Detrended Fluctuation Analysis Research Articles

How Gait Nonlinearities in Individuals Without Known Pathology Describe Metabolic Cost During Walking Using Artificial Neural Network and Multiple Linear Regression

This study uses Artificial Neural Networks (ANNs) and multiple linear regression (MLR) models to explore the relationship between gait dynamics and the metabolic cost. Six nonlinear metrics—Lyapunov Exponents based on Rosenstein’s algorithm (LyER), Detrended Fluctuation Analysis (DFA), the Approximate Entropy (ApEn), the correlation dimension (CD), the Sample Entropy (SpEn), and Lyapunov Exponents based on Wolf’s algorithm (LyEW)—were utilized to predict the metabolic cost during walking. Time series data from 10 subjects walking under 13 conditions, with and without hip exoskeletons, were analyzed. Six ANN models, each corresponding to a nonlinear metric, were trained using the Levenberg–Marquardt backpropagation algorithm and compared with MLR models. Performance was assessed based on the mean squared error (MSE) and correlation coefficients. ANN models outperformed MLR, with DFA and Lyapunov Exponent models showing higher R2 values, indicating stronger predictive accuracy. The results suggest that gait’s nonlinear characteristics significantly impact the metabolic cost, and ANNs are more effective for analyzing these dynamics than MLR models. The study emphasizes the potential of focusing on specific nonlinear gait variables to enhance assistive device optimization, particularly for hip exoskeletons. These findings support the development of personalized interventions that improve walking efficiency and reduce metabolic demands, offering insights into the design of advanced assistive technologies.

A Hybrid Approach to Enhanced Signal Denoising Using Data-Driven Multiresolution Analysis with Detrended-Fluctuation-Analysis-Based Thresholding and Stationary Wavelet Transform

In this work, a new method for denoising signals is developed that is based on variational mode decomposition (VMD) and a novel metric using detrended fluctuation analysis (DFA). The proposed method first decomposes the signal into band-limited intrinsic mode functions (BLIMFs) using VMD. Then, a DFA-based developed metric is employed to identify the ‘noisy’ BLIMFs (based on their DFA-based scaling exponent and frequency content). The existing DFA-based methods use a single-slope threshold to detect noise, assuming all signals have the same noise pattern and ignoring their unique characteristics. In contrast, the proposed DFA-based metric sets adaptive thresholds for each mode based on their specific frequency and correlation properties, making it more effective for diverse signals and noise types. These predominantly noisy BLIMFs are then denoised using shrinkage techniques in the framework of stationary wavelet transform (SWT). This step allows efficient denoising of components, mainly the noisy BLIMFs identified by the adaptive threshold, without losing important signal details. Extensive computer simulations have been carried out for both synthetic and real electrocardiogram (ECG) signals. It is demonstrated that the proposed method outperforms the state-of-the-art denoising methods and with a comparable computational complexity.

Crossover detection based on variances of slope differences for multi-fractal detrended fluctuation analysis (MF-DFA)

AbstractMulti-fractal detrended fluctuation analysis (MF-DFA) describes long-range correlations across scales in time series in terms of the linear fittings of the fluctuation functions $$F_q(s)$$ F q ( s ) for every moment q considered. It is relatively common to find time scales, also called crossovers, where different linearities are found at each side of the $$F_q(s)$$ F q ( s ) , leading to different (multi-)fractal properties. Generally, crossovers are manually identified by experts, subject then to their skills and possible bias. In this work, a novel methodology based on the variances of the slope differences (CDV-A) is proposed to automatically detect crossovers. A set of synthetic fluctuation functions with different mono-fractal and multi-fractal features, different noise levels and known crossovers is generated to assess the CDV-A. Moreover, the proposed methodology is also applied to state of the art fluctuation functions in order to compare the proposed manual and automated locations of the crossovers. Results show the proposed methodology is highly accurate, specially in low noise synthetic fluctuation functions and in most of the real cases.

