Fractal Exponent Research Articles

Initial fractal exponent of heart rate variability is associated with success of early resuscitation in patients with severe sepsis or septic shock: a prospective cohort study

IntroductionHeart rate variability (HRV) reflects autonomic nervous system tone as well as the overall health of the baroreflex system. We hypothesized that loss of complexity in HRV upon intensive care unit (ICU) admission would be associated with unsuccessful early resuscitation of sepsis. MethodsWe prospectively enrolled patients admitted to ICUs with severe sepsis or septic shock from 2009 to 2011. We studied 30 minutes of electrocardiogram, sampled at 500 Hz, at ICU admission and calculated heart rate complexity via detrended fluctuation analysis. Primary outcome was vasopressor independence at 24 hours after ICU admission. Secondary outcome was 28-day mortality. ResultsWe studied 48 patients, of whom 60% were vasopressor independent at 24 hours. Five (10%) died within 28 days. The ratio of fractal alpha parameters was associated with both vasopressor independence and 28-day mortality (P = .04) after controlling for mean heart rate. In the optimal model, Sequential Organ Failure Assessment score and the long-term fractal α parameter were associated with vasopressor independence. ConclusionsLoss of complexity in HRV is associated with worse outcome early in severe sepsis and septic shock. Further work should evaluate whether complexity of HRV could guide treatment in sepsis.

Heart Rate Variability in Risk Stratification of Cardiac Patients

Heart rate (HR) variability has been extensively studied in cardiac patients, especially in patients surviving an acute myocardial infarction (AMI) and also in patients with congestive heart failure (CHF) or left ventricular (LV) dysfunction. The majority of studies have shown that patients with reduced or abnormal HR variability have an increased risk of mortality within a few years after an AMI or after a diagnosis of CHF/LV dysfunction. Various measures of HR dynamics, such as time-domain, spectral, and non-linear measures of HR variability have been used in risk stratification. The prognostic power of various measures, except of those reflecting rapid R–R interval oscillations, has been almost identical, albeit some non-linear HR variability measures, such as short-term fractal scaling exponent have provided somewhat better prognostic information than the others. Abnormal HR variability predicts both sudden and non-sudden cardiac death. Because of remodeling of the arrhythmia substrate after AMI, early measurement of HR variability to identify those at high risk should likely be repeated later in order to assess the risk of fatal arrhythmia events. Future randomized trials using HR variability/turbulence as one of the pre-defined inclusion criteria will show whether routine measurement of HR variability/turbulence will become a routine clinical tool for risk stratification of cardiac patients.

A DFA approach in well-logs for the identification of facies associations

Well-log analysis is a useful tool for the lithological description of wells. Its adequate interpretation allows determining different rock properties such as permeability, density, resistivity and porosity among others. However, given the complexity inherent in the signals, the identification of lithological properties from well-log analysis is not an easy task. In this work, an alternative methodology for sedimentary facies identification based on the detrended fluctuation analysis (DFA) is presented. Our methodology has been calibrated using information from a reference well located at the Chicontepec formation of the Tampico-Misantla basin in Mexico. Its characterization includes the interpretation of different well-logs and cores. Our results indicate that well-log signals present fractal characteristics exhibiting long-range memory. For the gamma ray, resistive and sonic logs a direct relationship between the scaling exponent as a function of the depth and rock types is observed. In this way, the fractal scaling exponents estimated with DFA can be used to identify different sedimentary facies.

Nonlinear analysis of dynamic signature

Signature is a long trained motor skill resulting in well combination of segments like strokes and loops. It is a physical manifestation of complex motor processes. The problem, generally stated, is that how relative simplicity in behavior emerges from considerable complexity of perception–action system that produces behavior within an infinitely variable biomechanical and environmental context. To solve this problem, we present evidences which indicate that motor control dynamic in signing process is a chaotic process. This chaotic dynamic may explain a richer array of time series behavior in motor skill of signature. Nonlinear analysis is a powerful approach and suitable tool which seeks for characterizing dynamical systems through concepts such as fractal dimension and Lyapunov exponent. As a result, they can be analyzed in both horizontal and vertical for time series of position and velocity. We observed from the results that noninteger values for the correlation dimension indicates low dimensional deterministic dynamics. This result could be confirmed by using surrogate data tests. We have also used time series to calculate the largest Lyapunov exponent and obtain a positive value. These results constitute significant evidence that signature data are outcome of chaos in a nonlinear dynamical system of motor control.

