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Integrating running gait coordination assessment into athlete monitoring systems could provide unique insight into training tolerance and fatigue-related gait alterations. This study investigated the impact of an overload training intervention and recovery on running gait coordination assessed by field-based self-testing. Fifteen trained distance runners were recruited to perform 1-week of light training (baseline), 2 weeks of heavy training (high intensity, duration, and frequency) designed to overload participants, and a 10-day light taper to allow recovery and adaptation. Field-based running assessments using ankle accelerometry and online short recovery and stress scale (SRSS) surveys were completed daily. Running performance was assessed after each training phase using a maximal effort multi-stage running test-to-exhaustion (RTE). Gait coordination was assessed using detrended fluctuation analysis (DFA) of a stride interval time series. Two participants withdrew during baseline training due to changed personal circumstances. Four participants withdrew during heavy training due to injury. The remaining nine participants completed heavy training and were included in the final analysis. Heavy training reduced DFA values (standardised mean difference (SMD) = −1.44 ± 0.90; p = 0.004), recovery (SMD = −1.83 ± 0.82; p less than 0.001), performance (SMD = −0.36 ± 0.32; p = 0.03), and increased stress (SMD = 1.78 ± 0.94; p = 0.001) compared to baseline. DFA values (p = 0.73), recovery (p = 0.77), and stress (p = 0.73) returned to baseline levels after tapering while performance trended towards improvement from baseline (SMD = 0.28 ± 0.37; p = 0.13). Reduced DFA values were associated with reduced performance (r2 = 0.55) and recovery (r2 = 0.55) and increased stress (r2 = 0.62). Field-based testing of running gait coordination is a promising method of monitoring training tolerance in running athletes during overload training.

We present an innovative method harnessing multistability within a diode-pumped erbium-doped fiber laser to construct logic gates. Our approach involves manipulating the intensity of external noise to regulate the probability of transitioning among four concurrent attractors. In this manner, we facilitate the realization of OR, AND, NOR, and NAND logic operations, aligning with the coexisting period-1, period-3, period-4, and period-5 orbits. Employing detrended fluctuation analysis, we establish equilibrium in the probability distributions of these states. The obtained results denote a substantial advancement in the field of optical logic gate development, representing a pivotal stride toward the seamless integration of an all-optical logic gate within laser oscillator-based systems.

Continuous adaptations of the movement system to changing environments or task demands rely on superposed fractal processes exhibiting power laws, that is, multifractality. The estimators of the multifractal spectrum potentially reflect the adaptive use of perception, cognition, and action. To observe time-specific behavior in multifractal dynamics, a multiscale multifractal analysis based on DFA (MFMS-DFA) has been recently proposed and applied to cardiovascular dynamics. Here we aimed at evaluating whether MFMS-DFA allows identifying multiscale structures in the dynamics of human movements. Thirty-six (12 females) participants pedaled freely, after a metronomic initiation of the cadence at 60 rpm, against a light workload for 10 min: in reference to cycling (C), cycling while playing “Tetris” on a computer, alone (CT) or collaboratively (CTC) with another pedaling participant. Pedal revolution periods (PRP) series were examined with MFMS-DFA and compared to linearized surrogates, which attested to a presence of multifractality at almost all scales. A marked alteration in multifractality when playing Tetris was evidenced at two scales, τ ≈ 16 and τ ≈ 64 s, yet less marked at τ ≈ 16 s when playing collaboratively. Playing Tetris in collaboration attenuated these alterations, especially in the best Tetris players. This observation suggests the high sensitivity to cognitive demand of MFMS-DFA estimators, extending to the assessment of skill/demand interplay from individual behavior. So, by identifying scale-dependent multifractal structures in movement dynamics, MFMS-DFA has obvious potential for examining brain-movement coordinative structures, likely with sufficient sensitivity to find echo in diagnosing disorders and monitoring the progress of diseases that affect cognition and movement control.

The temporal evolution of the global mean sea level (GMSL) is investigated in the present analysis using the monthly mean values obtained from two sources: a reconstructed dataset and a satellite altimeter dataset. To this end, we use two well-known techniques, detrended fluctuation analysis (DFA) and multifractal DFA (MF-DFA), to study the scaling properties of the time series considered. The main result is that power-law long-range correlations and multifractality apply to both data sets of the global mean sea level. In addition, the analysis revealed nearly identical scaling features for both the 134-year and the last 28-year GMSL-time series, possibly suggesting that the long-range correlations stem more from natural causes. This demonstrates that the relationship between climate change and sea-level anomalies needs more extensive research in the future due to the importance of their indirect processes for ecology and conservation.

Reciprocating compressors and centrifugal pumps are rotating machines used in industry, where fault detection is crucial for avoiding unnecessary and costly downtime. A novel method for fault classification in reciprocating compressors and multi-stage centrifugal pumps is proposed. In the feature extraction stage, raw vibration signals are processed using multi-fractal detrended fluctuation analysis (MFDFA) to extract features indicative of different types of faults. Such MFDFA features enable the training of machine learning models for classifying faults. Several classical machine learning models and a deep learning model corresponding to the convolutional neural network (CNN) are compared with respect to their classification accuracy. The cross-validation results show that all models are highly accurate for classifying the 13 types of faults in the centrifugal pump, the 17 valve faults, and the 13 multi-faults in the reciprocating compressor. The random forest subspace discriminant (RFSD) and the CNN model achieved the best results using MFDFA features calculated with quadratic approximations. The proposed method is a promising approach for fault classification in reciprocating compressors and multi-stage centrifugal pumps.

