Search Results (108)

Nonlinear analysis provides a framework for understanding the complexity and stability of human locomotion by capturing dynamic patterns beyond linear methods. This study examined the effect of data length on seven nonlinear measures: Sample Entropy (SpEn), Approximate Entropy (ApEn), Lyapunov Exponents using Wolf’s (LyE_W) and Rosenstein’s (LyE_R) algorithms, Detrended Fluctuation Analysis (DFA), Correlation Dimension (CD), and the Hurst–Kolmogorov process (HK). A 3500-frame kinematic dataset from a healthy adult performing motor-assisted elliptical training and treadmill walking was segmented from 100 to 3500 frames in 10-frame increments. Data from treadmill and elliptical conditions were analyzed and presented in a combined manner to highlight general stabilization trends across locomotor tasks. Results revealed that increasing data length significantly affected all nonlinear metrics (p ≤ 0.0005). Stabilization occurred at varying minimum lengths: SpEn at ~4.5–8.8 s (540–1060 frames), ApEn at ~5.4–7.7 s (650–920 frames), LyE_W at ~19.1–29.2 s (2290–3500 frames), LyE_R at ~1.3–1.5 s (150–180 frames), DFA at ~29.2 s (3500 frames), CD at ~1.7–15.9 s (200–1910 frames), and HK at ~9.1–9.8 s (1090–1180 frames). Notably, HK achieved stable estimates in approximately one-third of the time required for DFA and substantially less than LyE_W, supporting its suitability for time-constrained or clinical settings. These findings suggest the need to tailor data collection to each nonlinear metric and to report data length explicitly to improve accuracy, reproducibility, and methodological rigor in gait variability research. However, these findings should be interpreted within the limitations of a single-participant, exploratory design. Full article

This study characterizes the multifractal structure of monthly precipitation in the semi-arid state of Guanajuato, Mexico, using Multifractal Detrended Fluctuation Analysis with quadratic detrending (MFDFA-2). We analyze 65 quality-controlled meteorological stations covering the period 1981–2016. All series exhibit multifractality, with generalized Hurst exponents $h\left(2\right)=0.568\pm 0.065$ indicating predominantly persistent dynamics and long-term positive autocorrelation (64.6% of stations). The multifractal spectrum width ($\Delta \alpha$) ranges from 0.15 to 0.72 (mean = 0.2423), revealing substantial spatial variability in scaling complexity. K-means clustering based on multifractal features identifies the following four hydroclimatic groups: one random cluster (29.2% of stations) and three persistence-dominated clusters (70.8%), with coherent spatial organization. These findings provide new insights into the temporal scaling properties of precipitation in semi-arid regions and have important implications for water resource management and regionalized drought-risk assessment. Full article

The dominant pores governing methane adsorption in coal are micropores (pore size < 2 nm). Their spatial heterogeneity can be quantitatively characterized using multifractal theory; however, the evolution patterns and mechanisms of microporous structures across different coalification degrees remain unclear. This research selected a series of coal samples from different ranks and identified the coalification degree using the maximum vitrinite reflectance (Rₒ_,max). By comprehensively employing low-temperature CO₂ adsorption experiments and multifractal analysis, the evolution patterns of the microporous structures and their multifractal spectral parameters were systematically revealed, and the underlying control mechanisms were explored. Results indicate that micropore volume (PV) and specific surface area (SSA) first exhibit a decrease and then increase as Rₒ_,max increases, with the trough occurring during the second coalification jump at Rₒ_,max = 1.2–1.4%. The pore sizes exhibit bimodal distributions, with the primary peak occurring in the range of 0.45–0.65 nm and the secondary peak occurring in the range of 0.8–0.9 nm. All microporous structures possess pronounced multifractal characteristics. The generalized dimension spectrum width (ΔD) and singularity spectrum width (Δα) exhibit an increasing–decreasing–increasing trend with Rₒ_,max, whereas the Hurst exponent (H) follows an inverted parabolic curve, first increases then decreases. This contrasts with the trends in PV and SSA, indicating that the evolution of pore-space heterogeneity and connectivity is independent of and lags the changes in micropore quantity. These patterns are governed by a structural phase transition within the coal macromolecular network. Marked by the second coalification jump, the microporous system shifts from a flexible degradation–polycondensation paradigm to a rigid ordering–construction paradigm. This transition drives the asynchronous, synergistic evolutions of pore quantity, spatial heterogeneity (ΔD and Δα), and topological connectivity (H). This research provides a theoretical basis for quantitatively evaluating pore heterogeneity in coal reservoirs. Full article