A Monte Carlo simulation model for surface evolution by plasma etching

A Monte Carlo simulation model, i.e. sputtering etch model, was established with Monte Carlo method to simulate plasma etching induced surface morphology evolution. The surface morphology images and fractal exponents were obtained, with α=0.5, β=0.27, z=1.76. Germanium samples were etched in SF6 plasma and the post-etch surface was analyzed in order to compare with the model. The surface morphology images and fractal exponents by simulation describe the experimental results very well.

Permeability from porosimetry measurements: Derivation for a tortuous and fractal tubular bundle

Abstract Permeability modeling of complex carbonate reservoirs is difficult. Porosity–permeability relationships are weak in carbonates and conventional porosity–permeability transforms give poor results. Even though the link between porosity and permeability in carbonates persists, other pore system properties, such as the largest connected pore-throat diameter, are more strongly linked to permeability. This important pore-throat diameter, as well as related porosity and other pore system architectural information, is determined by the analysis of mercury injection capillary pressure (MICP) porosimetry experiments. This paper explores the use of porosimetry data for the calculation of permeability as originally demonstrated by Purcell in 1949. We return to the tubular bundle model of Purcell and Burdine with a general mathematical form for the porosimetry data and a new tortuous and fractal relative tubular bundle. Using mathematical reasoning, without fitting parameters, we obtain a new formula for the computation of permeability based on the pore system architectural information of highly connected systems using the MICP porosimetry data. Moreover, we include the observation that the flow paths and the related tortuosity have a fractal aspect. The result is compared to an extensive porosimetry data set of the highly connected Arab D limestone, where vugs are absent. For porous media characterized by porosimetry data, the following expression emerges: κ ≈ 506 B v ∞ P d 2 e − 4.43 G , which is the permeability for a monomodal carbonate pore system characterized by a single Thomeer hyperbola with associated Thomeer parameters ( κ is in Darcy; B v ∞ in fractional bulk volume, and P d is the minimum entry pressure in psi and G is the pore-geometrical factor). The nearly equal sign is used here only because of an approximation used for the modified Bessel function of the second kind present in the general solution and approximate knowledge of the fractal exponent and the percolation path length ratio. There are no fitting factors. The exponents on the variables in our permeability formula demonstrate the significant shift in emphasis from porosity to the diameter of the largest connected pore throats, P d . Note that the presence of vugs are not considered in this work, since they do not form part of the Arab-D limestone matrix. This mathematical effort emphasizes the relative importance of pore system attributes on permeability as commonly found in carbonate porosimetry data. The approach can be readily extended to multimodal carbonate pore systems, to other sources of pore system architectural data and is shown to be equivalent to the operation of an incomplete Laplace transform on the porosimetry data. Importantly, and in contrast to previous permeability models to which we compare, this new formulation sets the stage for a complete and scale independent understanding of permeability in carbonate pore systems commonly encountered in the Arab D limestone and similar pore systems.

Heart rate variability in sciatica patients referred to spine surgery: a case control study

BackgroundA chronic pain condition may result in altered autonomic nervous system regulation in various patient populations. We evaluated whether autonomic regulation differs between sciatica patients referred to spine surgery and age-matched healthy controls analyzed with heart rate variability techniques (HRV).MethodsHRV of patients (n = 201) and healthy controls (n = 138) were measured in standing conditions (5 min). High frequency (HF) power as an index of cardiac vagal modulation and the low-to-high-frequency (LF/HF) ratio and short-term fractal scaling exponent α1 as indices of sympathovagal balance were analyzed. Pain intensity was assessed on a Visual Analog Scale (VAS) and perceived disability with Oswestry Disability Index.ResultsThe Oswestry and VAS scores were higher in the patients than in the controls (p < 0.0001 for both). HF power was markedly lower for the patients compared to the controls (p < 0.0001). The LF/HF ratio and α1 were higher in the patients than in the controls (p < 0.01 for both). After adjusting for sex, smoking, BMI, and leisure-time physical activity, HF power (p = 0.011) and α1 (p = 0.012) still differed between the groups. Among the patients, HF power was slightly associated with the duration of chronic pain (r = −.232, p = 0.003).ConclusionsSciatica patients referred to spine surgery had altered cardiac autonomic regulation expressed as decreased vagal activity and an increased sympathovagal balance toward sympathetic dominance when compared with age-matched healthy controls.

