Fractal Fluctuations Research Articles

A Novel Fourth-Order Finite Difference Scheme for European Option Pricing in the Time-Fractional Black–Scholes Model

This paper addresses the valuation of European options, which involves the complex and unpredictable dynamics of fractal market fluctuations. These are modeled using the α-order time-fractional Black–Scholes equation, where the Caputo fractional derivative is applied with the parameter α ranging from 0 to 1. We introduce a novel, high-order numerical scheme specifically crafted to efficiently tackle the time-fractional Black–Scholes equation. The spatial discretization is handled by a tailored finite point scheme that leverages exponential basis functions, complemented by an L1-discretization technique for temporal progression. We have conducted a thorough investigation into the stability and convergence of our approach, confirming its unconditional stability and fourth-order spatial accuracy, along with (2−α)-order temporal accuracy. To substantiate our theoretical results and showcase the precision of our method, we present numerical examples that include solutions with known exact values. We then apply our methodology to price three types of European options within the framework of the time-fractional Black–Scholes model: (i) a European double barrier knock-out call option; (ii) a standard European call option; and (iii) a European put option. These case studies not only enhance our comprehension of the fractional derivative’s order on option pricing but also stimulate discussion on how different model parameters affect option values within the fractional framework.

Comprehensive linear and nonlinear heart rate variability normative data in children.

The autonomic nervous system (ANS) is critical in regulating involuntary bodily functions, including heart rate. Heart rate variability (HRV) reflects the complex interplay between the ANS and humoral factors, making it a valuable noninvasive tool for assessing autonomic function. While HRV has been extensively studied in adults, normative data for HRV in children, primarily based on long-term rhythm recordings, are limited. This study aimed to establish comprehensive normative data for HRV in children. In this retrospective study, we examined 24-h Holter monitors of children aged 1day to 18years, divided into six age groups, at Nemours Children's Health in Orlando, Florida, spanning the years 2013-2023. HRV analysis encompassed time-domain, frequency-domain, and nonlinear indices. Holter data for a total of 247 patients in six age groups were included. An age-related uptrend was observed in all time- and frequency-domain variables except the normalized unit of low-frequency power. Entropy analysis revealed contradictory results among different entropy techniques. Sample and approximate entropy analyses were consistent and showed less complexity and more predictability of HRV with decreasing heart rate, while Shannon entropy analysis showed the opposite. Fractal detrended fluctuation analysis exhibited significant decreases across the age groups, suggestive of diminishing self-similarity of HRV patterns. Control of heart rate and HRV is a highly complex process and requires further study for a better understanding. It seems that no single parameter can fully elucidate the entire process. A combination of time-domain, frequency-domain, and nonlinear indices may be necessary to explain HRV behavior in the growing body.

Possible origin for the similar phase transitions in k-core and interdependent networks

The models of k-core percolation and interdependent networks (IN) have been extensively studied in their respective fields. A recent study has revealed that they share several common critical exponents. However, several newly discovered exponents in IN have not been explored in k-core percolation, and the origin of the similarity still remains unclear. Thus, in this paper, by considering k-core percolation on random networks, we first verify that the two newly discovered exponents (fractal fluctuation dimension, , and correlation length exponent, ) observed in d-dimensional IN spatial networks also exist with the same values in k-core percolation. That is, the fractality of the k-core giant component fluctuations is manifested by a fractal fluctuation dimension, , within a correlation size Nʹ that scales as , with . Here we define, and . This implies that both models, IN and k-core, feature the same scaling behaviors with the same critical exponents, further reinforcing the similarity between the two models. Furthermore, we suggest that these two models are similar since both have two types of interactions: short-range (SR) connectivity links and long-range (LR) influences. In IN the LR are the influences of dependency links while in k-core we find here that for k = 1 and k = 2 the influences are SR and in contrast for the influence is LR. In addition, analytical arguments for a universal hyper-scaling relation for the fractal fluctuation dimension of the k-core giant component and for IN as well as for any mixed-order transition are established. Our analysis enhances the comprehension of k-core percolation and supports the generalization of the concept of fractal fluctuations in mixed-order phase transitions.

A Multivariate Method for Dynamic System Analysis: Multivariate Detrended Fluctuation Analysis Using Generalized Variance.