Profound research on the characteristics of the long-term persistence of wind is greatly significant for understanding the characteristics of wind speed mechanisms as well as for avoiding disasters caused by wind. In the current study, we selected the hourly 10 min wind speed series between 2017 and 2021 from 105 nation-level meteorological stations in Xinjiang and investigated the spatiotemporal variations in the long-term persistence of wind speed in different regions of Xinjiang and in different seasons using detrended fluctuation analysis. The main findings are as follows: (1) The wind speed in Xinjiang shows noticeable annual and seasonal variations, exhibiting satisfactory long-term sustainability. Winter has the best long-term sustainability, followed sequentially by spring, autumn, and summer because of wind speed stability. (2) The long-term persistence of hourly wind speed in Xinjiang exhibits remarkable regionality, with regions with strong wind superior to the remaining regions. (3) The long-term persistence of wind speed within the same season is primarily associated with wind speed magnitude and the dispersion degree between 90% and 100% of the wind speed numerical values. A higher wind speed indicates better long-term persistence. At the same speed, the more discrete the numerical values in the 90–100% distribution range, the better the persistence.

In this paper, we thoroughly analyze the detection of sleep apnea events in the context of Obstructive Sleep Apnea (OSA), which is considered a public health problem because of its high prevalence and serious health implications. We especially evaluate patients who do not always show desaturations during apneic episodes (non-desaturating patients). For this purpose, we use a database (HuGCDN2014-OXI) that includes desaturating and non-desaturating patients, and we use the widely used Physionet Apnea Dataset for a meaningful comparison with prior work. Our system combines features extracted from the Heart-Rate Variability (HRV) and SpO2, and it explores their potential to characterize desaturating and non-desaturating events. The HRV-based features include spectral, cepstral, and nonlinear information (Detrended Fluctuation Analysis (DFA) and Recurrence Quantification Analysis (RQA)). SpO2-based features include temporal (variance) and spectral information. The features feed a Linear Discriminant Analysis (LDA) classifier. The goal is to evaluate the effect of using these features either individually or in combination, especially in non-desaturating patients. The main results for the detection of apneic events are: (a) Physionet success rate of 96.19%, sensitivity of 95.74% and specificity of 95.25% (Area Under Curve (AUC): 0.99); (b) HuGCDN2014-OXI of 87.32%, 83.81% and 88.55% (AUC: 0.934), respectively. The best results for the global diagnosis of OSA patients (HuGCDN2014-OXI) are: success rate of 95.74%, sensitivity of 100%, and specificity of 89.47%. We conclude that combining both features is the most accurate option, especially when there are non-desaturating patterns among the recordings under study.

Vehicle driving cycles have complex characteristics, but there are few publicly reported methods for their quantitative characterization. This paper innovatively investigates their multifractal characteristics using the fractal theory to characterize their complex properties, laying the foundation for applications such as vehicle driving cycle feature identification, vehicle energy management strategies (EMS), and so on. To explore the scale-invariance of the vehicle driving cycles, the four vehicle driving cycles were analyzed using the Multifractal Detrended Fluctuation Analysis (MF-DFA) method, three of which are standard vehicle test cycles: the New European Driving Cycle (NEDC), the World-wide harmonized Light-duty Test Cycle (WLTC) and the China Light-duty Vehicle Test Cycle for Passenger Car (CLTC-P), and the other is the Urban Road Real Driving Cycle (URRDC), which was obtained by analyzing and processing vehicle driving data collected in actual urban driving conditions. The fluctuation functions, the generalized Hurst exponents, the mass exponent spectra, the multifractal singularity spectra, and the multifractal characteristic parameters were calculated to verify the multifractal characteristics, and to quantify the fluctuation singularities of different driving cycles as the time series. The results show that the fluctuations of all four driving cycles have long-range anticorrelations and exhibit significant multifractal characteristics. The results can provide a basis for the analysis of the complexity of the vehicle driving cycles.

In this study we investigate possible long-range trends in the cryptocurrency market. We employed the Hurst exponent in a sample covering the period from 1 January 2016 to 26 March 2021. We calculated the Hurst exponent in three non-overlapping consecutive windows and in the whole sample. Using these windows, we assessed the dynamic evolution in the structure and long-range trend behavior of the cryptocurrency market and evaluated possible changes in their behavior towards an efficient market. The innovation of this research is that we employ the Hurst exponent to identify the long-range properties, a tool that is seldomly used in analysis of this market. Furthermore, the use of both the R/S and the DFA analysis and the use of non-overlapping windows enhance our research’s novelty. Finally, we estimated the Hurst exponent for a wide sample of cryptocurrencies that covered more than 80% of the entire market for the last six years. The empirical results reveal that the returns follow a random walk making it difficult to accurately forecast them.

Rangeland ecosystems comprise more than a third of the global land surface, sustaining essential ecosystem services and livelihoods. In Spain, Southeast Spain includes some of the driest regions; accordingly, rangelands from Murcia and Almeria provinces were selected for this study. We used time series metrics and the Hurst Exponent from rescale range and detrended fluctuation analysis to cluster different rangeland dynamics to classify temporally and spatially diverse rangelands. The metrics were only calculated for three time periods that showed significant NDVI changes: March to April, April to July, and September to December. Detrended fluctuation analysis was not previously employed to cluster vegetation. This study used it to improve rangeland classification. K-means and unsupervised random forest were used to cluster the pixels using time series metrics and Hurst exponents. The best clustering results were obtained when unsupervised random forest was used with the Hurst exponent calculated with detrended fluctuation analysis. We used the Silhouette Index to evaluate the clustering results and a spatial comparison with topographical data. Our results show that adding the Hurst exponent, calculated with detrended fluctuation analysis, provided a better classification when clustering NDVI time series, while classifications without the Hurst exponent or with the Hurst exponent calculated with the rescale range method showed lower silhouette values. Overall, this shows the importance of using detrending when calculating the Hurst exponent on vegetation time series, and its usefulness in studying rangeland dynamics for management and research.