Evolutions of speckles on rough glass/silver surfaces with film thickness

This paper reports experimental studies on speckles produced by the rough silver films. The speckles on the rough glass/silver surfaces are measured with a microscopic imaging system. The structures of speckle patterns have the characteristics of fractals and multi-scaled sizes. We find that with the increase of the silver film thickness, the contrast of the speckles increases, and the intensity probability density functions gradually transit to exponential decay. We calculate the global and the local correlation functions of the speckle patterns, and find that both the fractal exponent and correlation length of the small-sized speckles decrease with the thickness of the silver films. We use the mechanisms of rough dielectric interface scattering and random surface plasmon waves to give the preliminary explanations for the evolutions of the speckles.

Absence of the nonpercolating phase for percolation on the nonplanar Hanoi network

We investigate bond percolation on the nonplanar Hanoi network (HN-NP), which was studied previously [Boettcher et al. Phys. Rev. E 80, 041115 (2009)]. We calculate the fractal exponent of a subgraph of the HN-NP, which gives a lower bound for the fractal exponent of the original graph. This lower bound leads to the conclusion that the original system does not have a nonpercolating phase, where only finite-size clusters exist for $p&gt;0$, or equivalently, that the system exhibits either the critical phase, where infinitely many infinite clusters exist, or the percolating phase, where a unique giant component exists. Monte Carlo simulations support our conjecture.

Heart rate variability during sleep and subsequent sleepiness in patients with chronic fatigue syndrome.

We determined whether alterations in heart rate dynamics during sleep in patients with chronic fatigue syndrome (CFS) differed from controls and/or correlated with changes of sleepiness before and after a night in the sleep laboratory. We compared beat-to-beat RR intervals (RRI) during nocturnal sleep, sleep structure, and subjective scores on visual analog scale for sleepiness in 18 CFS patients with 19 healthy controls aged 25-55 after excluding subjects with sleep disorders. A short-term fractal scaling exponent (α1) of RRI dynamics, analyzed by the detrended fluctuation analysis (DFA) method, was assessed after stratifying patients into those who reported more or less sleepiness after the night's sleep (a.m. sleepier or a.m. less sleepy, respectively). Patients in the a.m. sleepier group showed significantly (p<0.05) higher fractal scaling index α1 during non-rapid eye movement (non-REM) sleep (Stages 1, 2, and 3 sleep) than healthy controls, although standard polysomnographic measures did not differ between the groups. The fractal scaling index α1 during non-REM sleep was significantly (p<0.05) higher than that during awake periods after sleep onset for healthy controls and patients in the a.m. less sleepy group, but did not differ between sleep stages for patients in the a.m. sleepier group. For patients, changes in self-reported sleepiness before and after the night correlated positively with the fractal scaling index α1 during non-REM sleep (p<0.05). These results suggest that RRI dynamics or autonomic nervous system activity during non-REM sleep might be associated with disrupted sleep in patients with CFS.

Multifractal scaling behavior analysis for existing dams

The fractal theory was used to describe long term behavior of dam structures by means of determining (mono-) fractal exponents. Many records do not exhibit a simple monofractal scaling behavior, which can be accounted for by a single scaling exponent. In this paper the multifractal detrended fluctuation analysis (MF-DFA) is employed to analyze the time series of in situ observed data of existing dam which intrinsically reflects its long term behavior and structural evolution law. Deformation analysis of one gravity dam is taken as an example, the multifractal characteristic of the time series is obtained. The results show that this method can reliably determine the multifractal scaling behavior of time series of existing dams. The fractal theory can be applied to predict and diagnose dam behavior.

SCD-HEFT: TURBULENCE SLOPE AND FRACTAL INFORMATION FROM BASELINE HOLTER MONITORING MAY AVOID UNNECESSARY IMPLANTABLE CARDIOVERTER DEFIBRILLATOR THERAPY DESPITE HEART FAILURE

SCD-HeFT included 2,521 patients with NYHA class II or III heart failure (HF) and ejection fraction less than or equal to 35%. We hypothesized that turbulence slope (TS) and alpha1 (short-term fractal exponent from Detrended Fluctuation Analysis) of HF patients can be correlated to the necessity of

Constant dimensionality of fault roughness from the scale of micro‐fractures to the scale of continents

AbstractMany faults and fractures in various natural and man‐made materials share a remarkable common fractal property in their morphology. We report on the roughness of faults in rocks by analyzing the out‐of‐plane fluctuations of slip surfaces. They display a statistical power‐law relationship with a nearly constant fractal exponent from millimeter scale micro‐fractures in fault zones to coastlines measuring thousands of kilometers that have recorded continental breakup. A possible origin of this striking fractal relationship over 11 orders of magnitude of length scales is that all faulting processes in rocks share common characteristics that play a crucial role in the shaping of fault surfaces, such as the effects of elastic long‐range stress interactions and stress screening by mechanical heterogeneities during quasi‐static fracture growth.

