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Background Understanding the underlying architecture of mood regulation in bipolar disorder (BD) is important, as we are starting to conceptualize BD as a more complex disorder than one of recurring manic or depressive episodes. Nonlinear techniques are employed to understand and model the behavior of complex systems. Our aim was to assess the underlying nonlinear properties that account for mood and energy fluctuations in patients with BD; and to compare whether these processes were different in healthy controls (HC) and unaffected first-degree relatives (FDR). We used three different nonlinear techniques: Lyapunov exponent, detrended fluctuation analysis and fractal dimension to assess the underlying behavior of mood and energy fluctuations in all groups; and subsequently to assess whether these arise from different processes in each of these groups. Results There was a positive, short-term autocorrelation for both mood and energy series in all three groups. In the mood series, the largest Lyapunov exponent was found in HC (1.84), compared to BD (1.63) and FDR (1.71) groups [F (2, 87) = 8.42, p Conclusions The underlying nature of mood variability is in keeping with that of a chaotic system, which means that fluctuations are generated by deterministic nonlinear process(es) in HC, BD, and FDR. The value of this complex modeling lies in analyzing the nature of the processes involved in mood regulation. It also suggests that the window for episode prediction in BD will be inevitably short.

Myocardial infarction (MI), usually termed as heart attack, is one of the main cardiovascular diseases that occur due to the blockage of coronary arteries. This blockage reduces the blood supply to heart muscles, and a prolonged deficiency of blood supply causes the death of heart muscles leading to a heart attack that may cause death. An electrocardiogram (ECG) is used to diagnose MI as it causes variations like ST-T changes in the recorded ECG. Manual inspection of these variations is a tedious task and also requires expertise as the variations produced by MI are often very short in duration with a low amplitude. Hence, these changes may be misinterpreted, leading to delayed diagnosis and appropriate treatment. Therefore, computer-aided analysis of ECG may help to detect MI automatically. In this study, a robust deep learning model is proposed to detect MI based on heart rate variability (HRV) analysis of ECG signals from a single lead. Ultrashort-term HRV analysis is performed to compute HRV analysis features from time-domain and frequency-domain parameters through power spectral density estimations. Nonlinear HRV parameters are also computed using Poincare Plot, Recurrence Analysis, and Detrended Fluctuation Analysis. A finely tuned customized artificial neural network (ANN) algorithm is applied on 23 HRV features for MI detection and classification. The K-fold validation method is used to avoid any biases in results and reported 99.1% accuracy, 100% sensitivity, 98.1% specificity, and 99.0% F1 for MI detection, whereas 98.85% accuracy, 97.40% sensitivity, 99.05% specificity, and 97.70% F1 score is achieved for classification. Furthermore, the ANN algorithm completed its execution in just 59 seconds that indicates the efficiency of the proposed ANN model. The overall performance in terms of computed evaluation matrices and execution time indicates the robustness and cost-effectiveness of the proposed methodology. Thus, the proposed model can be used for high-performance MI detection, even in wearable devices.

Present study is aimed at investigating the solar faculae area from 1990 to 2007 which partially covered the22nd and 23rd solar cycle. Rescaled Range Analysis (RRA) and Detrended Fluctuation Analysis (DFA) have beenadopted to evaluate the behaviour of nonlinear dynamics of solar faculae area. Results show that the value of Hurstexponent for solar faculae area from RRA and DFA is negatively correlated. It means it is non-persistent and longrange correlated. Obtained result is inaccurate so the only solution is to transform the data into stationary data by takingdifferencing. RRA is applied on residuals and RRA to evaluate the fractal property of the time series. Solar faculae areainvestigated in this study is fractal in nature and predictable ass well. Moreover, the time series of solar faculae area isnon-linear as established by the Brock – Dechert – Scheinkman (BDS) test results

Studies on entomopter’s performance under the influence of gust have received impetus in the past decade. There exists a dire need to ascertain the threshold of the frontal gusty conditions which would destabilize these anthropogenic flyers. This would help to devise methods to mitigate the detrimental effects of gust. In light of this aspect, the present study aims at analyzing the onsets of instability in a flapping wing system subjected to temporal gust by employing recurrence period density entropy (RPDE) and detrended fluctuation analysis (DFA). Simulation of the flapping wing along inclined stroke is carried out for a Reynolds number of 150. This Reynolds number lies in the typical operating regime of fruit flies and entomopters like the Pico aerial vehicle. Numerical simulations are carried out to solve the laminar, unsteady, and incompressible Navier–Stokes equations. The dynamic meshing technique is employed to model flapping kinematics. Nine gusts with a combination of frequency and velocity ratios of 0.1, 0.5, and 1.0 are considered. Instantaneous horizontal and vertical forces are estimated. Time series of these forces are analyzed using RPDE and DFA paradigms. These analyses indicate that gust frequency of an order of magnitude higher than flapping frequency and gust amplitude of the order of magnitude as the wing’s root mean square velocity induces a possible onset of instability.

