. In the univariate analysis of HR variability measurements of the first postoperative day (POD), the increased ratio between the short-term (SD1) and long-term (SD2) HR variability analyzed from the Poincaré plot and the decreased short- and intermediate-term fractal scaling exponents α 1 and α 2 were significantly associated with ischemia during the study period (p <0.01, p <0.05, and p <0.05, respectively). In the multivariate model, the increased SD1/SD2 ratio of the first POD was the most powerful independent predictor of all possible confounding variables for the occurrence of postoperative ischemia (corresponding to a change of 0.15 U; odds ratio 2.2 and 95% confidence interval 1.2 to 5.7; p <0.01). Altered HR dynamics have been associated with myocardial ischemic episodes in patients after CABG, suggesting that the autonomic nervous system has an important role in the pathogenesis of myocardial ischemia in the postoperative phase of CABG.