Fractal fluctuations are a core concept for inquiries into human behavior and cognition from a dynamic systems perspective. Here, we present a generalized variance method for multivariate detrended fluctuation analysis (mvDFA). The advantage of this extension is that it can be applied to multivariate time series and considers intercorrelation between these time series when estimating fractal properties. First, we briefly describe how fractal fluctuations have advanced a dynamic system understanding of cognition. Then, we describe mvDFA in detail and highlight some of the advantages of the approach for simulated data. Furthermore, we show how mvDFA can be used to investigate empirical multivariate data using electroencephalographic recordings during a time-estimation task. We discuss this methodological development within the framework of interaction-dominant dynamics. Moreover, we outline how the availability of multivariate analyses can inform theoretical developments in the area of dynamic systems in human behavior.

Fractal Fluctuations at Mixed-Order Transitions in Interdependent Networks.

We study the critical features of the order parameter's fluctuations near the threshold of mixed-order phase transitions in randomly interdependent spatial networks. Remarkably, we find that although the structure of the order parameter is not scale invariant, its fluctuations are fractal up to a well-defined correlation length ξ^{'} that diverges when approaching the mixed-order transition threshold. We characterize the self-similar nature of these critical fluctuations through their effective fractal dimension d_{f}^{'}=3d/4, and correlation length exponent ν^{'}=2/d, where d is the dimension of the system. By analyzing percolation and magnetization, we demonstrate that d_{f}^{'} and ν^{'} are the same for both, i.e., independent of the symmetry of the process for any d of the underlying networks.

Four Methods to Distinguish between Fractal Dimensions in Time Series through Recurrence Quantification Analysis.

Fractal properties in time series of human behavior and physiology are quite ubiquitous, and several methods to capture such properties have been proposed in the past decades. Fractal properties are marked by similarities in statistical characteristics over time and space, and it has been suggested that such properties can be well-captured through recurrence quantification analysis. However, no methods to capture fractal fluctuations by means of recurrence-based methods have been developed yet. The present paper takes this suggestion as a point of departure to propose and test several approaches to quantifying fractal fluctuations in synthetic and empirical time-series data using recurrence-based analysis. We show that such measures can be extracted based on recurrence plots, and contrast the different approaches in terms of their accuracy and range of applicability.

Research on Fractal Evolution Characteristics and Safe Mining Technology of Overburden Fissures under Gully Water Body

A fractal realizes the quantitative characterization of complex and disordered mining fracture networks, and it is of great significance to grasp the fractal characteristics of rock movement law to guide mine production. To prevent the water-conducting fracture (WF) under the gullies from conducting the surface water body, and to realize the purpose of safe production and surface water body protection. The evolution of overburden fissures in the working face with shallow buried gulley landform and thick bedrock conditions is studied. The development height of water-conducting fracture (DHWF) is theoretically analyzed. The evolution characteristics of overlying fissures with different mining heights were observed by similarity simulation, and the observation results were analyzed by fractal theory. The results show that the main factor that determines the height of WF is mining height. The working face is mined at different mining heights, and the corresponding indexes such as the height of the WF, the area of the caving zone and the fractal dimension are related to engineering phenomena. In particular, the appearance and disappearance of the separation space correspond to the fractal dimension fluctuation phase. The safe mining technology under a gully water body, which mainly reduces mining height, is adopted, and the fissures of the working face are not connected to the surface water body after mining.

Fractal characteristics analysis and fluctuation trend prediction of commercial bank funding liquidity

ABSTRACT Funding liquidity has been considered as the lifeline of commercial banks. If there are abnormal fluctuations in funding liquidity, it may lead to bank runs, which may bring about the destruction of commercial banks and even spread to cause a financial crisis. Therefore, exploring the fluctuation characteristics and trend prediction of commercial bank funding liquidity has traditionally attracted much attention. Based on this, this paper analyzes the fluctuation characteristics of commercial bank funding liquidity by Multi-Fractal Detrended Fluctuation Analysis (MF-DFA), and innovatively uses the Moving Trend Entropy Dimension (MTED) for commercial bank funding liquidity fluctuation trend prediction. The results of the study show that the commercial bank funding liquidity is multifractal, so it has predictability; MTED can not only accurately predict the fluctuation trend of commercial bank funding liquidity, but also has robustness. Through effective monitoring of funding liquidity, commercial banks can develop their funding liquidity risk management programs and improve the effectiveness of their own funding liquidity risk control. At the same time, regulators can also achieve the goal of preventing and resolving funding liquidity risks as well as maintaining financial stability.