This study aimed to apply different complexity-based methods to surface electromyography (EMG) in order to detect neuromuscular changes after realistic warm-up procedures that included stretching exercises. Sixteen volunteers conducted two experimental sessions. They were tested before, after a standardized warm-up, and after a stretching exercise (static or neuromuscular nerve gliding technique). Tests included measurements of the knee flexion torque and EMG of biceps femoris (BF) and semitendinosus (ST) muscles. EMG was analyzed using the root mean square (RMS), sample entropy (SampEn), percentage of recurrence and determinism following a recurrence quantification analysis (%Rec and %Det) and a scaling parameter from a detrended fluctuation analysis. Torque was significantly greater after warm-up as compared to baseline and after stretching. RMS was not affected by the experimental procedure. In contrast, SampEn was significantly greater after warm-up and stretching as compared to baseline values. %Rec was not modified but %Det for BF muscle was significantly greater after stretching as compared to baseline. The a scaling parameter was significantly lower after warm-up as compared to baseline for ST muscle. From the present results, complexity-based methods applied to the EMG give additional information than linear-based methods. They appeared sensitive to detect EMG complexity increases following warm-up.

Lidar is an active remote sensing technology that has many advantages, but the echo lidar signal is extremely susceptible to noise and complex atmospheric environment, which affects the effective detection range and retrieval accuracy. In this paper, a wavelet transform (WT) and locally weighted scatterplot smoothing (LOWESS) based on ensemble empirical mode decomposition (EEMD) for Rayleigh lidar signal denoising was proposed. The WT method was used to remove the noise in the signal with a signal-to-noise ratio (SNR) higher than 16 dB. The EEMD method was applied to decompose the remaining signal into a series of intrinsic modal functions (IMFs), and then detrended fluctuation analysis (DFA) was conducted to determine the threshold for distinguishing whether noise or signal was the main component of the IMFs. Moreover, the LOWESS method was adopted to remove the noise in the IMFs component containing the signal, and thus, finely extract the signal. The simulation results showed that the denoising effect of the proposed WT-EEMD-LOWESS method was superior to EEMD-WT, EEMD-SVD and VMD-WOA. Finally, the use of WT-EEMD-LOWESS on the measured lidar signal led to significant improvement in retrieval accuracy. The maximum error of density and temperature retrievals was decreased from 1.36% and 125.79 K to 1.1% and 13.84 K, respectively.

Background: Previous studies have associated slowed ventricular conduction with the arrhythmogenesis mediated by the gap junction and sodium channel inhibitor heptanol in mouse hearts. However, they did not study the propagation patterns that might contribute to the arrhythmic substrate. This study used a multi-electrode array mapping technique to further investigate different conduction abnormalities in Langendorff-perfused mouse hearts exposed to 0.1 or 2 mM heptanol. Methods: Recordings were made from the left ventricular epicardium using multi-electrode arrays in spontaneously beating hearts during right ventricular 8 Hz pacing or S1S2 pacing. Results: In spontaneously beating hearts, heptanol at 0.1 and 2 mM significantly reduced the heart rate from 314 ± 25 to 189 ± 24 and 157 ± 7 bpm, respectively (ANOVA, p < 0.05 and p < 0.001). During regular 8 Hz pacing, the mean LATs were increased by 0.1 and 2 mM heptanol from 7.1 ± 2.2 ms to 19.9 ± 5.0 ms (p < 0.05) and 18.4 ± 5.7 ms (p < 0.05). The standard deviation of the mean LATs was increased from 2.5 ± 0.8 ms to 10.3 ± 4.0 ms and 8.0 ± 2.5 ms (p < 0.05), and the median of phase differences was increased from 1.7 ± 1.1 ms to 13.9 ± 7.8 ms and 12.1 ± 5.0 ms by 0.1 and 2 mM heptanol (p < 0.05). P5 took a value of 0.2 ± 0.1 ms and was not significantly altered by heptanol at 0.1 or 2 mM (1.1 ± 0.9 ms and 0.9 ± 0.5 ms, p > 0.05). P50 was increased from 7.3 ± 2.7 ms to 24.0 ± 12.0 ms by 0.1 mM heptanol and then to 22.5 ± 7.5 ms by 2 mM heptanol (p < 0.05). P95 was increased from 1.7 ± 1.1 ms to 13.9 ± 7.8 ms by 0.1 mM heptanol and to 12.1 ± 5.0 ms by 2 mM heptanol (p < 0.05). These changes led to increases in the absolute inhomogeneity in conduction (P5–95) from 7.1 ± 2.6 ms to 31.4 ± 11.3 ms, 2 mM: 21.6 ± 7.2 ms, respectively (p < 0.05). The inhomogeneity index (P5–95/P50) was significantly reduced from 3.7 ± 1.2 to 3.1 ± 0.8 by 0.1 mM and then to 3.3 ± 0.9 by 2 mM heptanol (p < 0.05). Conclusion: Increased activation latencies, reduced CVs, and the increased inhomogeneity index of conduction were associated with both spontaneous and induced ventricular arrhythmias.

A better understanding of past climate change is vital to our ability to predict possible future environmental dynamics. This study attempts to investigate the dynamic features of the temporal variability of peat humification, water table depth and air temperature by analyzing palaeoecological data from the Valdai Uplands region (Central European Russia). The regression analysis revealed the presence of a periodicity of about 6000 years in the reconstructed peat humification timeseries. Nonlinear analysis showed that humification time variability, water table depth and air temperature exhibit persistent long-range correlations of 1/f type. This indicates that a fluctuation in these variables in the past is very likely to be followed by a similar one in the future, but is magnified by 1/f power-law. In addition, it dictates that humification, water table depth and temperature are key parameters of a system that implies the existence of a special structure, such as self-organized criticality, operating close to a minimum stability configuration, and achieves it without any fine adjustment by external forcing. These conclusions point to new avenues for modeling future ecosystem disturbances and, in particular, for predicting relevant extreme events.