We aimed to analyse the complexity and fractal nature of heartbeat during constant exercise, at three different intensities, and recovery.Fourteen healthy men underwent 4 separate sessions. The first session was an incremental treadmill test to determine ventilatory thresholds (VT1 and VT2) and maximal aerobic speed (MAS). Each subject ran at VT1 and VT2 speeds and MAS (second, third and fourth day). The duration of VT1 and VT2 loads were selected in such a way that the product intensity-duration (training load) was the same. Sample Entropy (SampEn) and slope of Detrended Fluctuation Analysis (DFA α1) were measured during the whole session.DFA α1 declines with exercise, being less in the VT1 trial than in the other two.SampEn shows no significant change during exercise. The three tests induce the same decline in SampEn, but at the highest intensity (MAS) tends to decline during the exercise itself, whereas at lower intensities (VT1, VT2) the decline is delayed (10 min of recovery). Subsequently, SampEn at VT1 gradually recovers, whereas at VT2 and MAS it remains stable during recovery.In conclusion, exercise produces a loss of heartbeat complexity, but not fractal nature, during recovery and it depends on intensity.

Graphical abstract, Highlights • The dynamics of the cryptocurrencies is analyzed. • Several tools for investigating the dynamic behavior of the cryptocurrencies are adopted. • Characteristics of the cryptocurrencies price time series such as persistence, randomness, predictability and chaoticity are addressed. • With exception of for the Bitcoin, the other five cryptocurrencies analyzed are mean reverting., Introduction Cryptocurrencies have been attracting the attention from media, investors, regulators and academia during the last years. In spite of some scepticism in the financial area, cryptocurrencies are a relevant subject of academic research. Objectives In this paper, several tools are adopted as an instrument that can help market agents and investors to more clearly assess the cryptocurrencies price dynamics and, thus, guide investment decisions more assertively while mitigating risks. Methods We consider three methods, namely the Auto-Regressive Integrated Moving Average (ARIMA), Auto-Regressive Fractionally Integrated Moving Average (ARFIMA) and Detrended Fluctuation Analysis, and three indices given by the Hurst and Lyapunov exponents or the Fractal Dimension. This information allows assessing the behaviour of the time series, such as their persistence, randomness, predictability and chaoticity. Results The results suggest that, except for the Bitcoin, the other cryptocurrencies exhibit the characteristic of mean reverting, showing a lower predictability when compared to the Bitcoin. The results for the Bitcoin also indicate a persistent behavior that is related to the long memory effect. Conclusions The ARFIMA reveals better predictive performance than the ARIMA for all cryptocurrencies. Indeed, the obtained residual values for the ARFIMA are smaller for the auto and partial auto correlations functions, as well as for confidence intervals.

The influence of ARFIMA model parameters on the accuracy of financial time series forecasting on the example of artificially generated long memory series and daily log returns of RTS index is investigated. The investigated parameters are deviation of the integration order value from its «true» value, as well as the memory «length» considered by the model. Based on the research results, some practical recommendations for modeling using ARFIMA have been formulated.

Control system performance assessment is significant, especially in practical applications. One of the most important indices for the performance assessment of control systems is minimum variance. Calculating the minimum variance index in multivariate systems requires prior knowledge of system parameters and models, and is therefore an obstacle in practical applications. In this article, an index is proposed for the performance assessment of multivariate control loops, evaluating the system performance with the minimum variance criterion and using only the system’s routine operation data. This index can quantify the performance using neither any prior knowledge of system parameters nor the system’s optimal operation data. The proposed index is based on the Hurst exponent, a parameter for measuring correlations in time series data. In this article, detrended fluctuation analysis and rescaled range analysis are used to estimate the Hurst exponents of system outputs. Using a combination of these Hurst exponents, an index is defined for the performance assessment of multivariate systems. The results of simulation examples illustrate that the proposed index can assess the performance efficiently.

Long-term memory has been studied for decades and it has long been acknowledged that hydrological and hydro-meteorological time series exhibit this property. Physically, long-term memory is explained by saying that the stronger the long-term memory the more likely a series will ‘remember’ its previous value, in other words, the longer a time series is likely to persist in the proximity of a certain value. To increase the benefit of the study of long-term memory, we investigate the extent to which this explanation is accurate and descriptive of long-term memory by developing a ‘persistence measure’ that quantifies the intuitive description of long-term memory. The ‘persistence measure’ is compared to the scaling exponent (α) which quantifies long-term memory using detrended fluctuation analysis. A total of 17,359 series, including hydrological and hydro-meteorological series, downloaded from online sources and 130 synthetic series are analyzed. The main outcome is that two regimes are found. The first is for series with α ≲ 1.05 where the persistence measure is not sensitive to the scaling exponent. Hence, changes in α does not lead to an increase in the memory strength (i.e. the persistence measure). The second regime is for series with α ≳ 1.05 where a statistically significant positive correlation was found. Hence, for a certain range of values of α the physical explanation of long-term memory holds. Nevertheless, it is proposed that adding additional factors may create a more robust ‘persistence measure’ that can be easily understood and incorporated in hydrologists’ work.

Purpose The new COVID-19 pandemic caused by SARS-CoV-2 initiated in the world a large quarantine, due to the exponential capacity of the virus in spreading from human contact. In the present work, the dynamics of such spreading was evaluated by the analysis of the growth factor of the disease. Method It was applied the space phase of the time series, the detrended fluctuation analysis (DFA) of the series, and the fractal dimension of the space phase. Results It was possible to notice a strange attractor in the space phase of the growth factor indicating that the process is a deterministic chaos. The value of the alpha coefficient by DFA showed to be less than 0.5, characteristic of anti-persistent long-range memory, in which large events alternate with small events, and vice-versa. The fractal dimension of the phase space set was a fractal number, between 1 and 2, another indicator that the growth factor of the disease is not random. Conclusion This global analysis is pointing that the spread of COVID-19 is a deterministic chaos, with long-range memory. Understanding such dynamics may help the control of this disease.