Autocorrelation and Cross-Correlation of Hospitalizations for Syphilis and HIV/AIDS in the State of Bahia

Objective: To identify and estimate the autocorrelation and cross-correlation of time series of hospitalization rates for syphilis and HIV/AIDS in the State of Bahia, in the period from 2000 to 2020. Theoretical Framework: It is based on the characterization and factors associated with notifications of syphilis and HIV/AIDS infection, as well as the social determinants that influence access to health programs and policies. Method: This is a time-based ecological study, carried out using data from the Hospital Information System (SIH) and the Brazilian Institute of Geography and Statistics (IBGE). The analyses were performed using the Detrended Fluctuation Analysis (DFA) method and the cross-correlation coefficient ρDCCA. Results and Discussion: There was persistent behavior for the time series of hospitalizations due to syphilis and HIV/AIDS (αDFA > 0.50), in addition to a weak negative cross-correlation (< -0.20), with only the autocorrelation of syphilis being statistically significant. It is believed that the behavior described tends to occur in the long term and may be repeated in the coming years, if interventions that contribute to the reversal are not carried out, both for the diseases in isolation and for co-infection. Research Implications: The findings of this research may contribute to updating knowledge about syphilis and HIV/AIDS infection in adults, but mainly to the construction of indicators with an emphasis on health surveillance systems in the State of Bahia. Originality/Value: The study aims to analyze hospitalization cases in which syphilis or HIV/AIDS was the main diagnosis, thus corroborating the planning and strengthening of public policies.

Effect of the uncertainty of autocorrelation estimation on trend analysis

Autocorrelation is ubiquitous in real-life time series. For trend analysis, positive autocorrelation mainly increases the statistical uncertainty of the detected trend. The theoretical framework has been established to quantify how the autocorrelation can impact trend analysis if its characteristic is known precisely. However, if the autocorrelation is also uncertain like that estimated in real-life data, then the consequences are yet unclear. The situation is further complicated by the concurrence of both short- and long-range correlation (SLRC), e.g., in air temperature. In this study, we use an appropriate minimal model with one short-range autoregressive parameter and one long-range fractional parameter to account for such SLRC, and then accordingly propose a framework to integrate the effect of the uncertainty of autocorrelation estimation into trend analysis, on the basis of the detrended fluctuation analysis and Akaike weights. This framework can also obtain a joint probability density function of the estimated autoregressive and fractional parameters for measuring their uncertainty, identifying their combined optimal estimates, uncovering their strong interdependence, and determining their two-dimensional confidence region. The effectiveness and applicability of the proposed framework are verified by a numerical experiment and a case study of daily air temperature anomalies at the Potsdam station. This study can provide a solid theoretical basis to underpin our understanding of the autocorrelation effect on trend analysis in theory and applications. Published by the American Physical Society 2024

Fully multivariate detrended fluctuation analysis using Mahalanobis norm with application to multivariate signal denoising

Detrended fluctuation analysis (DFA) has become an important tool for the long-range correlation and local regularity fluctuation analysis of nonstationary time series data. While the method is well-established and well-understood for single time series data, its extensions for multivariate data (comprising multiple channels) are still emerging. A major challenge in that regard is to incorporate inherent inter-channel dependencies within the DFA analysis. We propose a novel method to address that challenge through Mahalanobis distance (MD) norm that provides an analytical way to incorporate covariance matrix within the computation of the proposed multichannel fluctuation function. Through analytical analysis and experimental results, we show that incorporation of cross-channel correlations within the fluctuation function makes the rendered long-range correlation analysis more accurate for the multivariate correlated data. Next, we next demonstrate the utility of the proposed generic multichannel DFA (GMDFA) within the multivariate signal denoising problem(s). To this end, our denoising approach first obtains data driven multiscale signal representation by multi-stage use of multivariate variational mode decomposition (MVMD) method. Then, proposed GMDFA is used to reject the predominantly noisy modes based on their randomness scores.

Running Gait Complexity During an Overground, Mass-Participation Five-Kilometre Run.

