Effect of trends on detrended fluctuation analysis

Potential influence of geomagnetic activity on blood pressure statistical fluctuations at mid-magnetic latitudes

BACKGROUND

Solar activity and the consequent geomagnetic activity (GMA) profoundly influence human biological rhythms and cardiovascular system functions. Although the response of blood pressure (BP) to GMA has attracted considerable attention, it is unclear whether the GMA can have an influence alone and how it occurs.

METHODS

In this six-year time series analysis, we collated over 500,000 BP measurements from two representative cities (Qingdao and Weihai) at mid-magnetic latitudes in China. Using various statistical methods, we analyzed the correlation between BP and the GMA (represented by Ap index) and their quasi-periodic fluctuations. Additionally, we conducted a comparative analysis of the influence of other environmental factors (air temperature and PM2.5) on BP.

RESULTS

The statistical BP level fluctuations correlate with the GMA. Both BP and the GMA index exhibit similar annual bimodal patterns and multiple periodicities, including 12-month and 6-month cycles, and an intermittent 3-month cycle. In contrast, other known environmental factors influencing BP such as air temperature and PM2.5 do not exhibit similar periodicities, particularly they lack 3-month cycles. In years with higher GMA levels, the BP shows stronger correlations with the Ap index and responds on a shorter timescale. Additionally, BP in females appears to be more strongly correlated with GMA.

CONCLUSIONS

Our findings highlight potential risks to individuals with hypertension with elevated GMA levels, deepen our understanding of GMA's role in human health, and offer insights for healthcare policymakers on the clinical significance of the geomagnetic environment.

Multi-Scale Analysis of Knee Joint Acoustic Signals for Cartilage Degeneration Assessment

This study focuses on the diagnostic analysis of cartilage damage in the knee joint based on acoustic signals generated by the joint. The research utilizes a combination of advanced signal processing techniques, specifically empirical mode decomposition (EEMD) and detrended fluctuation analysis (DFA), alongside convolutional neural networks (CNNs) for classification and detection tasks. Acoustic signals, often reflecting the mechanical behavior of the joint during movement, serve as a non-invasive diagnostic tool for assessing the cartilage condition. EEMD is applied to decompose the signals into intrinsic mode functions (IMFs), which are then analyzed using DFA to quantify the scaling properties and detect irregularities indicative of cartilage damage. The separation of individual frequency components allows for multi-scale analysis of the signals, with each of the functions resulting from the analysis reflecting local variations in the amplitude and frequency over time and allowing for effective removal of noise present in the signal. The CNN model is trained on features extracted from these signals to accurately classify different stages of cartilage degeneration. The proposed method demonstrates the potential for early detection of knee joint pathology, providing a valuable tool for preventive healthcare and reducing the need for invasive diagnostic procedures. The results suggest that the combination of EEMD-DFA for feature extraction and CNN for classification offers a promising approach for the non-invasive assessment of cartilage damage.

Fractal Analysis of Electrodermal Activity for Emotion Recognition: A Novel Approach Using Detrended Fluctuation Analysis and Wavelet Entropy

The field of emotion recognition from physiological signals is a growing area of research with significant implications for both mental health monitoring and human-computer interaction. This study introduces a novel approach to detecting emotional states based on fractal analysis of electrodermal activity (EDA) signals. We employed detrended fluctuation analysis (DFA), Hurst exponent estimation, and wavelet entropy calculation to extract fractal features from EDA signals obtained from the CASE dataset, which contains physiological recordings and continuous emotion annotations from 30 participants. The analysis revealed significant differences in fractal features across five emotional states (neutral, amused, bored, relaxed, and scared), particularly those derived from wavelet entropy. A cross-correlation analysis showed robust correlations between fractal features and both the arousal and valence dimensions of emotion, challenging the conventional view of EDA as a predominantly arousal-indicating measure. The application of machine learning for emotion classification using fractal features achieved a leave-one-subject-out accuracy of 84.3% and an F1 score of 0.802, surpassing the performance of previous methods on the same dataset. This study demonstrates the potential of fractal analysis in capturing the intricate, multi-scale dynamics of EDA signals for emotion recognition, opening new avenues for advancing emotion-aware systems and affective computing applications.