This paper investigates the existence of chaos in concentration dynamics of particulate matter with an aerodynamic diameter less than or equal to $$10\,\upmu \hbox {m}$$ ( $$\hbox {PM}_{10}$$ ) in the greater Athens area (GAA), Greece. It reports findings on three 16-year $$\hbox {PM}_{10}$$ time series recorded by three different air pollution monitoring stations located in GAA and examines if critical fractal epochs with long memory exist. Detrended Fluctuation Analysis (DFA) and Rescaled Range (R/S) Analysis were used via sliding windows of approximately 1-month duration. In all $$\hbox {PM}_{10}$$ time series, several segments were found with critical fractal behaviour and hidden long-memory patterns. All these segments exhibited Hurst exponents above 0.75 and DFA exponents above 1.75. Twelve $$\hbox {PM}_{10}$$ segments with fractality and long memory were commonly identified by both techniques. In one case, long memory was identified concurrently across all three air pollution monitoring stations and in another case, across two stations. The importance of the agreement between two different and independent chaos-analysis techniques is discussed in association with the proper selection of threshold values. This is the second time to address chaos in $$\hbox {PM}_{10}$$ data series in GAA, and the first time to combine two widely accepted techniques, DFA and R/S analysis.

This study analysed long-term rainfall data (1851–2006) over seven climatic zones of India at seasonal and annual scales based on three techniques: (i) linear regression, (ii) multifractal detrende...

The robustness of two widespread multifractal analysis methods, one based on detrended fluctuation analysis and one on wavelet leaders, is discussed in the context of time-series containing non-uniform structures with only isolated singularities. Signals generated by simulated and experimentally-realized chaos generators, together with synthetic data addressing particular aspects, are taken into consideration. The results reveal essential limitations affecting the ability of both methods to correctly infer the non-multifractal nature of signals devoid of a cascade-like hierarchy of singularities. Namely, signals harboring only isolated singularities are found to artefactually give rise to broad multifractal spectra, resembling those expected in the presence of a well-developed underlying multifractal structure. Hence, there is a real risk of incorrectly inferring multifractality due to isolated singularities. The careful consideration of local scaling properties and the distribution of H\"older exponent obtained, for example, through wavelet analysis, is indispensable for rigorously assessing the presence or absence of multifractality., Comment: To appear in Nonlinear Dynamics

This paper proposes a strategy for suppressing noise via empirical mode decomposition (EMD)-based hierarchical multiresolution analysis. First, an intrinsic mode function (IMF) index that can differentiate the noise-dominant IMFs and the information-dominant IMFs in the first level of decomposition is determined by the conventional EMD-based denoising method. Next, the EMD is applied to those discarded IMFs in the first level of decomposition to obtain the IMFs in the second level of decomposition. Then, the detrended fluctuation analysis (DFA) is employed to define the IMF index separating the noise-dominant IMFs and the information-dominant IMFs in the second level of decomposition. Information-dominant IMFs in both the first and second levels of decompositions are summed together to obtain the denoised signal. The study is limited to signals that were corrupted by additive white Gaussian noise (AWGN). Extensive computer numerical simulations are conducted on both synthetic signals and practical signals. The results show that our proposed method is promising in some cases, especially at high-input SNR levels. For example, in the case of the CH4 signal with 12-dB input SNR, the processed SNR based on our proposed strategy with the consecutive mean squared error (CMSE) is 23.27 dB, while that based on the conventional CMSE method is only 23.14 dB. Also, our proposed method can extract information from more than one level of decomposition to reconstruct the denoised signal compared to only one level of decomposition in the conventional EMD-based denoising method.

We consider a method of detrended fluctuation analysis (DFA method), which enables the study of long-range correlations in nonstationary processes. Its modification is proposed, including the calculation of additional quantity, that is, a scaling index, characterizing the effects of nonstationarity in the experimental data. Using the dynamics of the blood flow velocity in cerebral vessels as an example, the possibilities of a quantitative description of changes in the signal structure using the proposed modification of the DFA method have been demonstrated.

The present study examined how prevalent methods for determining maximal voluntary contraction (MVC) impact the experimentally derived functions of graded force-force variability. Thirty-two young healthy subjects performed continuous isometric force tracking (20 s trials) at 10 target percentages (5–95% MVC) normalized to a conventional discrete-point (n = 16), or sustained (n = 16) MVC calculation. Distinct rates and magnitudes of change were observed for absolute variability (standard deviation (SD), root mean squared error (RMSE)), tracking error (RMSE, constant error (CE)), and complexity (detrended fluctuation analysis (DFA)) (all p &lt, 0.05) of graded force fluctuations between the MVC groups. Differential performance strategies were observed beyond ~65% MVC, with the discrete-point group minimizing their SD at force values below that of the criterion target (higher CE/RMSE). Moreover, the sustained group’s capacity to minimize SD/RMSE/CE corresponded to a more complex structure in their force fluctuations. These findings reveal that the time component of MVC estimation has a direct influence on the corrective strategies supporting near-maximal manual force control. While discrete MVC protocols predominate in the study of manual strength/endurance/precision, a 1:1 MVC-task mapping appears more to be ecologically valid if visuo-motor precision outcomes are of central importance.