Human locomotion contains innate variability which may provide health insights. Detrended fluctuation analysis (DFA) has been used to quantify the temporal structure of variability for treadmill running, although it has been less commonly applied to uncontrolled overground running. This study aimed to determine how running gait complexity changes in response to gradient and elapsed exercise duration during uncontrolled overground running. Sixty-eight participants completed an overground, mass-participation five-kilometre run (a parkrun). Stride times were recorded using an inertial measurement unit mounted on the distal shank. Data were divided into four consecutive intervals (uphill lap 1, downhill lap 1, uphill lap 2, downhill lap 2). The magnitude (SD) and structure (DFA) of stride time variability were compared across elapsed exercise duration and gradient using a repeated-measures ANOVA. Participants maintained consistent stride times throughout the run. Stride time DFA-α displayed a moderate decrease (d = |0.39| ± 0.13) during downhill running compared to uphill running. DFA-α did not change in response to elapsed exercise duration, although a greater stride time SD was found during the first section of lap 1 (d = |0.30| ± 0.12). These findings suggest that inter- and intra-run changes in gait complexity should be interpreted in the context of course elevation profiles before conclusions on human health are drawn.

Multifractal Analysis of Standardized Precipitation Evapotranspiration Index in Serbia in the Context of Climate Change

A better understanding of climate change impact on dry/wet conditions is crucial for agricultural planning and the use of renewable energy, in terms of sustainable development and preservation of natural resources for future generations. The objective of this study was to investigate the impact of climate change on temporal fluctuations of dry/wet conditions in Serbia on multiple temporal scales through multifractal analysis of the standardized precipitation evapotranspiration index (SPEI). We used the well-known method of multifractal detrended fluctuation analysis (MFDFA), which is suitable for the analysis of scaling properties of nonstationary temporal series. The complexity of the underlying stochastic process was evaluated through the parameters of the multifractal spectrum: position of maximum α0 (persistence), spectrum width W (degree of multifractality) and skew parameter r dominance of large/small fluctuations). MFDFA was applied on SPEI time series for the accumulation time scale of 1, 3, 6 and 12 months that were calculated using the high-resolution meteorological gridded dataset E-OBS for the period from 1961 to 2020. The impact of climate change was investigated by comparing two standard climatic periods (1961–1990 and 1991–2020). We found that all the SPEI series show multifractal properties with the dominant contribution of small fluctuations. The short and medium dry/wet conditions described by SPEI-1, SPEI-3, and SPEI-6 are persistent (0.5<α0<1); stronger persistence is found at higher accumulation time scales, while the SPEI-12 time series is antipersistent (0<α0−1<0.5). The degree of multifractality increases from SPEI-1 to SPEI-6 and decreases for SPEI-12. In the second period, the SPEI-1, SPEI-3, and SPEI-6 series become more persistent with weaker multifractality, indicating that short and medium dry/wet conditions (which are related to soil moisture and crop stress) become easier to predict, while SPEI-12 changed toward a more random regime and stronger multifractality in the eastern and central parts of the country, indicating that long-term dry/wet conditions (related to streamflow, reservoir levels, and groundwater levels) become more difficult for modeling and prediction. These results indicate that the complexity of dry/wet conditions, in this case described by the multifractal properties of the SPEI temporal series, is affected by climate change.

Approaching Multifractal Complexity in Decentralized Cryptocurrency Trading

Multifractality is a concept that helps compactly grasp the most essential features of financial dynamics. In its fully developed form, this concept applies to essentially all mature financial markets and even to more liquid cryptocurrencies traded on centralized exchanges. A new element that adds complexity to cryptocurrency markets is the possibility of decentralized trading. Based on the extracted tick-by-tick transaction data from the Universal Router contract of the Uniswap decentralized exchange, from 6 June 2023 to 30 June 2024, the present study using multifractal detrended fluctuation analysis (MFDFA) shows that even though liquidity on these new exchanges is still much lower compared to centralized exchanges, convincing traces of multifractality are already emerging in this new trading as well. The resulting multifractal spectra are, however, strongly left-side asymmetric, which indicates that this multifractality comes primarily from large fluctuations, and small ones are more of the uncorrelated noise type. What is particularly interesting here is the fact that multifractality is more developed for time series representing transaction volumes than rates of return. On the level of these larger events, a trace of multifractal cross-correlations between the two characteristics is also observed.