The interest in photoplethysmography (PPG) for sleep monitoring is increasing because PPG may allow assessing heart rate variability (HRV), which is particularly important in breathing disorders. Thus, we aimed to evaluate how PPG wearable systems measure HRV during sleep at high altitudes, where hypobaric hypoxia induces respiratory disturbances. We considered PPG and electrocardiographic recordings in 21 volunteers sleeping at 4554 m a.s.l. (as a model of sleep breathing disorder), and five alpine guides sleeping at sea level, 6000 m and 6800 m a.s.l. Power spectra, multiscale entropy, and self-similarity were calculated for PPG tachograms and electrocardiography R–R intervals (RRI). Results demonstrated that wearable PPG devices provide HRV measures even at extremely high altitudes. However, the comparison between PPG tachograms and RRI showed discrepancies in the faster spectral components and at the shorter scales of self-similarity and entropy. Furthermore, the changes in sleep HRV from sea level to extremely high altitudes quantified by RRI and PPG tachograms in the five alpine guides tended to be different at the faster frequencies and shorter scales. Discrepancies may be explained by modulations of pulse wave velocity and should be considered to interpret correctly autonomic alterations during sleep from HRV analysis.

The purpose of this study is to understand the characteristics of the spatial distribution of forest fire occurrences with the local indicators of temporal burstiness in Korea. Forest fire damage data were produced in the form of areas by combining the forest fire damage ledger information with VIIRS-based forest fire occurrence data. Then, detrended fluctuation analysis and the local indicator of temporal burstiness were applied. In the results, the forest fire occurrence follows a self-organized criticality mechanism, and the temporal irregularities of fire occurrences exist. When the forest fire occurrence time series in Gyeonggi-do Province, which had the highest value of the local indicator of temporal burstiness, was checked, it was found that the frequency of forest fires was increasing at intervals of about 10 years. In addition, when the frequencies of forest fires and the spatial distribution of the local indicators of forest fire occurrences were compared, it was found that there were spatial differences in the occurrence of forest fires. This study is meaningful in that it analyzed the time series characteristics of the distribution of forest fires in Korea to understand that forest fire occurrences have long-term temporal correlations and identified areas where the temporal irregularities of forest fire occurrences are remarkable with the local indicators of temporal burstiness.

Detrended Fluctuation Analysis (DFA) has become a standard method to quantify the correlations and scaling properties of real-world complex time series. For a given scale ℓ of observation, DFA provides the function F(ℓ), which quantifies the fluctuations of the time series around the local trend, which is substracted (detrended). If the time series exhibits scaling properties, then F(ℓ)∼ℓα asymptotically, and the scaling exponent α is typically estimated as the slope of a linear fitting in the logF(ℓ) vs. log(ℓ) plot. In this way, α measures the strength of the correlations and characterizes the underlying dynamical system. However, in many cases, and especially in a physiological time series, the scaling behavior is different at short and long scales, resulting in logF(ℓ) vs. log(ℓ) plots with two different slopes, α1 at short scales and α2 at large scales of observation. These two exponents are usually associated with the existence of different mechanisms that work at distinct time scales acting on the underlying dynamical system. Here, however, and since the power-law behavior of F(ℓ) is asymptotic, we question the use of α1 to characterize the correlations at short scales. To this end, we show first that, even for artificial time series with perfect scaling, i.e., with a single exponent α valid for all scales, DFA provides an α1 value that systematically overestimates the true exponent α. In addition, second, when artificial time series with two different scaling exponents at short and large scales are considered, the α1 value provided by DFA not only can severely underestimate or overestimate the true short-scale exponent, but also depends on the value of the large scale exponent. This behavior should prevent the use of α1 to describe the scaling properties at short scales: if DFA is used in two time series with the same scaling behavior at short scales but very different scaling properties at large scales, very different values of α1 will be obtained, although the short scale properties are identical. These artifacts may lead to wrong interpretations when analyzing real-world time series: on the one hand, for time series with truly perfect scaling, the spurious value of α1 could lead to wrongly thinking that there exists some specific mechanism acting only at short time scales in the dynamical system. On the other hand, for time series with true different scaling at short and large scales, the incorrect α1 value would not characterize properly the short scale behavior of the dynamical system.

The lymphatic system of the brain meninges and head plays a crucial role in the clearance of amyloid-β protein (Aβ), a peptide thought to be pathogenic in Alzheimer’s disease (AD), from the brain. The development of methods to modulate lymphatic clearance of Aβ from the brain coild be a revolutionary step in the therapy of AD. The opening of the blood–brain barrier (OBBB) by focused ultrasound is considered as a possible tool for stimulation of clearance of Aβ from the brain of humans and animals. Here, we propose an alternative method of noninvasive music-induced OBBB that is accompanied by the activation of clearance of fluorescent Aβ (Fαβ) from the mouse brain. Using confocal imaging, fluorescence microscopy, and magnetic resonance tomography, we clearly demonstrate that OBBB by music stimulates the movement of Fαβ and Omniscan in the cerebrospinal fluid and lymphatic clearance of Fαβ from the brain. We propose the extended detrended fluctuation analysis (EDFA) as a promising method for the identification of OBBB markers in the electroencephalographic (EEG) patterns. These pilot results suggest that music-induced OBBB and the EDFA analysis of EEG can be a noninvasive, low-cost, labeling-free, clinical perspective and completely new approach for the treatment and monitoring of AD.