Application of Improved MFDFA and D-S Evidence Theory in Fault Diagnosis

To improve the accuracy of centrifugal pump fault diagnosis, a novel fault diagnosis method based on improved multiple fractal detrended fluctuation analysis (MFDFA), the fusion of multi-sensing information derived from the back propagation (BP) neural network and the Dempster–Shafter (D-S) evidence theory, is accordingly proposed. Firstly, the multifractal spectral parameters of four sensor signals under four different operating conditions were extracted as centrifugal pump fault feature vectors using improved MFDFA and input to the BP neural network. Then, the basic trust assignment function was constructed by calculating trustworthiness (both local and global) as priori information, which is based on the output results of the neural networks specific to of each group of sensors. Finally, the basic trust assignment function was fused with decision processing in accordance with the D-S evidence combination rule in order to effectively achieve the multi-sensor information fusion centrifugal pump fault diagnosis. The experimental results show the multiple fractal spectrum feature parameters extracted by the improved MFDFA can accurately reflect the signal essence, and can be used as the fault feature vector. On this basis, this multi-sensor fault diagnosis reduces the uncertainty of fault classification and demonstrates improved accuracy compared to the single-sensor fault diagnosis thanks to being based on a combination of the BP neural networks and D-S evidence theory. Thereby, this method can facilitate accurate diagnosis of the centrifugal pump fault type with high confidence, subsequently providing a novel and alternative method to existing methods of diagnosis.

Comparing adaptive fractal and detrended fluctuation analyses of stride time variability: Tests of equivalence

BackgroundFractal analyses quantify self-similarities in stride-to-stride fluctuations over different time scales. Fractal exponents can be measured with adaptive fractal analysis (AFA) or detrended fluctuation analysis (DFA), though measurements obtained with the algorithms have not been directly compared. Research questionAre stride time fractal exponents measured with AFA and DFA algorithms equivalent? MethodsData from 50 participants with Parkinson’s Disease (n = 15), age-similar healthy adults (n = 15) and healthy young adults (n = 20) were analyzed in this cross-sectional, observational study. Participants completed 6-min walks at self-selected speeds overground on a straight walkway and on a treadmill. Stride times were measured with inertial measurement units. Fractal exponents in stride time data were processed using AFA and DFA algorithms and compared with two one-sided tests of equivalence. Mixed ANOVAs were used to compare exponents between groups and conditions. ResultsFractal exponents computed with AFA and DFA were equivalent neither in the overground (0.796 & 0.830, respectively, p = .587) nor treadmill conditions (0.806 & 0.882, respectively, p = .122). Fractal exponents measured with DFA were higher than when measured with AFA. Standard errors were 22% lower when measured with AFA. Additionally, a group × condition interaction was statistically significant when fractal exponents were processed with the AFA algorithm (F(2,47) = 11.696, p < .001), whereas the group × condition interaction was not statistically significant when DFA exponents were compared (F(2, 47) = 2.144, p = .129). SignificanceAFA and DFA do not produce equivalent estimates of the fractal exponent α in stride time dynamics. Estimates of the fractal exponent α obtained with AFA or DFA algorithms therefore should not be used interchangeably. Standard errors were lower when derived with AFA. Fractal exponents calculated with AFA may be more sensitive to conditions that influence stride time fractal dynamics than are measures calculated with DFA.

Research on the portfolio model based on Mean-MF-DCCA under multifractal feature constraint

In order to incorporate the multifractal characteristics of the capital market into the research framework of portfolios and overcome the shortcomings of existing research results that had not considered the existence of multiple fractal fluctuation characteristics and multiple fractal correlation characteristics of asset prices, in this paper, multifractal detrended cross-correlation analysis (MF-DCCA) was embedded into the mean–variance criterion, the Mean-MF-DCCA portfolio model was constructed under the constraints of multiple fractal features, and the analytical solution of the model was given. On this basis, the effectiveness of Mean-MF-DCCA portfolio model was tested using empirical analysis. The results showed that the Mean-MF-DCCA portfolio model was effective, and compared with the mean–variance portfolio model, it is more conducive to investors to construct a sophisticated portfolio under the multiple fluctuation importance degree and multiple time scales, so as to improve their portfolio performance.