Deep Neural Network Model for Hurst Exponent: Learning from R/S Analysis

This paper proposes a deep neural network (DNN) model to estimate the Hurst exponent, a crucial parameter in modelling stock market price movements driven by fractional geometric Brownian motion. We randomly selected 446 indices from the S&P 500 and extracted their price movements over the last 2010 trading days. Using the rescaled range (R/S) analysis and the detrended fluctuation analysis (DFA), we computed the Hurst exponent and related parameters, which serve as the target parameters in the DNN architecture. The DNN model demonstrated remarkable learning capabilities, making accurate predictions even with small sample sizes. This addresses a limitation of R/S analysis, known for biased estimates in such instances. The significance of this model lies in its ability, once trained, to rapidly estimate the Hurst exponent, providing results in a small fraction of a second.

Correlations and Fractality in Sentence-Level Sentiment Analysis Based on VADER for Literary Texts

We perform a sentence-level sentiment analysis study of different literary texts in English language. Each text is converted into a series in which the data points are the sentiment value of each sentence obtained using the sentiment analysis tool (VADER). By applying the Detrended Fluctuation Analysis (DFA) and the Higuchi Fractal Dimension (HFD) methods to these sentiment series, we find that they are monofractal with long-term correlations, which can be explained by the fact that the writing process has memory by construction, with a sentiment evolution that is self-similar. Furthermore, we discretize these series by applying a classification approach which transforms the series into a one on which each data point has only three possible values, corresponding to positive, neutral or negative sentiments. We map these three-states series to a Markov chain and investigate the transitions of sentiment from one sentence to the next, obtaining a state transition matrix for each book that provides information on the probability of transitioning between sentiments from one sentence to the next. This approach shows that there are biases towards increasing the probability of switching to neutral or positive sentences. The two approaches supplement each other, since the long-term correlation approach allows a global assessment of the sentiment of the book, while the state transition matrix approach provides local information about the sentiment evolution along the text.

Preoperative autonomic failure in neonates with critical congenital heart disease

BackgroundNeonates with critical congenital heart disease (cCHD) undergo a complicated transition to ex-utero life. However, continuous monitoring of autonomic tone using heart rate variability is currently lacking. Materials and methodsWe retrieved continuous electrocardiograms from the time of admission or from 10 days prior to surgery for neonates with dextro-transposition of the great arteries (d-TGA) and hypoplastic left heart syndrome (HLHS). Beat-to-beat intervals (RRi) were calculated and divided into 10-min epochs. Spectral metrics, including low-frequency (LF) and high-frequency (HF) powers, as well as detrended fluctuation analysis metrics (αS, αL, RMSS, and RMSL), were computed for RRi within each epoch and averaged over 24 h. The relationship between RRi metrics and time to surgery was analyzed using linear mixed-effects models, adjusting for prenatal and postnatal factors. ResultsThe study included 10 neonates with HLHS and 23 with d-TGA. RRi metrics were available for 110 days. In the unadjusted models, LF power (Estimate: −4.4×10−3, P = 0.02), HF power (−4.1 x 10−2, 5 x 10−5), RMSS(−3.7 x 10−4, 8.7 x 10−3), and RMSL(−1.4× 10−3, 0.02) were all negatively associated with time to surgery, with HF power showing the strongest association. After adjusting for covariates, HF power retained its significant negative association with time (−0.04, P = 0.03). The cCHD diagnosis did not significantly influence RRi metrics. ConclusionsIn neonates with cCHD, there is a progressive decline in autonomic function leading up to surgery. RRi metrics may serve as valuable indicators of deteriorating physiology in these patients.

Exercise intensity measurement using fractal analysis of heart rate variability: Reliability, agreement and influence of sex and cardiorespiratory fitness

ABSTRACT The study aimed to establish the test-retest reliability of detrended fluctuation analysis of heart rate variability (DFA-α1) based exercise intensity thresholds, assess its agreement with ventilatory- and lactate-derived thresholds and the moderating effect of sex and cardiorespiratory fitness (CRF) on the agreement. Intensity thresholds for thirty-seven participants (17 females) based on blood lactate (LT1/LT2), gas-exchange (VT1/VT2) and DFA-α1 (αTh1/αTh2) were assessed. Heart rate (HR) at αTh1 and αTh2 showed good test-retest reliability (coefficient of variation [CV] < 6%), and moderate to high agreement with LTs (r = 0.40 – 0.57) and VTs (r = 0.61 – 0.66) respectively. Mixed effects models indicated bias magnitude depended on CRF, with DFA-α1 overestimating thresholds versus VTs for lower fitness levels (speed at VT1 <8.5 km⋅hr−1), while underestimating for higher fitness levels (speed at VT2 >15 km⋅hr−1; VO2max >55 mL·kg−1·min−1). Controlling for CRF, sex significantly affected bias magnitude only at first threshold, with males having higher mean bias (+2.41 bpm) than females (−1.26 bpm). DFA-α1 thresholds are practical and reliable intensity measures, however it is unclear if they accurately represent LTs/VTs from the observed limits of agreement and unexplained variance. To optimise DFA-α1 threshold estimation across different populations, bias should be corrected based on sex and CRF.