Brain electrical activity recorded as electroencephalogram data provides relevant information that can contribute to a better understanding of pathologies and human behaviour. This study explores extant electroencephalogram (EEG) signals in search of patterns that could differentiate subjects undertaking mental tasks and reveals insights on said data. We estimated the power spectral density of the signals and found that the subjects showed stronger gamma brain waves during activity while presenting alpha waves at rest. We also found that subjects who performed better in those tasks seemed to present less power density in high-frequency ranges, which could imply decreased brain activity during tasks. In a time-domain analysis, we used Hall–Wood and Robust–Genton estimators along with the Hurst exponent by means of a detrented fluctuation analysis and found that the first two fractal measures are capable of better differentiating signals between the rest and activity datasets. The statistical results indicated that the brain region corresponding to Fp channels might be more suitable for analysing EEG data from patients conducting arithmetic tasks. In summary, both frequency- and time-based methods employed in the study provided useful insights and should be preferably used together in EEG analysis.

In this paper, we have modified the Detrended Fluctuation Analysis (DFA) using the ternary Cantor set. We propose a modification of the DFA algorithm, Cantor DFA (CDFA), which uses the Cantor set theory of base 3 as a scale for segment sizes in the DFA algorithm. An investigation of the phenomena generated from the proof using real-world time series based on the theory of the Cantor set is also conducted. This new approach helps reduce the overestimation problem of the Hurst exponent of DFA by comparing it with its inverse relationship with α of the Truncated Lévy Flight (TLF). CDFA is also able to correctly predict the memory behavior of time series.

Pulmonary hypertension (PH) is a fatal disease—even with state-of-the-art medical treatment. Non-invasive clinical tools for risk stratification are still lacking. The aim of this study was to investigate the clinical utility of heart rhythm complexity in risk stratification for PH patients. We prospectively enrolled 54 PH patients, including 20 high-risk patients (group A; defined as WHO functional class IV or class III with severely compromised hemodynamics), and 34 low-risk patients (group B). Both linear and non-linear heart rate variability (HRV) variables, including detrended fluctuation analysis (DFA) and multiscale entropy (MSE), were analyzed. In linear and non-linear HRV analysis, low frequency and high frequency ratio, DFAα1, MSE slope 5, scale 5, and area 6–20 were significantly lower in group A. Among all HRV variables, MSE scale 5 (AUC: 0.758) had the best predictive power to discriminate the two groups. In multivariable analysis, MSE scale 5 (p = 0.010) was the only significantly predictor of severe PH in all HRV variables. In conclusion, the patients with severe PH had worse heart rhythm complexity. MSE parameters, especially scale 5, can help to identify high-risk PH patients.

Estimates suggest that more than 70% of the world’s rangelands are degraded. The Normalized Difference Vegetation Index (NDVI) is commonly used by ecologists and agriculturalists to monitor vegetation and contribute to more sustainable rangeland management. This paper aims to explore the scaling character of NDVI and NDVI anomaly (NDVIa) time series by applying three fractal analyses: generalized structure function (GSF), multifractal detrended fluctuation analysis (MF-DFA), and Hurst index (HI). The study was conducted in four study areas in Southeastern Spain. Results suggest a multifractal character influenced by different land uses and spatial diversity. MF-DFA indicated an antipersistent character in study areas, while GSF and HI results indicated a persistent character. Different behaviors of generalized Hurst and scaling exponents were found between herbaceous and tree dominated areas. MF-DFA and surrogate and shuffle series allow us to study multifractal sources, reflecting the importance of long-range correlations in these areas. Two types of long-range correlation appear to be in place due to short-term memory reflecting seasonality and longer-term memory based on a time scale of a year or longer. The comparison of these series also provides us with a differentiating profile to distinguish among our four study areas that can improve land use and risk management in arid rangelands.

In rotary machinery, bearings are susceptible to different types of mechanical faults, including ball, inner race, and outer race faults. In condition-based monitoring (CBM), several techniques have been proposed in fault diagnostics based on the vibration measurements. For this paper, we studied the fractal characteristics of non-stationary vibration signals collected from bearings under different health conditions. Using the detrended fluctuation analysis (DFA), we proposed a novel method to diagnose the bearing faults based on the scaling exponent (α1) of vibration signal at the short-time scale. In vibration data with high sampling rate, our results showed that the proposed measure, scaling exponent, provides an accurate identification of the health state of the bearing. At the end, we evaluated the performance of the proposed method under different data quality issues, data loss and induced noise.

Sleep plays a crucial role in maintaining brain health. Insufficient sleep leads to an enhanced permeability of the blood–brain barrier and the development of diseases of small cerebral vessels. In this study, we discuss the possibility of detecting changes in the electrical activity of the brain associated with sleep deficit, using an extended detrended fluctuation analysis (EDFA). We apply this approach to electroencephalograms (EEG) in mice to identify signs of changes that can be caused by short-term sleep deprivation (SD). Although the SD effect is usually subject-dependent, analysis of a group of animals shows the appearance of a pronounced decrease in EDFA scaling exponents, describing power-law correlations and the impact of nonstationarity as a fairly typical response. Using EDFA, we revealed an SD effect in 9 out of 10 mice (Mann–Whitney test, p<0.05) that outperforms the DFA results (7 out of 10 mice). This tool may be a promising method for quantifying SD-induced pathological changes in the brain.

Recent study points to the value of a non-linear heart rate variability (HRV) biomarker using detrended fluctuation analysis (DFA a1) for aerobic threshold determination (HRVT). Significance of recording artefact, correction methods and device bias on DFA a1 during exercise and HRVT is unclear. Gas exchange and HRV data were obtained from 17 participants during an incremental treadmill run using both ECG and Polar H7 as recording devices. First, artefacts were randomly placed in the ECG time series to equal 1, 3 and 6% missed beats with correction by Kubios software’s automatic and medium threshold method. Based on linear regression, Bland Altman analysis and Wilcoxon paired testing, there was bias present with increasing artefact quantity. Regardless of artefact correction method, 1 to 3% missed beat artefact introduced small but discernible bias in raw DFA a1 measurements. At 6% artefact using medium correction, proportional bias was found (maximum 19%). Despite this bias, the mean HRVT determination was within 1 bpm across all artefact levels and correction modalities. Second, the HRVT ascertained from synchronous ECG vs. Polar H7 recordings did show an average bias of minus 4 bpm. Polar H7 results suggest that device related bias is possible but in the reverse direction as artefact related bias.