Tapping strength variability in sensorimotor experiments on rhythmic tapping

We report psychophysical experiments and time series analyses to investigate sensorimotor tapping strength fluctuations in human periodic tapping with and without a metronome. The power spectral density of tapping strength fluctuations typically decays in an inverse power law (1/fβ-noise) associated with long-range correlations, i.e., with a slow power-law decay of tapping strength autocorrelations and scale-free behavior. The power-law exponents β are scattered around β=1 ranging from 0.67 to 1.8. A log-linear representation of the power spectral densities reveals rhythmic peaks at frequencies f=0.25 (and f=0.5) and a tendency to slightly accentuate every fourth (and second) stroke when subjects try to synchronize their tapping with a metronome.

Role of the Locus Coeruleus Arousal Promoting Neurons in Maintaining Brain Criticality across the Sleep-Wake Cycle

Sleep control depends on a delicate interplay among brain regions. This generates a complex temporal architecture with numerous sleep-stage transitions and intermittent fluctuations to micro-states and brief arousals. These temporal dynamics exhibit hallmarks of criticality, suggesting that tuning to criticality is essential for spontaneous sleep-stage and arousal transitions. However, how the brain maintains criticality remains not understood. Here, we investigate θ- and δ-burst dynamics during the sleep-wake cycle of rats (Sprague-Dawley, adult male) with lesion in the wake-promoting locus coeruleus (LC). We show that, in control rats, θ- and δ-bursts exhibit power-law (θ-bursts, active phase) and exponential-like (δ-bursts, quiescent phase) duration distributions, as well as power-law long-range temporal correlations (LRTCs)-typical of non-equilibrium systems self-organizing at criticality. Furthermore, consecutive θ- and δ-bursts durations are characterized by anti-correlated coupling, indicating a new class of self-organized criticality that emerges from underlying feedback between neuronal populations and brain areas involved in generating arousals and sleep states. In contrast, we uncover that LC lesion leads to alteration of θ- and δ-burst critical features, with change in duration distributions and correlation properties, and increase in θ-δ coupling. Notably, these LC-lesion effects are opposite to those observed for lesions in the sleep-promoting ventrolateral preoptic (VLPO) nucleus. Our findings indicate that critical dynamics of θ- and δ-bursts arise from a balanced interplay of LC and VLPO, which maintains brain tuning to criticality across the sleep-wake cycle-a non-equilibrium behavior in sleep micro-architecture at short timescales that coexists with large-scale sleep-wake homeostasis.

Relationship of Cycling Power and Non-Linear Heart Rate Variability from Everyday Workout Data: Potential for Intensity Zone Estimation and Monitoring

The short-term scaling exponent alpha1 of detrended fluctuation analysis (DFA-a1) of heart rate variability (HRV) has been shown to be a sensitive marker for assessing global organismic demands. The wide dynamic range within the exercise intensity spectrum and the relationship to established physiologic threshold boundaries potentially allow in-field use and also open opportunities to provide real-time feedback. The present study expands the idea of using everyday workout data from the AI Endurance app to obtain the relationship between cycling power and DFA-a1. Collected data were imported between September 2021 and August 2023 with an initial pool of 3123 workouts across 21 male users. The aim of this analysis was to further apply a new method of implementing workout group data considering representative values of DFA-a1 segmentation compared to single workout data and including all data points to enhance the validity of the internal-to-external load relationship. The present data demonstrate a universal relationship between cycling power and DFA-a1 from everyday workout data that potentially allows accessible and regular tracking of intensity zone demarcation information. The analysis highlights the superior efficacy of the representative-based approach of included data in most cases. Validation data of the performance level and the up-to-date relationship are still pending.

Measures of fragmentation of rest activity patterns: mathematical properties and interpretability based on accelerometer real life data

Accelerometers, devices that measure body movements, have become valuable tools for studying the fragmentation of rest-activity patterns, a core circadian rhythm dimension, using metrics such as inter-daily stability (IS), intradaily variability (IV), transition probability (TP), and self-similarity parameter (named α ). However, their use remains mainly empirical. Therefore, we investigated the mathematical properties and interpretability of rest-activity fragmentation metrics by providing mathematical proofs for the ranges of IS and IV, proposing maximum likelihood and Bayesian estimators for TP, introducing the activity balance index (ABI) metric, a transformation of α , and describing distributions of these metrics in real-life setting. Analysis of accelerometer data from 2,859 individuals (age=60-83 years, 21.1% women) from the Whitehall II cohort (UK) shows modest correlations between the metrics, except for ABI and α . Sociodemographic (age, sex, education, employment status) and clinical (body mass index (BMI), and number of morbidities) factors were associated with these metrics, with differences observed according to metrics. For example, a difference of 5 units in BMI was associated with all metrics (differences ranging between -0.261 (95% CI -0.302, -0.220) to 0.228 (0.18, 0.268) for standardised TP rest to activity during the awake period and TP activity to rest during the awake period, respectively). These results reinforce the value of these rest-activity fragmentation metrics in epidemiological and clinical studies to examine their role for health. This paper expands on a set of methods that have previously demonstrated empirical value, improves the theoretical foundation for these methods, and evaluates their empirical use in a large dataset.