Econophysics is an interdisciplinary field that applies physics concepts to economic and financial systems. By utilizing tools such as statistical physics, including fractal analysis and entropy measures, econophysics helps model the complex and non-linear dynamics of equity markets. This paper examines the intrinsic dynamics and regularity in information content in green finance markets (carbon, clean energy, and sustainability markets) by means of range scale analysis (R/S), detrended fluctuation analysis (DFA), fractionally integrated generalized auto-regressive conditionally heteroskedastic (FIGARCH) process, and Shannon entropy (SE). The empirical results can be summarized as follows. First, prices in all markets are persistent; however, returns are likely random as estimated Hurst exponents are close to 0.5. Second, the FIGARCH process shows that volatility series in carbon and sustainability markets are persistent, whilst volatility in clean energy is anti-persistent. Third, in carbon and sustainability markets, entropy is high in prices compared to returns and volatility series. On the contrary, the clean energy market shows lower entropy for prices than for returns and volatility. In sum, it is concluded that price and volatility series are predictable, whilst return series are not. Finally, based on a rolling window framework, it is concluded that the COVID-19 pandemic and the Russia–Ukraine war have altered long memory and randomness in all three green finance markets. Full article

The nature of nonlinear dynamics of financial markets results in fractal geometry and chaotic behavior that can be viewed on a variety of scales in time. This paper conducts research on the fractal characteristics of the stock network and its contribution to the price of assets based on the Fractal Market Hypothesis (FMH). A multiscale network centrality measure is built based on high-frequency return dependencies to measure the self-similar, scale-invariant nature of inter-stock dependencies. The network factor and portfolio returns are then broken down with the wavelet packet decomposition (WPD) to obtain frequency-domain profiles, which characterize the variability of risk transmission in relation to investment horizons. The profiles are consistent with scaling properties of fractal, but the decomposition does not identify causal pathways on its own. Estimation of fractal dimension by use of the box-counting technique aided by the Hurst exponent analysis reveals that the A-share of China market exhibited long-range dependence and multifractal scaling. Network factor has the largest explanatory power in mid-frequency between the D5 and D6 bands of 32 to 128 days. This intermediary frequency concentration is consistent with the hypothesis of heterogeneous markets, in which the groups of investors with varying time horizons generate scale-related price dynamics. The addition of the network factor to a 6-factor specification lowers the GRS under the 5-factor specification by 31.45 to 17.82 on the same test-asset universe, indicating better cross-sectional coverage in the sample. The estimates of the Lyapunov exponents (0.039) as well as the correlation dimension ($D_2=4.7$) confirm the presence of low-dimensional chaotic processes of the network factor series, but these values are specific to the Chinese A-share market over the 2005–2023 sample period. These results provide a frequency-disaggregated use of network-based factor modeling and suggest that it can be applicable in multiscale portfolio risk management where the investor horizon is not uniform. Full article

Timber price dynamics are most often analysed using trends, seasonality, and classical measures of volatility, which describe the magnitude of fluctuations but only to a limited extent capture the temporal structure of the price-generating process. The aim of this study is to identify the structural complexity and long-term memory of quarterly prices of WC0 pine timber in the regional timber market in Poland. The analysis is based on nominal net prices (PLN/m³) from 16 forest districts of the Regional Directorate of State Forests in Kraków over the period 2005–2024, with reference to nationally averaged timber prices. Long-term dependence is assessed using the Hurst exponent estimated by detrended fluctuation analysis (DFA) applied to log returns, while the geometric complexity of price trajectories is characterised by the fractal dimension and additionally validated using the Higuchi estimator. Cross-sectional results reveal substantial spatial heterogeneity in scaling properties, indicating the coexistence of persistent (trend-following) and corrective (anti-persistent) dynamics across forest districts. Rolling-window analysis (40 quarters) demonstrates temporal variability in price dynamics, with particularly pronounced shifts observed in 2019–2021. Cluster analysis based on time-varying Hurst exponent values identifies two groups of forest districts with distinct persistence trajectories, corresponding to more trend-dominated and corrective price dynamics. In contrast, national-level prices generally exhibit higher persistence than local prices, reflecting the effects of price aggregation. Overall, the results show that fractal analysis uncovers persistent spatial and temporal differences in timber price structures that remain invisible when relying solely on variance-based measures, with direct implications for the choice of planning horizons and timber sale strategies in regional markets. Full article