International audience; The detrended fluctuation analysis (DFA) and its variants are popular methods to analyze the self-similarity of a signal. Two steps characterize them: firstly, the trend of the centered integrated signal is estimated and removed. Secondly, the properties of the so-called fluctuation function which is an approximation of the standard deviation of the resulting process is analyzed. However, it appears that the statistical mean was assumed to be equal to zero to obtain it. As there is no guarantee that this assumption is true a priori, this hypothesis is debatable. The purpose of this paper is to propose two alternative definitions of the fluctuation function. Then, we compare all of them based on a matrix formulation and the filter-based interpretation we recently proposed. This analysis will be useful to show that the approach proposed in the original paper remains a good compromise in terms of accuracy and computational cost.

Analysis of the complexity and variability from the biomedical physiological time series data raises significant interest as a promising and sensitive marker of abnormality or impairment assessment in muscle physiology, especially in electromyography (EMG) signal. This paper aimed to measure subject-specific (i.e., individual) fractal dimension as a quantitative measure of complexity of EMG signal (i.e., detecting long-range correlations in noisy signal) from upper limb bicep brachii (BB) muscle during five elbow joint angles movement (at 0°, 30°, 60°, 90° and 120°). The EMG signal was recorded from ten healthy (mean±SD age: 22.4±1.5 years) participants using wearable sensor. The fractal scaling (α-values) of the EMG time series was assessed using a non-linear technique called detrended fluctuation analysis (DFA). Majority of the results show that DFA α-values at each angle exhibit anti-correlated (i.e., DFA α < 0.05) behavior. Few results show positive correlation (i.e., DFA α between 0.53 to 0.77), but none of the α values have 1.0 (strongly correlated/pink noise). No significant difference exits between the elbow angles except one case, i.e., 0° vs. 30° (p < 0.05). This DFA-based complexity measuring results from EMG signal holds promise for rehabilitation of control of upper limb muscle activation patterns.

The detrended fluctuation analysis (DFA) is extensively useful in stochastic processes to unveil the long-term correlation. Here, we apply the DFA to point processes that mimick earthquake data. The point processes are synthesized by a model similar to the Epidemic-Type Aftershock Sequence model, and we apply the DFA to time series $N(t)$ of the point processes, where $N(t)$ is the cumulative number of events up to time $t$. Crossover phenomena are found in the DFA for these time series, and extensive numerical simulations suggest that the crossover phenomena are signatures of non-stationarity in the time series. We also find that the crossover time represents a characteristic time scale of the non-stationary process embedded in the time series. Therefore, the DFA for point processes is especially useful in extracting information of non-stationary processes when time series are superpositions of stationary and non-stationary signals. Furthermore, we apply the DFA to the cumulative number $N(t)$ of real earthquakes in Japan, and we find a crossover phenomenon similar to that found for the synthesized data., Comment: 9 pages, 5 figures

Research on complexity and uncertainty of nonlinear signals has great significance in dynamic system analysis. In order to further analyze the detailed information from the financial data to acquire deeper insight into the complex system and improve the ability of prediction, we propose the entropy of scale exponential spectrum (EOSES) as a new measure. Combined with multiscale theory, we get multiscale EOSES. The scale exponential spectrum (SES) is derived from the scale exponent of detrended fluctuation analysis. As for the entropy, we choose Renyi entropy and fractional cumulative residual entropy to compare and analyze the results. Simulated data and financial time series are used to obtain further in-depth information on the EOSES. Compared with traditional methods, we find that Renyi EOSES over moving window can provide more details of complexity which include fractal structure and scale properties. Also, it reduces the influence of degree of fitting polynomial and has higher noise immunity. In addition, through the SES and EOSES, we can better research the properties and stages of stock markets and distinguish stock markets with different characteristics.

This study is aimed to develop a post-processing method to regularize amplitude of PPG waveforms and evaluate the performance of the algorithm. The proposed procedure for regularizing amplitude consists of extracting the signal envelopes based on time-domain filtering, generating a compensation curve, and reconstructing the waveform. The proposed technique was evaluated with respect to dispersion and waveform fluctuation by means of coefficients of variation (CV) and a scaling exponent of detrended fluctuation analysis (DFA). Other criteria included the detection performances of major features such as upper and lower fiducial points and maximum dV/dt. In developing this method, we used a clinical dataset that comprised 74,225 beats. By applying reconstruction algorithm, the %CV and the scaling exponent of DFA were significantly decreased from 115.4 to 71.4 and from 1.029 to 0.836, respectively in peak amplitude. In valley amplitude, they were also decreased from 38.3 to 13.2 and from 0.993 to 0.713, respectively. Moreover, sensitivity of maximum dV/dt point, upper and lower peak detection are increased by 8.50, 2.72 and 5.21% after reconstruction, respectively. Our results clearly showed that the proposed reconstruction technique could reduce amplitude fluctuation and improve detection of fiducial points.