Fractal analysis of leaves: are all leaves self-similar along the cane?

AbstractThe fractal dimension can be used to quantify the shape of a natural curve. Curves with similar degrees of irregularity will tend to have the same fractal dimension. The fractal exponent describes the complexity of a shape and characterizes the scale-dependency of the pattern. This article presents an application of the fractal dimension in the analysis of leaves shape. In this paper I attempt to ask question if leaves of blackberry characterized by fractal dimension differ significantly in relation to the leaf ’s position along the cane. The fractal dimension of 49 leaves of blackberry from 8 primocanes, and 53 leaves from 19 lateral canes, from 9 individuals was estimated. The mean of D of a leaf is 1.12. There are no significant differences between D for leaves from two different cane types. Previous studies were focused on measurements of fractal dimension of leaves randomly chosen from one or a few individuals so there was necessity to measure fractal dimension all leaves growing along the same shoot, because usually leaf shape and size change more or less along a shoot. This research confirmed that fractal dimension is much more related to the shape complexity than to the size of leaves.

Classifying depression patients and normal subjects using machine learning techniques and nonlinear features from EEG signal

Diagnosing depression in the early curable stages is very important and may even save the life of a patient. In this paper, we study nonlinear analysis of EEG signal for discriminating depression patients and normal controls. Forty-five unmedicated depressed patients and 45 normal subjects were participated in this study. Power of four EEG bands and four nonlinear features including detrended fluctuation analysis (DFA), higuchi fractal, correlation dimension and lyapunov exponent were extracted from EEG signal. For discriminating the two groups, k-nearest neighbor, linear discriminant analysis and logistic regression as the classifiers are then used. Highest classification accuracy of 83.3% is obtained by correlation dimension and LR classifier among other nonlinear features. For further improvement, all nonlinear features are combined and applied to classifiers. A classification accuracy of 90% is achieved by all nonlinear features and LR classifier. In all experiments, genetic algorithm is employed to select the most important features. The proposed technique is compared and contrasted with the other reported methods and it is demonstrated that by combining nonlinear features, the performance is enhanced. This study shows that nonlinear analysis of EEG can be a useful method for discriminating depressed patients and normal subjects. It is suggested that this analysis may be a complementary tool to help psychiatrists for diagnosing depressed patients.

Morphological characterization of carbon nanofiber aerosol using tandem mobility and aerodynamic size measurements

Characterizing microstructural and transport properties of non-spherical particles, such as carbon nanofibers (CNF), is important for understanding their transport and deposition in human respiratory system and engineered devices such as particle filters. We describe an approach to obtain morphological information of non-spherical particles using a tandem system of differential mobility analyzer (DMA) and an electrical low-pressure impactor (ELPI). Effective density, dynamic shape factors (DSF), particle mass, and fractal dimension-like mass-scaling exponent of nanofibers were derived using the measured mobility and aerodynamic diameters, along with the known material density of CNF. Multiple charging of particles during DMA classification, which tends to bias the measured shape factors and particle mass toward higher values, was accounted for using a correction procedure. Particle mass derived from DMA–ELPI measurements agreed well with the direct mass measurements using an aerosol particle mass analyzer. Effective densities, based on mobility diameters, ranged from 0.32 to 0.67 g cm−3. The DSF of the CNF ranged from 1.8 to 2.3, indicating highly non-spherical particle morphologies.