Assessment of mountain river streamflow patterns and flood events using information and complexity measures

The availability of distinctly interpretable assessments to characterize and describe river discharge for mountainous rivers for different climatic events can improve our understanding of the various dynamics related to hydrological processes. Furthermore, information and complexity metrics can reveal invaluable information about the unseen processes that occur within a system. In this study, hourly streamflow records obtained from five gauging stations of a mountainous river were analyzed to quantify different patterns and characterize system states at both low and high frequencies using increasing aggregation lengths. In addition, we propose a new extension for the information and complexity theory, allowing it to be customized for flood assessment. Moreover, we clarify how a pattern (i.e., a word length) can be suitably defined by means of information and complexity metrics. Regarding low-frequency analyses, our results related to information and complexity metrics indicate two scaling regimes for river discharge, one of which may describe river memory characteristics. Regarding high-frequency analyses, our findings indicate the presence of an additional scaling regime that occurs along an hourly scale, captured by streamflow data, and is obtained using a novel hydroacoustic system. Additionally, power spectral density results confirmed our findings. A further significant result from our study is the clear correlation between complexity and fractal fluctuations, which should be addressed in future studies. In summary, this research focuses on new aspects of information and complexity metrics to be customized for the detection and understanding of temporal structures of streamflow patterns during both standard and extreme events.

Global broadcasting of local fractal fluctuations in a bodywide distributed system supports perception via effortful touch

A long history of research has pointed to the importance of fractal fluctuations in physiology, but so far, the physiological evidence of fractal fluctuations has been piecemeal and without clues to bodywide integration. What remains unknown is how fractal fluctuations might interact across the body and how those interactions might support the coordination of goal-directed behaviors. We demonstrate that a complex interplay of fractality in mechanical fluctuations across the body supports a more accurate perception of heaviness and length of occluded handheld objects via effortful touch in blindfolded individuals. For a given participant, the flow of fractal fluctuation through the body indexes the flow of perceptual information used to derive perceptual judgments. Effortful touch depends on compression of high-dimensional flux of mechanotransduction into low-dimensional perceptual information specifying properties of hefted occluded objects. These patterns in the waxing and waning of fluctuations across disparate anatomical locations provide novel insights into the form of this compression.

The scaling of biomass variance across trophic levels in stream species communities: a macroecological approach

Much has been published about different aspects of body-size distribution resting on the assumptions of metabolic scaling, although a number of studies in aquatic ecosystems have questioned its generality. This study considers the effects of individual body-mass and biomass variability on scaling properties of multi-species communities (protists, meio- and macroinvertebrates), and their intrinsic variations in assemblage structure. We examine how size traits within communities are distributed on local and regional scales and assess the potential sources of variation affecting whole ecosystems. Our results, built upon seven river catchment communities including 1204 species, revealed micro-meiofauna-dominated biomass distributions driven by stochastic hydrophysical processes that induce a fractal fluctuation scaling, irrespective of trophic levels, shaping local and regional scaling relations. Fractal-scaling differences are largely generated by the frequency of high flow events that influence the biomass assemblage configurations, which are significantly better represented by the Power Fraction model compared to single statistical random models. We conclude that environmental random variability contributes to the decoupling of total biomass and body mass per site from assemblage size, resulting in scale-invariant body-size traits among assemblages and systems. Generally, these findings emphasize that ignoring small-sized species and, thus, the wide range of body sizes makes accurate ecological model predictions, impossible.

Fractal biomarker of activity in patients with bipolar disorder.