Prognostic techniques play a critical role in predicting upcoming faults and failures in machinery or a system by monitoring any deviation in the operation. This paper presents a novel method to analyze multidimensional sensory data and use its characteristics in bearing health prognostics. Firstly, detrended fluctuation analysis (DFA) is exploited to evaluate the long-range correlations in ball bearing vibration data. The results reveal the existence of the crossover phenomenon in vibration data with two scaling exponents at the short-range and long-range scales. Among several data sets, applying the DFA method to vibration signals shows a consistent increase in the short-range scaling exponent toward bearing failure. Finally, Kendall’s tau is used as a ranking coefficient to quantify the trend in the scaling exponent. It was found that the Kendall’s tau coefficient of the vibration scaling exponent could provide an early warning signal (EWS) for bearing failure.

Rolling element bearings are widely employed in almost every rotating machine. The health status of bearings plays an important role in the reliability of rotating machines. This paper deals with the principle and application of an effective multi-sensor data fusion fault diagnosis approach for rolling element bearings. In particular, two single-axis accelerometers are employed to improve classification accuracy. By applying the improved detrended fluctuation analysis (IDFA), the corresponding fluctuations detrended by the local fit of vibration signals are evaluated. Then the polynomial fitting coefficients of the fluctuation function are selected as the fault features. A multi-sensor data fusion classification method based on linear discriminant analysis (LDA) is presented in the feature classification process. The faults that occurred in the inner race, cage, and outer race are considered in the paper. The experimental results show that the classification accuracy of the proposed diagnosis method can reach 100%.

We analyze trend and persistence in Standardized Precipitation Index (SPI) time series derived from monthly rainfall data at 133 gauging stations in Pernambuco state, Brazil, using a suite of complementary methods to address the spatially explicit tendencies, and persistence. SPI was calculated for 1-, 3-, 6-, and 12-month time scales from 1950 to 2012. We use Mann–Kendall test and Sen’s slope to determine sign and magnitude of the trend, and detrended fluctuation analysis (DFA) method to quantify long-term correlations. For all time scales significant negative trends are obtained in the Sertão (deep inland) region, while significant positive trends are found in the Agreste (intermediate inland), and Zona da Mata (coastal) regions. The values of DFA exponents show different scaling behavior for different time scales. For short-term conditions described by SPI-1 the DFA exponent is close to 0.5 indicating weak persistency and low predictability, while for medium-term conditions (SPI-3 and SPI-6) DFA exponents are greater than 0.5 and increase with time scale indicating stronger persistency and higher predictability. For SPI-12 that describes long-term precipitation patterns, the values of DFA exponents for inland regions are around 1, indicating strong persistency, while in the shoreline the value of the DFA exponent is between 1.0 and 1.5, indicating anti-persistent fractional Brownian motion. These results should be useful for agricultural planning and water resource management in the region.

The problem of revealing age-related distinctions in multichannel electroencephalograms (EEGs) during the execution of motor tasks in young and elderly adults is addressed herein. Based on the detrended fluctuation analysis (DFA), differences in long-range correlations are considered, emphasizing changes in the scaling exponent α. Stronger responses in elderly subjects are confirmed, including the range and rate of increase in α. Unlike elderly subjects, young adults demonstrated about 2.5 times more pronounced differences between motor task responses with the dominant and non-dominant hand. Knowledge of age-related changes in brain electrical activity is important for understanding consequences of healthy aging and distinguishing them from pathological changes associated with brain diseases. Besides diagnosing age-related effects, the potential of DFA can also be used in the field of brain–computer interfaces.

Financial and geophysical data, like many other low and high frequency time series, are known to exhibit some memory effects. These memory effects may be long or short, permanent or temporal depending on the event that is being modeled. The purpose of this study is to investigate the memory effects characterized by the financial market closing values and volcanic eruption time series as well as to investigate the relation between the self-similar models used and the Lévy process. This paper uses highly effective scaling methods including Lévy processes, Detrended Fluctuation Analysis (DFA) and Diffusion Entropy Analysis (DEA) to examine long-range persistence behavior in time series by estimating their respective parameters. We use the parameter of the Lévy process (α) characterizing the data and the scaling parameters of DFA (H) and DEA (δ) characterizing the self-similar property to generate a relationship between the three (3) aforementioned scaling methods. Findings from the numerical simulations confirm the existence of long-range persistence (long-memory behavior) in both the financial and geophysical time series. Furthermore, the numerical results from this study indicates an approximate inverse relationship between the parameter of the Lévy process and the scaling parameters of the DFA and DEA (i.e., H , δ ≈ 1 α ), which we prove analytically.

This work presents an analysis of low-visibility event persistence and prediction at Villanubla Airport (Valladolid, Spain), considering Runway Visual Range (RVR) time series in winter. The analysis covers long- and short-term persistence and prediction of the series, with different approaches. In the case of long-term analysis, a Detrended Fluctuation Analysis (DFA) approach is applied in order to estimate large-scale RVR time series similarities. The short-term persistence analysis of low-visibility events is evaluated by means of a Markov chain analysis of the binary time series associated with low-visibility events. We finally discuss an hourly short-term prediction of low-visibility events, using different approaches, some of them coming from the persistence analysis through Markov chain models, and others based on Machine Learning (ML) techniques. We show that a Mixture of Experts approach involving persistence-based methods and Machine Learning techniques provides the best results in this prediction problem.