Real-time fractal signal processing in the time domain

Fractal analysis has proven useful for the quantitative characterization of complex time series by scale-free statistical measures in various applications. The analysis has commonly been done offline with the signal being resident in memory in full length, and the processing carried out in several distinct passes. However, in many relevant applications, such as monitoring or forecasting, algorithms are needed to capture changes in the fractal measure real-time. Here we introduce real-time variants of the Detrended Fluctuation Analysis (DFA) and the closely related Signal Summation Conversion (SSC) methods, which are suitable to estimate the fractal exponent in one pass. Compared to offline algorithms, the precision is the same, the memory requirement is significantly lower, and the execution time depends on the same factors but with different rates. Our tests show that dynamic changes in the fractal parameter can be efficiently detected. We demonstrate the applicability of our real-time methods on signals of cerebral hemodynamics acquired during open-heart surgery.

Strong anticipation: complexity matching in interpersonal coordination

A subtle coordination occurs within complex systems, between multiple nested sub-systems. This intra-system coordination can be detected by the presence of 1/f fluctuations produced by the system. But coordination can occur also between systems. Interpersonal coordination has been studied from a local point of view until now, focusing on macroscopic interactions. But the recent concept of strong anticipation introduced by Dubois (Lect Notes Comput Sci 2684:110-132, 2003) suggests that interactions could occur on multiple levels between complex systems. The hypothesis is that time series in interpersonal synchronization present a matching of the complexity index (fractal exponent). Moreover, it is argued that this matching is not a consequence of short-term adaptations but reveals a global coordination between participants. Eleven pairs of participants oscillated a hand-held pendulum in the in-phase pattern for 11 min, in three conditions where the coupling strength was manipulated by the perceptual feedbacks. The results show a high correlation between fractal exponents irrespective of the coupling strength, and a very low percentage of local cross-correlations between time series appear at lag 0 and lag 1. These results suggest that interpersonal coordination, and more globally synchronization of participants with natural environments, is based on non-local time scales.

Fractal-based linear model of resting state hemodynamic response in fMRI

Understanding endogenous dynamics of the brain has been one of crucial issues in neuroscience since it will disclose a huge default-mode functional network hidden behind resting state signals. Recent studies have shown that a resting state fMRI time series tend to exhibit the fractal property such as 1/f-type spectral density which is characterized by a fractal exponent. There exist indirect evidences supporting that the fractal exponent is associated with neurophysiological activities, however the relevance of fractal behavior with functional connectivity has been still unclear. Recently it was observed that the spontaneous fluctuation of cerebral blood volume also exhibits fractal properties. Since fMRI is a non-invasive measurement of hemodynamic activities, it leads us to consider attributing the fractal behavior of BOLD signals to cerebral hemodynamics as well as neuronal activities. Motivated by this idea, we propose a fractal-based linear model of resting state hemodynamic response function (rs-HRF) whose behavior is summarized by a fractal exponent. It comprises the infinite or large number of density functions with slowly decaying coefficients according to a fractal exponent while the classical HRF consists of just two density functions. We showed that the special condition of rs-HRF causes long memory in BOLD signals. The rs-HRF model also implies that a resting state BOLD signal can be well approximated as a fractionally integrated process (FIP) rather than the typical fractional Gaussian noise (FGN). The simulation studies based on the rs-HRF model enables us to figure out the influence of hemodynamic fractal behavior on functional connectivity of resting state BOLD signals. First, we simulated a multivariate autoregressive process as an approximation of stationary resting state neuronal activity, and obtained its corresponding BOLD signals by convolving them with rs-HRF. Then, we observed the dissimilarity of wavelet correlation matrices between neuronal activities and BOLD signals on the basis of the symmetric Kullback-Leibler divergence which can be utilized as a measure of difference between two distributions. The dissimilarity of wavelet correlation increases in high frequency scales as the variance of fractal exponents increases while the dissimilarity approaches to zero in low frequency scales. These results suggest that the difference of fractal exponents between brain regions may cause apparent discrepancy of functional connectivity between neuronal activities and BOLD signals in high frequency scales. All of these results may give us insight into the dependence of functional connectivity on fractal behavior, and direct us to the default-mode functional network dominated by neuronal activities beyond fractal behavior of BOLD signals. While the linear relationship between neuronal activities and BOLD signals in evoked state has been well represented by HRF, it is questionable that the classical HRF is appropriate even for resting state since the model does not well reflect the fractal properties of cerebral hemodynamics as well as BOLD signals. In the future work, it will be valuable to constitute the nonlinear model of fractal behavior induced by cerebral hemodynamics as an extension of the classical Balloon model.

Fractal Lyapunov Exponent Based Anomaly Detection of Network Traffic

Fractal Lyapunov Exponent Based Anomaly Detection of Network Traffic