Multifractal perturbations to multiplicative cascades promote multifractal nonlinearity with asymmetric spectra

Biological and psychological processes have been conceptualized as emerging from intricate multiplicative interactions among component processes across various spatial and temporal scales. Among the statistical models employed to approximate these intricate nonlinear interactions across scales, one prominent framework is that of cascades. Despite decades of empirical work using multifractal formalisms, several fundamental questions persist concerning the proper interpretations of multifractal evidence of nonlinear cross-scale interactivity. Does multifractal spectrum width depend on multiplicative interactions, constituent noise processes participating in those interactions, or both? We conducted numerical simulations of cascade time series featuring component noise processes characterizing a range of nonlinear temporal correlations: nonlinearly multifractal, linearly multifractal (obtained via the iterative amplitude adjusted wavelet transform of nonlinearly multifractal), phase-randomized linearity (obtained via the iterative amplitude adjustment Fourier transform of nonlinearly multifractal), and phase and amplitude randomized (obtained via shuffling of nonlinearly multifractal). Our findings show that the multiplicative interactions coordinate with the nonlinear temporal correlations of noise components to dictate emergent multifractal properties. Multiplicative cascades with stronger nonlinear temporal correlations make multifractal spectra more asymmetric with wider left sides. However, when considering multifractal spectral differences between the original and surrogate time series, even multiplicative cascades produce multifractality greater than in surrogate time series, even with linearized multifractal noise components. In contrast, additivity among component processes leads to a linear outcome. These findings provide a robust framework for generating multifractal expectations for biological and psychological models in which cascade dynamics flow from one part of an organism to another.

Impact of Correlated Disorder on Surface Superconductivity: Revealing the Robustness of the Surface Ordering Effect

Surface superconductivity, wherein electron pairing occurs at material surfaces or interfaces, has attracted a remarkable amount of attention since its discovery. Recent theoretical predictions have unveiled increased critical temperatures, especially at the surfaces of certain compounds and/or structures. The notion of "surface ordering" has been advanced to elucidate this phenomenon. Employing the framework of self-consistent Bogoliubov-de Gennes equations and a model incorporating correlated disorder, our study demonstrates the persistence of the surface ordering effect in the presence of weak to moderate bulk disorder. Intriguingly, our findings indicate that under moderate disorder conditions the surface critical temperature can be further increased, depending on the intensity and correlation of the disorder.

Exploring disorder correlations in superconducting systems: spectroscopic insights and matrix element effects

Understanding the intricate interplay between disorder and superconductivity has become a key area of research in condensed matter physics, with profound implications for materials science. Recent studies have shown that spatial correlations of disorder potential can improve superconductivity, prompting a re-evaluation of some theoretical models. This paper explores the influence of disorder correlations on the fundamental properties of superconducting systems, going beyond the traditional assumption of spatially uncorrelated disorder. In particular, we investigate the influence of disorder correlations on key spectroscopic superconductor properties, including the density of states, as well as on the matrix elements of the superconducting coupling constant and their impact on the localization length. Our findings offer valuable insights into the role of disorder correlations in shaping the behavior of superconducting materials.

Multifractal Multiscale Analysis of Human Movements during Cognitive Tasks

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.

General spectral characteristics of human activity and its inherent scale-free fluctuations

The scale-free nature of daily human activity has been observed in different aspects; however, the description of its spectral characteristics is incomplete. General findings are complicated by the fact that-although actigraphy is commonly used in many research areas-the activity calculation methods are not standardized; therefore, activity signals can be different. The presence of 1/f noise in activity or acceleration signals was mostly analysed for short time windows, and the complete spectral characteristic has only been examined in the case of certain types of them. To explore the general spectral nature of human activity in greater detail, we have performed Power Spectral Density (PSD) based examination and Detrended Fluctuation Analysis (DFA) on several-day-long, triaxial actigraphic acceleration signals of 42 healthy, free-living individuals. We generated different types of activity signals from these, using different acceleration preprocessing techniques and activity metrics. We revealed that the spectra of different types of activity signals generally follow a universal characteristic including 1/f noise over frequencies above the circadian rhythmicity. Moreover, we discovered that the PSD of the raw acceleration signal has the same characteristic. Our findings prove that the spectral scale-free nature is generally inherent to the motor activity of healthy, free-living humans, and is not limited to any particular activity calculation method.