Implication of heart rhythm complexity in predicting long-term outcomes in pulmonary hypertension

Pulmonary hypertension (PH) is a serious disease, however simple tools to predict outcomes are lacking. In our previous investigation, we found that heart rate variability (HRV) and heart rhythm complexity (HRC) were associated with the detection and severity of PH, however their association with PH mortality remains unclear.The aim of this study was to investigate these metrics as a tool for determining long-term outcomes in PH patients. We enrolled 74 Asian PH patients with WHO PH group 1 or 4 at a single hospital in Taiwan between March 2012 and June 2018. After a median follow-up duration of 58 months (to January 2023), 22 patients had died. The patients who died had a significantly lower lean body mass index (BMI), impaired renal function, higher N-terminal pro B-type natriuretic peptide (NT-proBNP) level, lower very low-frequency (VLF), lower short-term detrended fluctuation analysis α1 (DFAα1), and lower multiscale entropy scale 5 value. In multivariable analysis, BMI, VLF and multiscale entropy scale 5 were significantly associated with survival. The best cut-off VLF and scale 5 values were 115.13 and 0.738, respectively. We then categorized the study population into three groups: both elevated VLF/scale 5 (group 1), either depressed VLF or depressed scale 5 (group 2), and both depressed VLF/scale 5 (group 3). The results showed that group 1 had the best outcomes, whereas group 3 had the worst survival (P < 0.001).Combining HRV and HRC metrics appears to be a good non-invasive tool to predict the long-term outcomes of patients with PH.

Ordinal language of antipersistent binary walks

This paper explores the effectiveness of using ordinal pattern probabilities to evaluate antipersistency in the sign decomposition of long-range anti-correlated Gaussian fluctuations. It is numerically shown that ordinal patterns are able to effectively measure both persistent and antipersistent dynamics by analyzing the sign decomposition derived from fractional Gaussian noise. These findings are crucial given that traditional methods such as Detrended Fluctuation Analysis are unsuccessful in detecting anti-correlations in such sequences. The numerical results are supported by physiological and environmental data, illustrating its applicability in real-world situations.

Dynamic Black–Litterman Portfolios Incorporating Asymmetric Fractal Uncertainty

This study investigates the profitability of portfolios that integrate asymmetric fractality within the Black–Litterman (BL) framework. It predicts 10-day-ahead exchange-traded fund (ETF) prices using recurrent neural networks (RNNs) based on historical price information and technical indicators; these predictions are utilized as BL views. While constructing the BL portfolio, the Hurst exponent obtained from the asymmetric multifractal detrended fluctuation analysis is employed to determine the uncertainty associated with the views. The Hurst exponent describes the long-range persistence in time-series data, which can also be interpreted as the uncertainty in time-series predictions. Additionally, uncertainty is measured using asymmetric fractality to account for the financial time series’ asymmetric characteristics. Then, backtesting is conducted on portfolios comprising 10 countries’ ETFs, rebalanced on a 10-day basis. While benchmarking to a Markowitz portfolio and the MSCI world index, profitability is assessed using the Sharpe ratio, maximum drawdown, and sub-period analysis. The results reveal that the proposed model enhances the overall portfolio return and demonstrates particularly strong performance during negative trends. Moreover, it identifies ongoing investment opportunities, even in recent periods. These findings underscore the potential of fractality in adjusting uncertainty for diverse portfolio optimization applications.