As the net gain of carbon by plants after accounting for respiration, vegetation net primary productivity (NPP) plays a central role in the terrestrial carbon cycle. However, a systematic and quantitative analysis of the spatiotemporal evolution and driving mechanisms of vegetation NPP on Hainan Island, a tropical region, is still lacking. Focusing on Hainan Island, this study employs an integrated approach—including the coefficient of variation, Mann–Kendall test, Hurst exponent, geographical detector, and PLS-SEM—to investigate the spatiotemporal dynamics of vegetation NPP and its underlying drivers from 2001 to 2022. The main conclusions as follows: (1) Vegetation NPP on Hainan Island showed a fluctuating upward trend from 2001 to 2022, with a mean annual increase of 3.6 g C·m⁻²·yr⁻¹, and displayed a spatial pattern of decrease from the central-southern mountainous areas toward the coastal regions. (2) NPP changes were generally stable; historically, areas showing an increasing trend exceeded those with a decreasing trend by 30.55%. In the future, the predominant projected trends are “persistent decrease” and “increase to decrease,” which together account for over 80% of the total area. (3) Topography and climate were the dominant drivers of NPP spatial heterogeneity. Elevation had the strongest explanatory power, followed by evapotranspiration and temperature. A significant, nonlinear enhancement effect was observed in the interaction between any two factors. (4) Topographic, climatic, anthropogenic, and vegetation factors all exerted direct positive effects on vegetation NPP. Anthropogenic activities also indirectly promoted NPP by influencing pathways such as vegetation growth. The conclusions of this research provide support for the implementation and evaluation of land-use planning, afforestation projects, and ecological protection and restoration measures on Hainan Island. Full article

The sustainable use of freshwater resources includes balancing between human demand for water and the long-term health of river systems. Although dams and reservoirs are essential for water supply, flood control and energy generation, they can induce significant hydrological alterations, affecting water quality, sediment transport, downstream water availability, and aquatic and riparian ecosystems. In this study, we employed multifractal analysis to investigate hydrological changes in the São Francisco River basin, Brazil, resulting from the construction of a cascade of dams and reservoirs. We applied multifractal detrended fluctuation analysis (MFDFA) to daily streamflow time-series spanning the period from 1929 to 2016, at locations both upstream and downstream of cascade dams, and for periods before and after dam construction. We calculated multifractal spectra f(α) and analyzed key complexity parameters: the position of the spectrum maximum α_0, representing the overall Hurst exponent H; the spectrum width W indicating the degree of multifractality; and the asymmetry parameter r, which reflects the dominance of small (r > 1) and large (r < 1) fluctuations. We found that after the construction of Sobradinho dam, located in the Sub-Middle São Francisco region, streamflow dynamics shifted towards a regime characterized by uncorrelated increments (H~0.5) and stronger multifractality (larger W), with the dominance of small fluctuations (r > 1). In contrast, the cumulative effect of all cascade dams downstream, in the Lower São Francisco region, led to streamflow regime with similarly uncorrelated increments (H~0.5), but with weaker multifractality (smaller W) and a dominance of large fluctuations (r < 1). The novelty of this work is the use of a sliding-window MFDFA approach to explore the temporal evolution of streamflow multifractality. This method uncovered otherwise hidden aspects of hydrological alterations, such as increasing tendency in spectrum width, indicating stronger multifractality and higher complexity of streamflow dynamics after the dam construction. These results demonstrate that multifractal analysis is a powerful tool for assessing the complexity of hydrological changes induced by human activities. Full article

Estimation of remaining useful life (RUL) of roller bearings is a prevalent problem for predictive maintenance in manufacturing. However, roller bearings are subject to a variety of factors during their operation. As a result, we deal with a slow nonlinear degradation process, which is long-range dependent, self-similar and has non-Gaussian characteristics. Proper data pre-processing enables us to use Pareto’s probability density function (PDF), Generalized Pareto motion (GPm) and its fractional-order extension (fGPm) as the degradation predictive model. Estimation of the Hurst exponent shows that this model has a long-range correlation and self-similarity. Through the analysis of the uncertainty of the end point of the bearing’s RUL and the prediction process, not only did it verify the high adaptability of fGPm in simulating complex degradation processes but also the criteria for judging self-similarity, and LRD characteristics were established. The case study mainly proves the validity of the theory, providing an effective analytical tool for a deeper understanding of the degradation mechanism. Full article