Research question The current study investigated stride-to-stride fluctuations of step rate and contact time in response to enforced step frequency perturbations as well as adaptation and de-adaptation behavior. Methods Forty distance runners ran at a self-selected speed and were asked to match five different enforced step frequencies (150, 160, 170, 180, and 190 beats per min). The influence of experience was explored, because running is a skill that presumably gets better with practice, and increased years of running experience is protective against injury. Detrended fluctuation analysis was used to determine the strength of long-range correlations in gait fluctuations at baseline, during the perturbation, and post-perturbation. Adaptive response was measured by the ability to match, rate of matching, and aftereffect of step frequency perturbations. Results The structure of stride-to-stride fluctuations for step rate and contact time did not change during the perturbation or post-perturbation compared to baseline. However, fluctuations in step rate were affected by the level of perturbation. Runners with the most experience had a less persistent structural gait pattern for both step rate and contact time at baseline. Highly experienced runners also demonstrated the best adaptive response. They better matched the enforced step frequency, reached the enforced step frequency sooner, and returned to preferred step frequency more quickly following removal of the perturbation. Significance These findings indicate baseline locomotor flexibility may be beneficial to achieve task demands and return to a stable state once the task is complete. Increased locomotor flexibility may also be a contributing factor for reduced injury risk in experienced runners.

The knowledge of geological strata sequences and their exact boundaries depths is of a great importance in the characterization of oil and gas reservoirs. The aim of automated well-to-well correlation approaches is to facilitate and accelerate the procedure of geological boundary detection. In this study, we implement an automatic well-to-well correlation approach based on pattern extraction from well-logs. The well-log patterns are recognized by calculating several statistical and fractal parameters. As a fractal parameter, we select the wavelet standard deviation exponent, calculated by the discrete wavelet transform. Furthermore, to select the proper wavelet function, as the heart of wavelet transform, the energy to Shannon entropy ratio criterion is implemented. The statistical pattern recognition parameters of this study include average value, maximum to minimum ratio, coefficient of variation, and the trend angle of well-log data (i.e., Gamma ray log) in a window around the geological boundary. Moreover, the analysis of variance (ANOVA) tool and the Tukey multiple comparison method are implemented to evaluate the effectiveness of each parameter (i.e., fractal or statistical parameters) during the determination of boundary depths. The gamma ray logs from three wells of an oil field are used as a dataset for the evaluation of our algorithm. The outputs of our methodology are also compared with the new detrended fluctuation analysis (DFA) method, which show promising outcomes over it. The results show that the average value of the signal, among the analyzed parameters, is the most effective parameter; however, implementing the combination of all fractal and statistical parameters can improve the accuracy of the geological boundary detection.

Among the sports fans beliefs about "hot hands" and "winning streaks" are widely spread, while the scientific debate about these effects is still ongoing. Recently in a paper by P. Ferreira [Physica A 500: 92-96] detrended fluctuation analysis was applied to the NBA teams' win records. It was shown that 28 considered NBA teams exhibit persistence in the win record time series. In this paper we take the same data set and compare the obtained results against various random models. We find that the empirical results are consistent with the results obtained from various simple random models., Comment: 10 pages, 4 figures, 3 tables

Based on multi-fractal detrended fluctuation analysis (MF-DFA) method, the extended self-similarity (ESS) is incorporated to develop a novel method to quantify multi-fractal characteristics. Detailed results show that compared with MF-DFA this new extended self-similarity based MF-DFA (ESS-MF-DFA) method can significantly extend scaling range and reduce uncertainties in estimating the exponents. Moreover, although ESS-MF-DFA method is developed from the DFA method with a fundamental assumption of a definite scaling range between fluctuation function and scale, ESS-MF-DFA can still work well even when DFA fails due to no scaling range between fluctuation function and scale. Furthermore, a criterion without estimating the generalized Hurst exponents is developed based on ESS-MF-DFA to distinguish multi-fractal from mono-fractal behavior and to quantify multi-fractal strength. All these results indicate that ESS-MF-DFA outperforms MF-DFA in reliably handling multi-fractal quantifications for much wider fields.

Investigation of the nonlinear dynamics of the heart rate is crucial for understanding the physiological mechanisms of various functional states and understanding the adaptation capacity of the human body. The purpose of this study was to evaluate some quantitative parameters of the nonlinear parameters of resonance in the heart rate variability. The study included 70 healthy students who underwent paced breath training. The heart rate was recorded during breathing at different respiratory rates (spontaneous breathing, 4.5, 5, 5.5, 6, and 6.5 breaths/min). The resonant frequency was determined using heart-rate spectral analysis. Nonlinear analysis of HRV was performed using entropy measures (ApEn, SampEn, and MSE), Correlation dimension, Detrended Fluctuation Analysis (DFA), Recurrence Plot (REC, DET, Lmean, Lmax, and ShanEn), the Poincare plot (SD1 and SD2), and time asymmetry (GI and P). We observed a decrease in the level of complexity, an increase in the self-similarity of the RR intervals, a decrease in the chaotic component, and an increase in rhythm asymmetry during breathing at resonant frequency.