Multifractal detrended fluctuation analysis, cross-correlation and clustering of global 7Be activity concentration

Multifractal detrended fluctuation analysis, cross-correlation and clustering of global ⁷<i>Be</i> activity concentration

Dynamic speculation and efficiency in European natural gas markets during the COVID-19 and Russia-Ukraine crises

Efficiency and dynamic speculation literature focused on financial and crude oil markets whereas empirical studies on natural gas markets are limited. This study examines market efficiency and dynamic speculation in major European natural gas hubs (the Zeebrugge (ZEE) hub of Belgium, Title Transfer Facility (TTF) hub of the Netherlands, and National Balancing Point (NBP) hub of the UK) during the COVID-19 pandemic and Russia-Ukraine conflict. Using the Asymmetric multifractal detrended fluctuation analysis (A-MFDFA) and the power law exponent (PLE) approach, we show that European natural gas markets exhibit an asymmetric multifractal behavior and long-memory effects, with variations observed during crises. The ZEE hub in Belgium and the NBP hub in the UK demonstrate a higher multifractality under different trends during the COVID-19 pandemic and the Russia-Ukraine conflict, respectively. Notably, the TTF hub in the Netherlands emerges as the most efficient market during all subperiods. Moreover, our results, using the PLE approach, suggest that European gas market inefficiencies during crises can be attributed to speculative strategies that create self-perpetuating dynamics and diminish market transparency, leading to heightened risk and volatility in gas prices. Our findings underscore the need for regulatory intervention to mitigate excessive speculation and enhance transparency in gas markets. Diversifying gas supplies and fortifying European gas hubs emerge as pivotal strategies to bolster resilience and stability during crises.

Comparative Analysis of Linear and Nonlinear sEMG Methods for Detecting Muscle Fatigue During Dynamic Biceps Curls

Muscle fatigue, a key concern in sports science, rehabilitation, and occupational health, influences performance, injury risk, and provides insights into muscle functionality and endurance. Surface electromyography (sEMG) has emerged as a vital tool for non-invasively tracking muscle electrical activity and gauging health. As its application for muscle fatigue assessment grows, identifying the most accurate analytical methods is essential. Current sEMG analyses employ both linear and nonlinear metrics to measure fatigue onset and progression, yet research is ongoing to determine which method is most effective in the context of dynamic contractions. The study was aimed to evaluate the efficacy of established linear and nonlinear methods in measuring muscle fatigue caused by dynamic contractions through surface electromyography (sEMG) signals. A group of twelve healthy individuals completed biceps curls at a consistent pace of one repetition per four seconds, which constituted 75% of their 10-repetition maximum. Concurrently, sEMG signals were captured from the biceps brachii muscle at 1000 Hz. To assess the sEMG signals during the initial, middle, and final sets of 10 repetitions, three linear metrics—mean frequency, median frequency, and spectral moment ratio (SMR)—along with two nonlinear approaches, namely sample entropy and detrended fluctuation analysis (DFA), were utilized. The study's outcomes indicated notable shifts in the SMR values and the two DFA-derived scaling exponents across the exercise sets. These results indicated that SMR, sample entropy, and DFA are effective in gauging muscle fatigue, with sample entropy and DFA demonstrating heightened sensitivity to the fatigue levels when compared to the linear metrics.

Decoding Market Efficiency: A Multifractal Approach to Financial Market Dynamics

Financial markets were traditionally analyzed using the Efficient Market Hypothesis (EMH), which argued that asset prices fully reflected all available information. This led to the belief that price movements were random, making it impossible to consistently outperform the market. However, emerging research had highlighted that markets could exhibit multifractality, where inefficiencies existed across different time scales, challenging the traditional EMH framework. Our study built on these insights by investigating the market efficiency of several asset classes, including stock market indices, cryptocurrencies (Bitcoin, Ethereum, USDT, and Binance), commodities (gold and crude oil), fixed income (U.S. 10-Year &amp; 30-Year Treasury Bonds), and foreign exchange (EUR/USD &amp; GBP/USD). Utilizing the Multifractal Detrended Fluctuation Analysis (MFDFA) method, we analyzed the multifractal properties and efficiency of each asset class. The results revealed that traditional assets exhibited monofractality, adhering more closely to the EMH. In contrast, cryptocurrencies displayed significant inefficiencies, likely due to their volatility, market structure, and susceptibility to external shocks. The findings provided valuable insights into portfolio diversification and risk management, suggesting that inefficiencies in certain asset classes might present both opportunities and risks for investors.