A variety of heart rate variability (HRV) indices have been reported to estimate sleep stages, but the associations are modest and lacking solid physiological basis. Non-REM (NREM) sleep is associated with increased regularity of respiratory frequency, which results in the concentration of high frequency (HF) HRV power into a narrow frequency range. Using this physiological feature, we developed a new HRV sleep index named Hsi to quantify the degree of HF power concentration. We analyzed 11,636 consecutive 5-min segments of electrocardiographic (ECG) signal of polysomnographic data in 141 subjects and calculated Hsi and conventional HRV indices for each segment. Hsi was greater during NREM (mean [SD], 75.1 [8.3]%) than wake (61.0 [10.3]%) and REM (62.0 [8.4]%) stages. Receiver-operating characteristic curve analysis revealed that Hsi discriminated NREM from wake and REM segments with an area under the curve of 0.86, which was greater than those of heart rate (0.642), peak HF power (0.75), low-to-high frequency ratio (0.77), and scaling exponent α (0.77). With a cutoff >70%, Hsi detected NREM segments with 77% sensitivity, 80% specificity, and a Cohen’s kappa coefficient of 0.57. Hsi may provide an accurate NREM sleep maker for ECG and pulse wave signals obtained from wearable sensors.

Recently, a multifractal-multiscale approach to detrended fluctuation analysis (DFA) was proposed to evaluate the cardiovascular fractal dynamics providing a surface of self-similarity coefficients α(q,τ), function of the scale τ, and moment order q. We hypothesize that this versatile DFA approach may reflect the cardiocirculatory adaptations in complexity and nonlinearity occurring during the day/night cycle. Our aim is, therefore, to quantify how α(q, τ) surfaces of cardiovascular series differ between daytime and night-time. We estimated α(q,τ) with −5 ≤ q ≤ 5 and 8 ≤ τ ≤ 2048 s for heart rate and blood pressure beat-to-beat series over periods of few hours during daytime wake and night-time sleep in 14 healthy participants. From the α(q,τ) surfaces, we estimated short-term (q. We found a long-term night/day modulation of α(q,τ) between 128 and 256 s affecting heart rate and blood pressure similarly, and multifractal short-term modulations at q < 0 for the heart rate and at q > 0 for the blood pressure. Consistent nonlinearity appeared at the shorter scales at night excluding q = 2. Long-term circadian modulations of the heart rate DFA were previously associated with the cardiac vulnerability period and our results may improve the risk stratification indicating the more relevant α(q,τ) area reflecting this rhythm. Furthermore, nonlinear components in the nocturnal α(q,τ) at q ≠ 2 suggest that DFA may effectively integrate the linear spectral information with complexity-domain information, possibly improving the monitoring of cardiac interventions and protocols of rehabilitation medicine.

In this study, we use the Diffusion Entropy Analysis (DEA) to analyze and detect the scaling properties of time series from both emerging and well established markets as well as volcanic eruptions recorded by a seismic station, both financial and volcanic time series data have high frequencies. The objective is to determine whether they follow a Gaussian or Lévy distribution, as well as establish the existence of long-range correlations in these time series. The results obtained from the DEA technique are compared with the Hurst R/S analysis and Detrended Fluctuation Analysis (DFA) methodologies. We conclude that these methodologies are effective in classifying the high frequency financial indices and volcanic eruption data—the financial time series can be characterized by a Lévy walk while the volcanic time series is characterized by a Lévy flight.

Vegetation is an important part of terrestrial ecosystems. Although vegetation dynamics have explicit spatial and temporal dimensions, the study of the temporal process is in its infancy. Evaluation of temporal scaling behavior could provide a unique perspective for exploring the temporal nature of vegetation dynamics. In this study, the Global Inventory Modeling and Mapping Studies (GIMMS) Normalized Difference Vegetation Index (NDVI) was used to reflect vegetation dynamics, and the temporal scaling behavior of the NDVI in China was determined via detrended fluctuation analysis (DFA). Our main objectives were to reveal the temporal scaling behavior of NDVI time series and to understand variation among vegetation types. First, DFA revealed similar exponents, which ranged from 0.6 to 0.9, for all selected pixels, implying that a long-range correlation was generally present in the NDVI time series at the individual pixel scale. We then extended the analysis to all of China and found that 99.30% of the pixel exponents ranged from 0.5 to 1. These results suggest that the NDVI time series displays strong long-range correlation throughout most of China; however, the exponents exhibited regional variability. To explain these differences, we further analyzed the exponents for 12 vegetation types based on a vegetation map of China. All of the vegetation types exhibited well-defined long-range correlation, with exponents ranging from 0.7189 to 0.8436. For all vegetation types, the maximum and average value and standard deviation of the exponents decreased with increasing annual maximum NDVI values, suggesting that low vegetation density is much more sensitive to external factors. These findings may be useful for understanding vegetation dynamics as a complex, temporally varying phenomenon.

The recent introduction of chronotaxic systems provides the means to describe nonautonomous systems with stable yet time-varying frequencies which are resistant to continuous external perturbations. This approach facilitates realistic characterization of the oscillations observed in living systems, including the observation of transitions in dynamics which were not considered previously. The novelty of this approach necessitated the development of a new set of methods for the inference of the dynamics and interactions present in chronotaxic systems. These methods, based on Bayesian inference and detrended fluctuation analysis, can identify chronotaxicity in phase dynamics extracted from a single time series. Here, they are applied to numerical examples and real experimental electroencephalogram (EEG) data. We also review the current methods, including their assumptions and limitations, elaborate on their implementation, and discuss future perspectives.