The aim of the present study was to investigate how age and sensory deprivation affect the temporal organization of CoP sway variability and the postural corrective commands during standing balance in typically developing preschoolers. A sample of 57 children aged 3–5 years participated in the study. Structural stabilometric descriptors of sample entropy (SEn), detrended fluctuation analysis (DFA), and sway density curve (SDC) analysis were employed to assess features of center of pressure sway. A force platform was used to collect center of pressure data during standing balance over 40 s in four conditions: standing on rigid and foam surfaces with eyes open and closed. The main results are as follows: (1) sample entropy decreased and DFA_coefficient increased with age, while the SDC variables remained unaltered among the 3-, 4-, and 5-year-old children; (2) as sensory conditions became more challenging, sample entropy decreased and DFA_coefficient increased, while MT and MD decreased and MD increased; age did not influence the responses to sensorial deprivation. In conclusion, 5-year-old children showed decreased variability of CoP sway during standing balance compared with the younger children, but all children used the same corrective torques to control for perturbations. More challenging sensory deprivation conditions resulted in decreased variability of postural sway, higher amplitudes and more frequent correcting torques for stabilization, but age did not influence these behaviors.

The purpose of the study was to compare the effects of a feedback-controlled treadmill (FeedbackTM) to a traditional fixed-speed treadmill (FixedTM) on spatiotemporal gait means, variability, and dynamics. The study also examined inter-session reliability when using the FeedbackTM. Ten young adults walked on the FeedbackTM for a 5-minute familiarization followed by a 16-minute experimental trial. They returned within one week and completed a 5-minute familiarization followed by a 16-minute experimental trial each for FeedbackTM and FixedTM conditions. Mean walking speed and step time, length, width, and speed means and coefficient of variation were calculated from all experimental conditions. Step time, length, width, and speed gait dynamics were analyzed using detrended fluctuation analysis. Mean differences between experimental trials were determined using ANOVAs and reliability between FeedbackTM sessions was determined by intraclass correlation coefficient. No difference was found in mean walking speed nor spatiotemporal variables, with the exception of step width, between the experimental trials. All mean spatiotemporal variables demonstrated good to excellent reliability between sessions, while coefficient of variation was not reliable. Gait dynamics of step time, length, width, and speed were significantly more persistent during the FeedbackTM condition compared to FixedTM, especially step speed. However, gait dynamics demonstrated fair to poor reliability between FeedbackTM sessions. When walking on the FeedbackTM, users maintain a consistent set point, yet the gait dynamics around the mean are different when compared to walking on a FixedTM. In addition, spatiotemporal gait dynamics and variability may not be consistent across separate days when using the FeedbackTM.

Satellite-derived Normalized Difference Vegetation index (NDVI) data records offer important sources for long term correlation modelling over West Africa. In this study, we assessed long range correlations in half monthly NDVI records over West Africa from 1982 to 2011 using GIMMS NDVI. In our analysis, we assessed (a) the annual and seasonal trends obtained using Ordinary Linear Regression, (b) the detrended lag-1-autocorrelation C(1), (c) the Detrended Fluctuation Analysis (DFA) scaling Hurst exponent h and (d) the Multifractal (MF) characteristics of NDVI. Results show that there exist some patterns or trends in the records that persist over time. The value of C(1) for NDVI was obtained as 0.989 is significant at 95% confidence interval. Consequently, the scaling h values of the Hurst DFA showed that about 37.4, 20.5, 41.7 and 0.5% of the vegetated areas are anti-correlated (h

Stochastic fractals of the 1/f noise type are an important manifestation of the brain’s electrical activity and other real-world complex systems. Fractal complexity can be successfully estimated by methods such as the Higuchi method and detrended fluctuation analysis (DFA). In this study, we show that if, as with the EEG, the signal is a combination of fractal and oscillation, the estimates of fractal characteristics will be inaccurate. On our test data, DFA overestimated the fractal dimension, while the Higuchi method led to underestimation in the presence of high-amplitude, densely sampled oscillations.

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.

It has been known that short-time auditory stimulation can contribute to the improvement of the balancing ability of the human body. The present study aims to explore the effects of white Gaussian noise (WGN) of different intensities and frequencies on dynamic balance performance in healthy young adults. A total of 20 healthy young participants were asked to stand at a dynamic balance force platform, which swung along the x-axis with an amplitude of ± 4° and frequency of 1 Hz. Their center of pressure (COP) trajectories were recorded when they were stimulated by WGN of different intensities (block 1) and different frequencies (block 2). A traditional method and detrended fluctuation analysis (DFA) were used for data preprocessing. The authors found that only with 75–85 dB WGN, the COP parameters improved. WGN frequency did not affect the dynamic balance performance of all the participants. The DFA results indicated stimulation with 75 dB WGN enhanced the short-term index and reduced the crossover point. Stimulation with 500 Hz and 2500 Hz WGN significantly enhanced the short-term index. These results suggest that 75 dB WGN and 500 Hz and 2500 Hz WGN improved the participants’ dynamic balance performance. The results of this study indicate that a certain intensity of WGN is indispensable to achieve a remarkable improvement in dynamic balance. The DFA results suggest that WGN only affected the short-term persistence, indicating the potential of WGN being considered as an adjuvant therapy in low-speed rehabilitation training.