Scale-Fractal Detrended Fluctuation Analysis for Fault Diagnosis of a Centrifugal Pump and a Reciprocating Compressor

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.

On Multiscaling of Parkinsonian Rest Tremor Signals and Their Classification

Self-similar stochastic processes and broad probability distributions are ubiquitous in nature and in many man-made systems. The brain is a particularly interesting example of (natural) complex system where those features play a pivotal role. In fact, the controversial yet experimentally validated "criticality hypothesis" explaining the functioning of the brain implies the presence of scaling laws for correlations. Recently, we have analyzed a collection of rest tremor velocity signals recorded from patients affected by Parkinson's disease, with the aim of determining and hence exploiting the presence of scaling laws. Our results show that multiple scaling laws are required in order to describe the dynamics of such signals, stressing the complexity of the underlying generating mechanism. We successively extracted numeric features by using the multifractal detrended fluctuation analysis procedure. We found that such features can be effective for discriminating classes of signals recorded in different experimental conditions. Notably, we show that the use of medication (L-DOPA) can be recognized with high accuracy.

Altered Motor Activity Patterns within 10-Minute Timescale Predict Incident Clinical Alzheimer's Disease

Background

Fractal motor activity regulation (FMAR), characterized by self-similar temporal patterns in motor activity across timescales, is robust in healthy young humans but degrades with aging and in Alzheimer's disease (AD).

Objective

To determine the timescales where alterations of FMAR can best predict the clinical onset of AD.

Methods

FMAR was assessed from actigraphy at baseline in 1,077 participants who had annual follow-up clinical assessments for up to 15 years. Survival analysis combined with deep learning (DeepSurv) was used to examine how baseline FMAR at different timescales from 3 minutes up to 6 hours contributed differently to the risk for incident clinical AD.

Results

Clinical AD occurred in 270 participants during the follow-up. DeepSurv identified three potential regions of timescales in which FMAR alterations were significantly linked to the risk for clinical AD: <10, 20-40, and 100-200 minutes. Confirmed by the Cox and random survival forest models, the effect of FMAR alterations in the timescale of <10 minutes was the strongest, after adjusting for covariates.

Conclusions

Subtle changes in motor activity fluctuations predicted the clinical onset of AD, with the strongest association observed in activity fluctuations at timescales <10 minutes. These findings suggest that short actigraphy recordings may be used to assess the risk of AD.

Circadian disturbances and frailty risk in older adults

Frailty is characterized by diminished resilience to stressor events. It is associated with adverse future health outcomes and impedes healthy aging. The circadian system orchestrates ~24-h rhythms in bodily functions in synchrony with the day-night cycle, and disturbed circadian regulation plays an important role in many age-related health consequences. We investigated prospective associations of circadian disturbances with incident frailty in over 1000 older adults who had been followed annually for up to 16 years. We found that decreased rhythm strength, reduced stability, or increased variation were associated with a higher risk of incident frailty and faster progress of frailty over time. Perturbed circadian rest-activity rhythms may be an early sign or risk factor for frailty in older adults.

Differential Fractal and Circadian Patterns in Motor Activity in Spontaneously Hypertensive Rats at the Stage of Prehypertension

One possible pathological mechanism underlying hypertension and its related health consequences is dysfunction of the circadian system-a network of coupled circadian clocks that generates and orchestrates rhythms of ≈24 h in behavior and physiology. To better understand the role of circadian function during the development of hypertension, circadian regulation of motor activity is investigated in spontaneously hypertensive rats (SHRs) before the onset of hypertension and in their age-matched controls-Wistar Kyoto rats (WKYs). Two complementary properties in locomotor activity fluctuations are examined to assessthe multiscale regulatory function of the circadian control network: 1) rhythmicity at ≈24 h and 2) fractal patterns-similar temporal correlation at different time scales (≈0.5-8 h). Compared to WKYs, SHRs have more stable and less fragmented circadian activity rhythms but the changes in the rhythms (e.g., period and amplitude) from constant dark to light conditions are reduced or opposite. SHRs also have altered fractal activity patterns, displaying activity fluctuations with excessive regularity at small timescales that are linked to rigid physiological states. These different rhythmicity/fractal patterns and their different responses to light in SHRs indicate that an altered circadian function may be involved in the development of hypertension.