The output of many healthy physiological systems displays fractal fluctuations with self-similar temporal structures. Altered fractal patterns are associated with pathological conditions. There is evidence that patients with bipolar disorder have altered daily behaviors. To test whether fractal patterns in motor activity are altered in patients with bipolar disorder, we analyzed 2-week actigraphy data collected from 106 patients with bipolar disorder type I in a euthymic state, 73 unaffected siblings of patients, and 76 controls. To examine the link between fractal patterns and symptoms, we analyzed 180-day actigraphy and mood symptom data that were simultaneously collected from 14 patients. Compared to controls, patients showed excessive regularity in motor activity fluctuations at small time scales (<1.5 h) as quantified by a larger scaling exponent (α1 > 1), indicating a more rigid motor control system. α1 values of siblings were between those of patients and controls. Further examinations revealed that the group differences in α1 were only significant in females. Sex also affected the group differences in fractal patterns at larger time scales (>2 h) as quantified by scaling exponent α2. Specifically, female patients and siblings had a smaller α2 compared to female controls, indicating more random activity fluctuations; while male patients had a larger α2 compared to male controls. Interestingly, a higher weekly depression score was associated with a lower α1 in the subsequent week. Our results show sex- and scale-dependent alterations in fractal activity regulation in patients with bipolar disorder. The mechanisms underlying the alterations are yet to be determined.

Fractal spike dynamics and neuronal coupling in the primate visual system.

We measured fractal (self-similar) fluctuations in ongoing spiking activity in subcortical (lateral geniculate nucleus, LGN) and cortical (area MT) visual areas in anaesthetised marmosets. Cells in the evolutionary ancient koniocellular LGN pathway and in area MT show high-amplitude fractal fluctuations, whereas evolutionarily newer parvocellular and magnocellular LGN cells do not. Spiking activity in koniocellular cells and MT cells shows substantial correlation to the local population activity, whereas activity in parvocellular and magnocellular cells is less correlated with local activity. We develop a model consisting of a fractal process and a global rate modulation which can reproduce and explain the fundamental relationship between fractal fluctuations and population coupling in LGN and MT. The model provides a unified account of apparently disparate aspects of neural spiking activity and can improve our understanding of information processing in evolutionary ancient and modern visual pathways. The brain represents and processes information through patterns of spiking activity, which is influenced by local and widescale brain circuits as well as intrinsic neural dynamics. Whether these influences have independent or linked effects on spiking activity is, however, not known. Here we measured spiking activity in two visual centres, the lateral geniculate nucleus (LGN) and cortical area MT, in marmoset monkeys. By combining the Fano-factor time curve, power spectral analysis and rescaled range analysis, we reveal inherent fractal fluctuations of spiking activity in LGN and MT. We found that the evolutionary ancient koniocellular (K) pathway in LGN and area MT exhibits strong fractal fluctuations at short (<1s) time scales. Parvocellular (P) and magnocellular (M) LGN cells show weaker fractal fluctuations at longer (multi-second) time scales. In both LGN and MT, the amplitude and time scale of fractal fluctuations can explain short and long time scale spiking dynamics. We further show differential neuronal coupling of LGN and MT cells to local population spiking activity. The population coupling is intrinsically linked to fractal fluctuations: neurons showing stronger fluctuations are more strongly correlated to the local population activity. To understand this relationship, we modelled spiking activity using a fractal inhomogeneous Poisson process with dynamic rate, which is the product of an intrinsic stochastic fractal rate and a global modulatory gain. Our model explains the intrinsic links between neuronal spike rate and population coupling in LGN and MT, and establishes a unified account of dynamic spiking properties in afferent visual pathways.

Bodywide fluctuations support manual exploration: Fractal fluctuations in posture predict perception of heaviness and length via effortful touch by the hand

The human haptic perceptual system respects a bodywide organization that responds to local stimulation through full-bodied coordination of nested tensions and compressions across multiple nonoverlapping scales. Under such an organization, the suprapostural task of manually hefting objects to perceive their heaviness and length should depend on roots extending into the postural control for maintaining upright balance on the ground surface. Postural sway of the whole body should thus carry signatures predicting what the hand can extract by hefting an object. We found that fractal fluctuations in Euclidean displacement in the participants' center of pressure (CoP) contributed to perceptual judgments by moderating how the participants' hand picked up the informational variable of the moment of inertia. The role of fractality in CoP displacement in supporting heaviness and length judgments increased across trials, indicating that the participants progressively implicate their fractal scaling in their perception of heaviness and length. Traditionally, we had to measure fractality in hand movements to predict perceptual judgments by manual hefting. However, our findings suggest that we can observe what is happening at hand in the relatively distant-from-hand measure of CoP. Our findings reveal the complex relationship through which posture supports manual exploration, entailing perception of the intended properties of hefted objects (heaviness or length) putatively through the redistribution of forces throughout the body.