Entropy estimation metrics have become a widely used method to identify subtle changes or hidden features in biomedical records. These methods have been more effective than conventional linear techniques in a number of signal classification applications, specially the healthy–pathological segmentation dichotomy. Nevertheless, a thorough characterization of these measures, namely, how to match metric and signal features, is still lacking. This paper studies a specific characterization problem: the influence of missing samples in biomedical records. The assessment is conducted using four of the most popular entropy metrics: Approximate Entropy, Sample Entropy, Fuzzy Entropy, and Detrended Fluctuation Analysis. The rationale of this study is that missing samples are a signal disturbance that can arise in many cases: signal compression, non-uniform sampling, or data transmission stages. It is of great interest to determine if these real situations can impair the capability of segmenting signal classes using such metrics. The experiments employed several biosignals: electroencephalograms, gait records, and RR time series. Samples of these signals were systematically removed, and the entropy computed for each case. The results showed that these metrics are robust against missing samples: With a data loss percentage of 50% or even higher, the methods were still able to distinguish among signal classes.

In recent years, increasing attention has been devoted to cryptocurrencies, owing to their great development and valorization. In this study, we propose to analyse four of the major cryptocurrencies, based on their market capitalization and data availability: Bitcoin, Ethereum, Ripple, and Litecoin. We apply detrended fluctuation analysis (the regular one and with a sliding windows approach) and detrended cross-correlation analysis and the respective correlation coefficient. We find that Bitcoin and Ripple seem to behave as efficient financial assets, while Ethereum and Litecoin present some evidence of persistence. When correlating Bitcoin with the other cryptocurrencies under analysis, we find that for short time scales, all the cryptocurrencies have statistically significant correlations with Bitcoin, although Ripple has the highest correlations. For higher time scales, Ripple is the only cryptocurrency with significant correlation.

The spatial and temporal variabilities of the daily Sunshine Duration (SSD) time series from the Chinese Meteorological Administration during the 1954⁻2009 period are examined by the Detrended Fluctuation Analysis (DFA) method. As a whole, weak long-range correlations (LRCs) are found in the daily SSD anomaly records over China. LRCs are also verified by shuffling the SSD records. The proportion of the stations with LRCs accounts for about 97% of the total. Many factors affect the scaling properties of the daily SSD records such as sea-land difference and Tibetan Plateau landform and so on. We find land use and land cover as one of the important factors closely links to LRCs of the SSD. Strong LRCs of the SSD mainly happen in underlying surface of deserts and crops, while weak LRCs occur in forest and grassland. Further studies of scaling behaviors are still necessary to be performed due to the complex underlying surface and climate system.

The non-linear analysis of the behavior of biological signals in humans is studied from different scientific disciplines. The aim of this study was to analyze the possible non-linear behavior present in eye movements during eye-tracking tasks in simulated sailing. Thirty young sailors were selected. Fuzzy entropy and detrended fluctuation analyses were applied to quantify the regularity and complexity of eye movements. The results show that neither experience nor ranking affect the regularity or the complexity of eye movement positions or velocities. Younger age is related to more regular visual behavior. At younger ages, eye positions present more complex behavior. Eye positions show more complex behavior than eye velocities. This complexity would allow for a more functional exploration of the environment by sailors. Eye movement velocity presents the greatest irregularity, with significantly higher values than eye movement position. This irregularity would facilitate the visual perception of the environment. All these findings could be related to the sailors’ functional behavior, based on complexity and stability, which has been associated with the ability of human beings to adapt to the environment.

Drought is a stochastic natural feature that arises due to intense and persistent shortage of precipitation. Its impact is mostly manifested as agricultural and hydrological droughts following an initial meteorological phenomenon. Drought prediction is essential because it can aid in the preparedness and impact-related management of its effects. This study considers the drought forecasting problem by developing a hybrid predictive model using a denoised empirical mode decomposition (EMD) and a deep belief network (DBN). The proposed method first decomposes the data into several intrinsic mode functions (IMFs) using EMD, and a reconstruction of the original data is obtained by considering only relevant IMFs. Detrended fluctuation analysis (DFA) was applied to each IMF to determine the threshold for robust denoising performance. Based on their scaling exponents, irrelevant intrinsic mode functions are identified and suppressed. The proposed method was applied to predict different time scale drought indices across the Colorado River basin using a standardized streamflow index (SSI) as the drought index. The results obtained using the proposed method was compared with standard methods such as multilayer perceptron (MLP) and support vector regression (SVR). The proposed hybrid model showed improvement in prediction accuracy, especially for multi-step ahead predictions.

Fluctuation Analysis (FA) and specially Detrended Fluctuation Analysis (DFA) are techniques commonly used to quantify correlations and scaling properties of complex time series such as the observable outputs of great variety of dynamical systems, from Economics to Physiology. Often, such correlated time series are analyzed using the magnitude and sign decomposition, i.e., by using FA or DFA to study separately the sign and the magnitude series obtained from the original signal. This approach allows for distinguishing between systems with the same linear correlations but different dynamical properties. However, here we present analytical and numerical evidence showing that FA and DFA can lead to spurious results when applied to sign and magnitude series obtained from power-law correlated time series of fractional Gaussian noise (fGn) type. Specifically, we show that: (i) the autocorrelation functions of the sign and magnitude series obtained from fGns are always power-laws; However, (ii) when the sign series presents power-law anticorrelations, FA and DFA wrongly interpret the sign series as purely uncorrelated; Similarly, (iii) when analyzing power-law correlated magnitude (or volatility) series, FA and DFA fail to retrieve the real scaling properties, and identify the magnitude series as purely uncorrelated noise; Finally, (iv) using the relationship between FA and DFA and the autocorrelation function of the time series, we explain analytically the reason for the FA and DFA spurious results, which turns out to be an intrinsic property of both techniques when applied to sign and magnitude series.