Long-term memory in Staphylococcus aureus α-hemolysin ion channel kinetics

The kinetics of an ion channel are classically understood as a random process. However, studies have shown that in complex ion channels, formed by multiple subunits, this process can be deterministic, presenting long-term memory. Staphylococcus aureus α-hemolysin (α-HL) is a toxin that acts as the major factor in Staphylococcus aureus virulence. α-HL is a water-soluble protein capable of forming ion channels into lipid bilayers, by insertion of an amphipathic  β-barrel. Here, the α-HL was used as an experimental model to study memory in ion channel kinetics. We applied the approximate entropy (ApEn) approach to analyze randomness and the Detrended Fluctuation Analysis (DFA) to investigate the existence of long memory in α-HL channel kinetics. Single-channel currents were measured through experiments with α-HL channels incorporated in planar lipid bilayers. All experiments were carried out under the following conditions: 1 M NaCl solution, pH 4.5; transmembrane potential of + 40 mV and temperature 25 ± 1 °C. Single-channel currents were recorded in real-time in the memory of a microcomputer coupled to an A/D converter and a patch-clamp amplifier. The conductance value of the α-HL channels was 0.82 ± 0.0025 nS (n = 128). The DFA analysis showed that the kinetics of α-HL channels presents long-term memory ([Formula: see text] = 0.63 ± 0.04). The ApEn outcomes showed low complexity to dwell times when open (ApEno = 0.5514 ± 0.28) and closed (ApEnc = 0.1145 ± 0.08), corroborating the results of the DFA method.

Fractal complexity alternations in paroxysmal atrial fibrillation patients with and without recurrence after pulmonary vein isolation

BACKGROUND

Pulmonary vein isolation (PVI) is a cornerstone therapy for paroxysmal atrial fibrillation (PAF). The variations in nonlinear heart rate variability (HRV) between patients with and without recurrences remain unclear. We aimed to characterize the nonlinear HRV before and after PVI in patients with and without recurrence.

METHODS

Twenty-five drug-refractory PAF patients (56.0 ± 9.1 years old, 20 males) who received PVI were enrolled. Holter electrocardiography were performed before, 1-3, and 6-12 months after PVI. After 8.2 ± 2.5 months of follow-ups after PVI, patients were divided into two groups: the recurrence (n = 8) and non-recurrence (n = 17) groups. Linear and nonlinear HRV variables were analyzed, including the Poincaré Plot analysis and the Detrended Fluctuation Analysis (DFA).

RESULTS

The non-recurrence group, but not the recurrence group, had decreased high-frequency component (HF), the root mean square of successive RR interval differences (RMSSD), and the Poincaré Plot index SD1 1-3 months after PVI and increased DFAslope2 6-12 months after PVI. The non-recurrence group's LF/HF ratio and DFAslope1 decreased significantly 1-3 and 6-12 months after PVI, respectively, whereas there was no significant change in the recurrence group after PVI.

CONCLUSIONS

Significantly reduced vagal tone 1-3 months after PVI, increased long-term fractal complexity 6-12 months after PVI, and decreased sympathetic tone as well as short-term fractal complexity 1-3 and 6-12 months after PVI led to a better AF-free survival after PVI. These findings suggest that neuromodulation and heart rate dynamics play crucial roles in AF recurrence following PVI.

Wavelet gated multiformer for groundwater time series forecasting

Developing accurate models for groundwater control is paramount for planning and managing life-sustaining resources (water) from aquifer reservoirs. Significant progress has been made toward designing and employing deep-forecasting models to tackle the challenge of multivariate time-series forecasting. However, most models were initially taught only to optimize natural language processing and computer vision tasks. We propose the Wavelet Gated Multiformer, which combines the strength of a vanilla Transformer with the Wavelet Crossformer that employs inner wavelet cross-correlation blocks. The self-attention mechanism (Transformer) computes the relationship between inner time-series points, while the cross-correlation finds trending periodicity patterns. The multi-headed encoder is channeled through a mixing gate (linear combination) of sub-encoders (Transformer and Wavelet Crossformer) that output trending signatures to the decoder. This process improved the model's predictive capabilities, reducing Mean Absolute Error by 31.26 % compared to the second-best performing transformer-like models evaluated. We have also used the Multifractal Detrended Cross-Correlation Heatmaps (MF-DCCHM) to extract cyclical trends from pairs of stations across multifractal regimes by denoising the pair of signals with Daubechies wavelets. Our dataset was obtained from a network of eight wells for groundwater monitoring in Brazilian aquifers, six rainfall stations, eleven river flow stations, and three weather stations with atmospheric pressure, temperature, and humidity sensors.