River discharge scaling is fundamental to the global hydrological cycle and to water resource assessment. This study investigates the existence of multiple scaling regimes and introduces a novel framework for clustering river discharge using multiscale fractal characteristics. We analyzed daily discharge data from 38 stations across continental Canada over an 80-year period. Multifractal characterization was performed at decadal and long-term scales using three key parameters: the singularity exponent (${\alpha }_0$), multifractal strength ($\alpha$), and asymmetry index (r). K-means clustering in the $\alpha -r$, ${\alpha }_0-r$, and $\alpha -{\alpha }_0$ planes revealed distinct clusters, with the asymmetric parameter (r) emerging as the strongest distinguishing factor. These clusters represent groups of rivers with similar dynamical structures: the $\alpha -r$ clusters categorize discharge based on scaling strength and fluctuation influence. Analysis of the generalized Hurst exponent revealed anti-persistent behavior at most stations, with exceptions at five specific locations. This multifractal clustering approach provides a powerful method for classifying river regimes based on intrinsic characteristics and identifying the physical drivers of discharge fluctuations. Full article

Understanding the temporal dynamics of urban air pollution is essential for effective environmental management, yet the nonlinear and scale-dependent behavior of key pollutants remains insufficiently explored. This study examines the multifractal properties of fine particulate matter ($P{M}_{2.5}$), ozone ($O_3$), and the Air Quality Index (AQI) across four major urban locations in California—Los Angeles, Orange, Riverside–Rubidoux, and Riverside–Mira Loma—regions characterized by persistent air-quality challenges and high population exposure. Using Multifractal Detrended Fluctuation Analysis (MF-DFA), we assess long-range dependence, heterogeneity, and cross-pollutant interactions to address the central question of whether these pollutants exhibit genuine multifractal behavior and how it varies across locations. The results reveal strong multifractality in all series, with spectrum widths ranging from 0.42 to 0.71 for $P{M}_{2.5}$-AQI and from 0.28 to 0.46 for $O_3$-AQI, indicating pronounced scale-dependent variability. Los Angeles consistently exhibited the widest spectra, reflecting greater temporal complexity. The generalized Hurst exponent at $q=2$ remained between 0.52 and 0.58 across all pollutant pairs, indicating persistent dynamics. Surrogate-data testing further confirmed that 60–75% of the observed multifractality arises from intrinsic long-range correlations rather than distributional effects. Overall, this study demonstrates that urban air pollutants in California display rich multifractal structures that differ systematically across regions, reflecting local emission patterns and atmospheric processes. These findings highlight the relevance of multifractal analysis as a powerful tool for improving air-quality modeling, forecasting, and policy design in densely populated environments. Full article

Background: Repetitive transcranial magnetic stimulation (rTMS) is an established neuromodulatory method, yet its multiscale neurophysiological effects in autism spectrum disorder (ASD) remain insufficiently characterized. Recent EEG analytic advances—such as spectral parameterization, long-range temporal correlation (LRTC) assessment, and connectivity modeling—enable quantitative evaluation of excitation–inhibition (E/I) balance and network organization. Objective: This study aimed to examine whether an eight-session, EEG-guided mixed-frequency rTMS protocol—combining inhibitory 1 Hz and excitatory 10 Hz trains individualized to quantitative EEG (qEEG) abnormalities—produces measurable changes in spectral dynamics, temporal correlations, and functional connectivity in a pediatric ASD case. Methods: An 11-year-old right-handed female with ASD (DSM-5-TR, ADOS-2) underwent resting-state EEG one week before and four months after intervention. Preprocessing used a validated automated pipeline, followed by spectral parameterization (FOOOF), detrended fluctuation analysis (DFA), and connectivity analyses (phase-lag index and Granger causality) in MATLAB (2023b). No inferential statistics were applied due to the single-case design. The study was conducted at Cosmos Healthcare (London, UK) with in-kind institutional support and approved by the Atlantic International University IRB (AIU-IRB-22-101). Results: Post-rTMS EEG showed (i) increased delta and reduced theta/alpha/beta power over central regions; (ii) steeper aperiodic slope and higher offset, maximal at Cz, suggesting increased inhibitory tone; (iii) reduced Hurst exponents (1–10 Hz) at Fz, Cz, and Pz, indicating decreased long-range temporal correlations; (iv) reorganization of hubs away from midline with marked Cz decoupling; and (v) strengthened parietal-to-central directional connectivity (Pz→Cz) with reduced Cz→Pz influence. Conclusions: Mixed-frequency, EEG-guided rTMS produced convergent changes across spectral, aperiodic, temporal, and connectivity measures consistent with modulation of cortical E/I balance and network organization. Findings are preliminary and hypothesis-generating. The study was supported by in-kind resources from Cosmos Healthcare, whose authors participated as investigators but had no influence on analysis or interpretation. Controlled trials are warranted to validate these exploratory results. Full article