Considerable disagreement exists on the linearity of the development of standing balance in children. This study aimed to use different traditional and nonlinear methods to investigate age-related changes in standing balance in preschoolers. A sample of 118 preschoolers took part in this study. A force platform was used to record the center of pressure during standing balance over 15 s in three conditions: eyes open, eyes closed, and/or head extended backward. Detrended fluctuation analysis (DFA), recurrence quantification analysis (RQA), and traditional measures were used to evaluate standing balance. The main results are as follows: (1) Higher range and SD in the anterior-posterior (AP) direction were observed for 5-year-old than for 4-year-old children, while higher DFA coefficient (at shorter time scales) and higher determinism and laminarity in the AP direction were found for 5-year-old children compared to 3- and 4-year-old children; and (2) as sensory conditions became more challenging, all traditional measures increased and DFA coefficients (at shorter and longer time scales) decreased in the AP and mediolateral directions, while determinism and laminarity significantly declined in the AP direction. In conclusion, although increased postural sway, 5-year-old preschool children’s balance performance improved, and their control strategy changed significantly compared with the younger preschoolers. Sensory perturbation (eye closure and/or head extension) changed preschoolers’ balance performance and control strategy. Moreover, both traditional and nonlinear methods provided complementary information on the control of standing balance in preschoolers.

This paper performs the Multiracial Detrended Fluctuation Analysis (MFDFA) of fine resolution (0.25°x0.25°) daily gridded rainfall of Kerala Meteorological subdivision, India. First the MFDFA method is applied for long term rainfall datasets of different grid points for 1901-2013 period. The results show that rainfall time series of majority of the grid points (∼66 %) exhibits short term persistence with strong multifractal characteristics. Except at three locations, the asymmetry index of rainfall time series displays right skewed spectra indicating local high fluctuations and fine structured series. Further, MFDF analysis was performed after splitting the time series into two with respect to the global climatic shift of 1976/77. The results of pre and post climatic shift period shows that 34 grids (out of 65) exhibited an increase multifractal spectral width while 41 grids exhibited a increase in persistence after the shift. The study further showed that the persistence property changed from short to long term in 23 % grids, which also indicated a change in pattern of rainfall within the subdivision. The results showed that a strong correlation 0.994 exists between Hurst exponent and Holder exponent, reinforcing the universal association between the two multifractal exponents.

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&rsquo, 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.

The short-term scaling exponent alpha1 of detrended fluctuation analysis (DFA a1), a nonlinear index of heart rate variability (HRV) based on fractal correlation properties, has been shown to steadily change with increasing exercise intensity. To date, no study has specifically examined using the behavior of this index as a method for defining a low intensity exercise zone. The aim of this report is to compare both oxygen intake (VO2) and heart rate (HR) reached at the first ventilatory threshold (VT1), a well-established delimiter of low intensity exercise, to those derived from a predefined DFA a1 transitional value. Gas exchange and HRV data were obtained from 15 participants during an incremental treadmill run. Comparison of both VO2 and HR reached at VT1 defined by gas exchange (VT1 GAS) was made to those parameters derived from analysis of DFA a1 reaching a value of 0.75 (HRVT). Based on Bland Altman analysis, linear regression, intraclass correlation (ICC) and t testing, there was strong agreement between VT1 GAS and HRVT as measured by both HR and VO2. Mean VT1 GAS was reached at 39.8 ml/kg/min with a HR of 152 bpm compared to mean HRVT which was reached at 40.1 ml/kg/min with a HR of 154 bpm. Strong linear relationships were seen between test modalities, with Pearson’s r values of 0.99 (p < 0.001) and.97 (p < 0.001) for VO2 and HR comparisons, respectively. Intraclass correlation between VT1 GAS and HRVT was 0.99 for VO2 and 0.96 for HR. In addition, comparison of VT1 GAS and HRVT showed no differences by t testing, also supporting the method validity. In conclusion, it appears that reaching a DFA a1 value of 0.75 on an incremental treadmill test is closely associated with crossing the first ventilatory threshold. As training intensity below the first ventilatory threshold is felt to have great importance for endurance sport, utilization of DFA a1 activity may provide guidance for a valid low training zone.

Extended detrended fluctuation analysis (EDFA) is a recently proposed modification of the conventional method, which provides a characterization of complex time series with varying nonstationarity. It evaluates two scaling exponents for a better quantification of inhomogeneous datasets. Here, we study the effect of different types of nonstationarity on these exponents, including trend, switching between processes with distinct statistical properties and energy variability. Using the simulated signals, we show that the first two types of nonstationarity have the strongest effect for anticorrelated processes and complicate their diagnosis. Nonstationarity in energy is more crucial for time series with positive long-range correlations. Next, we apply EDFA to rat experiments to study the activation of brain lymphatic drainage during sleep. Our analysis reveals significant distinctions in EDFA’s measures between the background electrical activity of the brain and the stage of sleep. The latter offers an indirect way to identify and characterize the nightly activation of the drainage and clearance of brain tissue.

Wind farms can be regarded as complex systems that are, on the one hand, coupled to the nonlinear, stochastic characteristics of weather and, on the other hand, strongly influenced by supervisory control mechanisms. One crucial problem in this context today is the predictability of wind energy as an intermittent renewable resource with additional non-stationary nature. In this context, we analyze the power time series measured in an offshore wind farm for a total period of one year with a time resolution of 10 min. Applying detrended fluctuation analysis, we characterize the autocorrelation of power time series and find a Hurst exponent in the persistent regime with cross-over behavior. To enrich the modeling perspective of complex large wind energy systems, we develop a stochastic reduced-form model ofpower time series. The observed transitions between two dominating power generation phases are reflected by a bistable deterministic component, while correlated stochastic fluctuations account for the identified persistence. The model succeeds to qualitatively reproduce several empirical characteristics such as the autocorrelation function and the bimodal probability density function., Comment: 20 pages, 8 figures