Visibility Graph Analysis of Heartbeat Time Series: Comparison of Young vs. Old, Healthy vs. Diseased, Rest vs. Exercise, and Sedentary vs. Active

Using the visibility graph algorithm (VGA), a complex network can be associated with a time series, such that the properties of the time series can be obtained by studying those of the network. Any value of the time series becomes a node of the network, and the number of other nodes that it is connected to can be quantified. The degree of connectivity of a node is positively correlated with its magnitude. The slope of the regression line is denoted by k-M, and, in this work, this parameter was calculated for the cardiac interbeat time series of different contrasting groups, namely: young vs. elderly; healthy subjects vs. patients with congestive heart failure (CHF); young subjects and adults at rest vs. exercising young subjects and adults; and, finally, sedentary young subjects and adults vs. active young subjects and adults. In addition, other network parameters, including the average degree and the average path length, of these time series networks were also analyzed. Significant differences were observed in the k-M parameter, average degree, and average path length for all analyzed groups. This methodology based on the analysis of the three mentioned parameters of complex networks has the advantage that such parameters are very easy to calculate, and it is useful to classify heartbeat time series of subjects with CHF vs. healthy subjects, and also for young vs. elderly subjects and sedentary vs. active subjects.

The Hurst Exponent as an Indicator to Anticipate Agricultural Commodity Prices

Anticipating and understanding fluctuations in the agri-food market is very important in order to implement policies that can assure fair prices and food availability. In this paper, we contribute to the understanding of this market by exploring its efficiency and whether the local Hurst exponent can help to anticipate its trend or not. We have analyzed the time series of the price for different agri-commodities and classified each day into persistent, anti-persistent, or white-noise. Next, we have studied the probability and speed to mean reversion for several rolling windows. We found that in general mean reversion is more probable and occurs faster during anti-persistent periods. In contrast, for most of the rolling windows we could not find a significant effect of persistence in mean reversion. Hence, we conclude that the Hurst exponent can help to anticipate the future trend and range of the expected prices in this market.

Circadian disturbances and frailty risk in older adults: a prospective cohort study

Frailty is characterized by diminished resilience to stressor events. It associates with adverse future health outcomes and impedes healthy aging. The circadian system orchestrates a ~24-h rhythm in bodily functions in synchrony with the day-night cycle, and disturbed circadian regulation plays an important role in many age-related health consequences. We investigated prospective associations of circadian disturbances with incident frailty in over 1,000 older adults who had been followed annually for up to 16 years. We found that decreased rhythm strength, reduced stability, or increased variation, were associated with a higher risk of incident frailty, and faster worsening of the overall frailty symptoms over time. Perturbed circadian rest-activity rhythms may be an early sign or risk factor for frailty in older adults.

Characterization of Anesthesia in Rats from EEG in Terms of Long-Range Correlations

Long-range correlations are often used as diagnostic markers in physiological research. Due to the limitations of conventional techniques, their characterizations are typically carried out with alternative approaches, such as the detrended fluctuation analysis (DFA). In our previous works, we found EEG-related markers of the blood-brain barrier (BBB), which limits the penetration of major drugs into the brain. However, anesthetics can penetrate the BBB, affecting its function in a dose-related manner. Here, we study two types of anesthesia widely used in experiments on animals, including zoletil/xylazine and isoflurane in optimal doses not associated with changes in the BBB. Based on DFA, we reveal informative characteristics of the electrical activity of the brain during such doses that are important for controlling the depth of anesthesia in long-term experiments using magnetic resonance imaging, multiphoton microscopy, etc., which are crucial for the interpretation of experimental results. These findings provide an important informative platform for the enhancement and refinement of surgery, since the EEG-based DFA analysis of BBB can easily be used during surgery as a tool for characterizing normal BBB functions under anesthesia.

Extended Detrended Fluctuation Analysis of Coarse-Grained Time Series

A coarse-graining procedure, which involves averaging time series in non-overlapping windows followed by processing of the obtained multiple data sets, is the initial step in the multiscale entropy computation method. In this paper, we discuss how this procedure can be applied with other methods of time series analysis. Based on extended detrended fluctuation analysis (EDFA), we compare signal processing results for data sets with and without coarse-graining. Using the simulated data provided by the interacting nephrons model, we show how this procedure increases, up to 48%, the distinctions between local scaling exponents quantifying synchronous and asynchronous chaotic oscillations. Based on the experimental data of electrocorticograms (ECoG) of mice, an improvement in differences in local scaling exponents up to 41% and Student's t-values up to 34% was revealed.