Monofractal analysis offers a promising framework for characterizing cardiac dynamics, particularly in the early detection of heart failure. However, most existing approaches rely on long-duration physiological signals and do not explore the classification of disease severity. In this study, we propose a hybrid CNN-ELM model trained exclusively on synthetic monofractal time series of short length (128 to 512 samples), aiming to assess its ability to distinguish between healthy individuals and varying degrees of heart failure defined by the NYHA functional classification. Our results show that Hurst exponent distributions reflect the progressive loss of complexity in cardiac rhythms as heart failure severity increases. The model successfully classified both binary (healthy vs. sick) and multiclass (NYHA I–IV) scenarios by grouping Hurst exponent values ($H_{0.1}$ to $H_{0.9}$) into clinical categories, achieving peak accuracy ranges of 97.3–98.9% for binary classification and 96.2–98.8% for multiclass classification across signal lengths of 128, 256, and 512 samples. Importantly, the CNN-ELM architecture demonstrated fast training times and robust generalization, outperforming previous approaches based solely on support vector machines. These findings highlight the clinical potential of monofractal indices as non-invasive biomarkers of cardiovascular health and support the use of short synthetic signals for scalable, low-cost screening applications. Future work will extend this framework to multifractal and real-world clinical data and explore its integration into intelligent diagnostic systems. Full article

Cryptocurrencies such as Bitcoin can be classified as commodities under the Commodity Exchange Act (CEA), giving the Commodity Futures Trading Commission (CFTC) jurisdiction over those cryptocurrencies deemed commodities, particularly in the context of futures trading. This paper presents a method for predicting both long- and short-term trends in selected cryptocurrencies based on the Fractal Market Hypothesis (FMH). The FMH applies the self-affine properties of fractal stochastic fields to model financial time series. After introducing the underlying theory and mathematical framework, a fundamental analysis of Bitcoin and Ethereum exchange rates against the U.S. dollar is conducted. The analysis focuses on changes in the polarity of the ‘Beta-to-Volatility’ and ‘Lyapunov-to-Volatility’ ratios as indicators of impending shifts in Bitcoin/Ethereum price trends. These signals are used to recommend long, short, or hold trading positions, with corresponding algorithms (implemented in Matlab R2023b) developed and back-tested. An optimisation of these algorithms identifies ideal parameter ranges that maximise both accuracy and profitability, thereby ensuring high confidence in the predictions. The resulting trading strategy provides actionable guidance for cryptocurrency investment and quantifies the likelihood of bull or bear market dominance. Under stable market conditions, machine learning (using the ‘TuringBot’ platform) is shown to produce reliable short-horizon estimates of future price movements and fluctuations. This reduces trading delays caused by data filtering and increases returns by identifying optimal positions within rapid ‘micro-trends’ that would otherwise remain undetected—yielding gains of up to approximately 10%. Empirical results confirm that Bitcoin and Ethereum exchanges behave as self-affine (fractal) stochastic fields with Lévy distributions, exhibiting a Hurst exponent of roughly 0.32, a fractal dimension of about 1.68, and a Lévy index near 1.22. These findings demonstrate that the Fractal Market Hypothesis and its associated indices provide a robust market model capable of generating investment returns that consistently outperform standard Buy-and-Hold strategies. Full article

This work analyzes the behavior of the boundary layer subjected to stresses by obstacles using hourly measurements, in the form of time series, of meteorological variables (temperature (T), relative humidity (RH), and magnitude of the wind speed (WS)) in a given period. The study region is Santiago, the capital of Chile. The measurement location is in a rugged basin geography with a nearly pristine atmospheric environment. The time series are analyzed through chaos theory, demonstrating that they are chaotic through the calculation of the parameters Lyapunov exponent (λ > 0), correlation dimension (D_C < 5), Kolmogorov entropy (S_K > 0), Hurst exponent (0.5 < H < 1), and Lempel–Ziv complexity (LZ > 0). These series are simultaneous measurements of the variables of interest, before and after, of three different volumetric geometries arranged as obstacles: a parallelepiped, a cylinder, and a miniature mountain. The three geometries are subject to the influence of the wind and present the same cross-sectional area facing the measuring instruments oriented in the same way. The entropies calculated for each variable in each geometry are compared. It is demonstrated, in a first approximation, that volumetric geometry impacts the magnitude of the entropic fluxes associated with the measured variables, which can affect micrometeorology and, by extension, the climate in general. Furthermore, the study examines which geometry favors greater information loss or greater fractality in the measured variables. Full article