Recent advancements in detrended fluctuation analysis (DFA) allow evaluating multifractal coefficients scale-by-scale, a promising approach for assessing the complexity of biomedical signals. The multifractality degree is typically quantified by the singularity spectrum width ( W SS ), a method that is critically unstable in multiscale applications. Thus, we aim to propose a robust multiscale index of multifractality, compare it with W SS and illustrate its performance on real biosignals. The proposed index is the cumulative function of squared increments between consecutive DFA coefficients at each scale n : α CF ( n ). We compared it with W SS calculated scale-by-scale considering monofractal/monoscale, monofractal/multiscale, multifractal/monoscale and multifractal/multiscale random processes. The two indices provided qualitatively similar descriptions of multifractality, but α CF ( n ) differentiated better the multifractal components from artefacts due to crossovers or detrending overfitting. Applied on 24 h heart rate recordings of 14 participants, the singularity spectrum failed to always satisfy the concavity requirement for providing meaningful W SS , while α CF ( n ) demonstrated a statistically significant heart rate multifractality at night in the scale ranges 16–100 and 256–680 s. Furthermore, α CF ( n ) did not reject the hypothesis of monofractality at daytime, coherently with previous reports of lower nonlinearity and monoscale multifractality during the day. Thus, α CF ( n ) appears a robust index of multiscale multifractality that is useful for quantifying complexity alterations of physiological series. This article is part of the theme issue ‘Advanced computation in cardiovascular physiology: new challenges and opportunities’.

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&rsquo, s tau is used as a ranking coefficient to quantify the trend in the scaling exponent. It was found that the Kendall&rsquo, s tau coefficient of the vibration scaling exponent could provide an early warning signal (EWS) for bearing failure.

This paper demonstrates the application of Multifractal Detrended Fluctuation Analysis (MFDFA) to portray the multifractal behavior of Standardized Precipitation Index (SPI) time series of four meteorological subdivisions from Western India, namely, Western Rajasthan, Saurashtra–Kutch-Diu (SK), Marathwada and Kerala at different temporal scales. The Hurst exponents obtained showed good agreement with those computed by Dispersional Analysis (DA) method and it varied from 0.53 to 0.75 in SPI series of different temporal scales of 3, 6 and 12 months. It is noticed that SPI series of different subdivisions are characterized by long-term persistence, which indicated the universal multifractal properties of drought within the subdivisions of the western India irrespective of their climatic conditions. Further, this study quantified the degree of multifractality (DoMF) from the plot of mass exponent and multifractal spectra of different time series, which showed that degree of multifractality shows an increase with accumulation time scale and the strongest DoMF is noted in the data of WR subdivision.

Recent developments of detrended fluctuation analysis (DFA) provide multifractal/multiscale (MFMS) descriptions of the heart rate self-similarity, a promising approach to cardiovascular complexity. However, it is unclear whether the MFMS DFA may also describe the nonlinear components of heart rate variability. Our aim is to define MFMS DFA indices for quantifying the short-term and long-term degree of the heart-rate nonlinearity and to apply these indices to detect possible sex-related differences.We recorded the inter-beat-interval (IBI) series in 42 male and in 42 female healthy participants sitting at rest for about 2 hours. For each series j, we generated 100 phase-randomized surrogate series. We applied the MFMS DFA to estimate the self-similarity coefficients α over scales τ between 8 and 512 s and moment orders q between –5 and +5, obtaining coefficients for the original series, αO,j (q, τ), and for each surrogate, αi,j (q, τ) with 1≤i≤100. We first evaluated πj(q, τ), percentile of αi,j (q, τ) distribution in which was αO,j (q, τ). Then we calculated the percentages of scales where πj(q, τ) was 1 (q)) and for 16≤τ≤512 s (long-term nonlinearity index NL 2 (q)). We found that NL 1 (q) was generally greater than 50% at all q≥0 but q=2 (i.e., moment order of the monofractal DFA), while at q

Aim Autonomic nervous system (ANS) activities during different types of stress could affect the electrocardiogram (ECG) signal. This study aimed to recognize the types of stress by using different ECG signals in order to prevent its actual physiological effects on the heart signal. Method The ECG signal recorded by portable wrist bracelets from 20 students in during seven phases which incorporated three different types of stress and four relaxation phases. After different forms of windowing the signal, we used linear and non-linear features such as detrended fluctuation analysis (DFA), Poincare plot, approximate and sample entropy, correlation dimension, and recurrence plot to extract various features of the heart rate variability (HRV). Then, different classifiers were used to identify the types of stress. Results The results showed a decrease in NN50, RMSSD, pNN50, and recurrence plot features, and an increase in the DFA method during stress stages, which show the effect of stress on heart rate. Also, by using the convolutional neural network (CNN), an average classification rate of 98 % was obtained in association with cognitive stress and that of 94.5 % in association with emotional stress. Conclusion This paper showed that features extracted from HRV can detect the stress and non-stress stages with high significance. Also, the accuracy of this paper proved that the proposed method is successful in preventing the dangerous effects of different types of stress on the heart (Tab. 5, Fig. 6, Ref. 34). Text in PDF www.elis.sk Keywords: stress, heart rate variability, non-linear features, convolutional neural network, classification.