Multi-fractal detrended cross-correlation heatmaps for time series analysis

Complex systems in biology, climatology, medicine, and economy hold emergent properties such as non-linearity, adaptation, and self-organization. These emergent attributes can derive from large-scale relationships, connections, and interactive behavior despite not being apparent from their isolated components. It is possible to better comprehend complex systems by analyzing cross-correlations between time series. However, the accumulation of non-linear processes induces multiscale structures, therefore, a spectrum of power-law exponents (the fractal dimension) and distinct cyclical patterns. We propose the Multifractal detrended cross-correlation heatmaps (MF-DCCHM) based on the DCCA cross-correlation coefficients with sliding boxes, a systematic approach capable of mapping the relationships between fluctuations of signals on different scales and regimes. The MF-DCCHM uses the integrated series of magnitudes, sliding boxes with sizes of up to 5% of the entire series, and an average of DCCA coefficients on top of the heatmaps for the local analysis. The heatmaps have shown the same cyclical frequencies from the spectral analysis across different multifractal regimes. Our dataset is composed of sales and inventory from the Brazilian automotive sector and macroeconomic descriptors, namely the Gross Domestic Product (GDP) per capita, Nominal Exchange Rate (NER), and the Nominal Interest Rate (NIR) from the Central Bank of Brazil. Our results indicate cross-correlated patterns that can be directly compared with the power-law spectra for multiple regimes. We have also identified cyclical patterns of high intensities that coincide with the Brazilian presidential elections. The MF-DCCHM uncovers non-explicit cyclic patterns, quantifies the relations of two non-stationary signals (noise effect removed), and has outstanding potential for mapping cross-regime patterns in multiple domains.

Homology groups of embedded fractional Brownian motion

A well-known class of nonstationary self-similar time series is the fractional Brownian motion (fBm) considered to model ubiquitous stochastic processes in nature. Due to noise and trends superimposed on data and even sample size and irregularity impacts, the well-known computational algorithm to compute the Hurst exponent (H) has encountered superior results. Motivated by this discrepancy, we examine the homology groups of high-dimensional point cloud data (PCD), a subset of the unit D-dimensional cube, constructed from synthetic fBm data as a pipeline to compute the H exponent. We compute topological measures for embedded PCD as a function of the associated Hurst exponent for different embedding dimensions, time delays, and amount of irregularity existing in the dataset in various scales. Our results show that for a regular synthetic fBm, the higher value of the embedding dimension leads to increasing the H dependency of topological measures based on zeroth and first homology groups. To achieve a reliable classification of fBm, we should consider the small value of time delay irrespective of the irregularity presented in the data. More interestingly, the value of the scale for which the PCD to be path connected and the postloopless regime scale are more robust concerning irregularity for distinguishing the fBm signal. Such robustness becomes less for the higher value of the embedding dimension. Finally, the associated Hurst exponents for our topological feature vector for the S&P500 were computed, and the results are consistent with the detrended fluctuation analysis method.

Lockdown during COVID-19 pandemic: A case study from Indian cities shows insignificant effects on persistent property of urban air quality

The influence of reduction in emissions on the inherent temporal characteristics of PM_2.5 and NO₂ concentration time series in six urban cities of India is assessed by computing the Hurst exponent using Detrended Fluctuation Analysis (DFA) during the lockdown period (March 24–April 20, 2020) and the corresponding period during the previous two years (i.e., 2018 and 2019). The analysis suggests the anticipated impact of confinement on the PM_2.5 and NO₂ concentration in urban cities, causing low concentrations. It is observed that the original PM_2.5 and NO₂ concentration time series is persistent but filtering the time series by fitting the autoregressive process of order 1 on the actual time series and subtracting it changes the persistence property significantly. It indicates the presence of linear correlations in the PM_2.5 and NO₂ concentrations. Hurst exponent of the PM_2.5 and NO₂ concentration during the lockdown period and previous two years shows that the inherent temporal characteristics of the short-term air pollutant concentrations (APCs) time series do not change even after withholding the emissions. The meteorological variations also do not change over the three time periods. The finding helps in developing the prediction models for future policy decisions to improve urban air quality across cities.