Effect of sampling frequency on fractal fluctuations during treadmill walking.

The temporal dynamics of stride-to-stride fluctuations in steady-state walking reveal important information about locomotor control and can be quantified using so-called fractal analyses, notably the detrended fluctuation analysis (DFA). Gait dynamics are often collected during treadmill walking using 3-D motion capture to identify gait events from kinematic data. The sampling frequency of motion capture systems may impact the precision of event detection and consequently impact the quantification of stride-to-stride variability. This study aimed i) to determine if collecting multiple walking trials with different sampling frequency affects DFA values of spatiotemporal parameters during treadmill walking, and ii) to determine the reliability of DFA values across downsampled conditions. Seventeen healthy young adults walked on a treadmill while their gait dynamics was captured using different sampling frequency (60, 120 and 240 Hz) in each condition. We also compared data from the highest sampling frequency to downsampled versions of itself. We applied DFA to the following time series: step length, time and speed, and stride length, time and speed. Reliability between experimental conditions and between downsampled conditions were measured with 1) intraclass correlation estimates and their 95% confident intervals, calculated based on a single-measurement, absolute-agreement, two-way mixed-effects model (ICC 3,1), and 2) Bland-Altman bias and limits of agreement. Both analyses revealed a poor reliability of DFA results between conditions using different sampling frequencies, but a relatively good reliability between original and downsampled spatiotemporal variables. Collectively, our results suggest that using sampling frequencies of 120 Hz or 240 Hz provide similar results, but that using 60 Hz may alter DFA values. We recommend that gait kinematics should be collected at around 120 Hz, which provides a compromise between event detection accuracy and processing time.

Asymmetric MF-DCCA Method Based on Fluctuation Conduction and its Application in Air Pollution in Hangzhou

The study of the relationship between the concentration of PM2.5 and the local air quality index (AQI) is significant for the improvement of urban air quality. This study not only considered multifractal cross-correlation but also the fluctuation conduction mechanism. An asymmetric multifractal detrended cross-correlation analysis (MF-DCCA) method based on fluctuation conduction is introduced here to empirically explore the causality and conduction time between air quality factors and PM2.5 concentration. The empirical results indicate the existence of a bidirectional fluctuation conduction effect between PM2.5 and PM10, SO2, and NO2 in Hangzhou, China, with a conduction time of 30 hours; this effect is non-existent between PM2.5 and O3. In addition, there is a unidirectional fractal fluctuation conduction between PM2.5 and CO with a conduction time of 21 hours.

Fractal properties and short-term correlations in motor control in cycling: influence of a cognitive challenge

Fluctuations in cyclic tasks periods is a known characteristic of human motor control. Specifically, long-range fractal fluctuations have been evidenced in the temporal structure of these variations in human locomotion and thought to be the outcome of a multicomponent physiologic system in which control is distributed across intricate cortical, spinal and neuromuscular regulation loops.Combined with long-range correlation analyses, short-range autocorrelations have proven their use to describe control distribution across central and motor components.We used relevant tools to characterize long- and short-range correlations in revolution time series during cycling on an ergometer in 19 healthy young adults. We evaluated the impact of introducing a cognitive task (PASAT) to assess the role of central structures in control organization.Autocorrelation function and detrending fluctuation analysis (DFA) demonstrated the presence of fractal scaling. PSD in the short range revealed a singular behavior which cannot be explained by the usual models of even-based and emergent timing.The main outcomes are that (1) timing in cycling is a fractal process, (2) this long-range fractal behavior increases in persistence with dual-task condition, which has not been previously observed, (3) short-range behavior is highly persistent and unaffected by dual-task.Relying on the inertia of the oscillator may be a way to distribute more control to the periphery, thereby allocating less resources to central process and better managing additional cognitive demands. This original behavior in cycling may explain the high short-range persistence unaffected by dual-task, and the increase in long-range persistence with dual-task.