Multifractal Conductance Fluctuations in High-Mobility Graphene in the Integer Quantum Hall Regime

We present the first experimental evidence for the multifractality of a transport property at a topological phase transition. In particular, we show that conductance fluctuations display multifractality at the integer quantum Hall plateau-to-plateau transitions in high-mobility mesoscopic graphene devices. The multifractality gets rapidly suppressed as the chemical potential moves away from these critical points. Our combination of experimental study and multifractal analysis provides a novel method for probing the criticality of wave functions at phase transitions in mesoscopic systems, and quantum criticality in several condensed-matter systems.

Associations between heart rate asymmetry expression and asymmetric detrended fluctuation analysis results

The relation between recently established asymmetry in Asymmetric Detrended Fluctuation Analysis (ADFA) and Heart Rate Asymmetry is studied. It is found that the ADFA asymmetric exponents are related both to the overall variability and to its asymmetric components at all studied time scales. We find that the asymmetry in scaling exponents, i.e., [Formula: see text] is associated with both variance-based and runs-based types of asymmetry. This observation suggests that the physiological mechanisms of both types are similar, even though their origins and mathematical methods are very different. The graphical abstract demonstrates strong, nonlinear association between the expression of Heart Rate Asymmetry measured using relative descriptors and the Asymmetric Detrended Fluctuation Analysis results. It is clear that there is a strong relation between the two theoretically disparate approaches to signal analysis. The technique to demonstrate the association is loess fit.

Metastability, fractal scaling, and synergistic information processing: What phase relationships reveal about intrinsic brain activity

Dynamic functional connectivity (dFC) in resting-state fMRI holds promise to deliver candidate biomarkers for clinical applications. However, the reliability and interpretability of dFC metrics remain contested. Despite a myriad of methodologies and resulting measures, few studies have combined metrics derived from different conceptualizations of brain functioning within the same analysis - perhaps missing an opportunity for improved interpretability. Using a complexity-science approach, we assessed the reliability and interrelationships of a battery of phase-based dFC metrics including tools originating from dynamical systems, stochastic processes, and information dynamics approaches. Our analysis revealed novel relationships between these metrics, which allowed us to build a predictive model for integrated information using metrics from dynamical systems and information theory. Furthermore, global metastability - a metric reflecting simultaneous tendencies for coupling and decoupling - was found to be the most representative and stable metric in brain parcellations that included cerebellar regions. Additionally, spatiotemporal patterns of phase-locking were found to change in a slow, non-random, continuous manner over time. Taken together, our findings show that the majority of characteristics of resting-state fMRI dynamics reflect an interrelated dynamical and informational complexity profile, which is unique to each acquisition. This finding challenges the interpretation of results from cross-sectional designs for brain neuromarker discovery, suggesting that individual life-trajectories may be more informative than sample means.

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

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

Characterizing stroke-induced changes in the variability of lower limb kinematics using multifractal detrended fluctuation analysis

Movement variability reflects the adaptation of the neuromuscular control system to internal or external perturbations, but its relationship to stroke-induced injury is still unclear. In this study, the multifractal detrended fluctuation analysis was used to explore the stroke-induced changes in movement variability by analyzing the joint angles in a treadmill-walking task. Eight healthy subjects and ten patients after stroke participated in the experiment, performing a treadmill-walking task at a comfortable speed. The kinematics data of the lower limbs were collected by the motion-capture system, and two indicators, the degree of multifractality (α) and degree of correlation [h(2)], were used to investigate the mechanisms underlying neuromuscular control. The results showed that the knee and ankle joint angles were multifractal and persistent at various scales, and there was a significant difference in the degree of multifractality and the degree of correlation at the knee and ankle joint angles among the three groups, with the values being ranked in the following order: healthy subjects < non-paretic limb < paretic limb. These observations highlighted increased movement variability and multifractal strength in patients after stroke due to neuromotor defects. This study provided evidence that multifractal detrended analysis of the angles of the knee and ankle joints is useful to investigate the changes in movement variability and multifractal after stroke. Further research is needed to verify and promote the clinical applications.

Multifractal Characterization and Modeling of Blood Pressure Signals

In this paper, a multi-fractal analysis on a diastolic blood pressure signal is conducted. The signal is measured in a time span of circa one day through the multifractal detrended fluctuation analysis framework. The analysis is performed on asymptotic timescales where complex regulating mechanisms play a fundamental role in the blood pressure stability. Given a suitable frequency range and after removing non-stationarities, the blood pressure signal shows interesting scaling properties and a pronounced multifractality imputed to long-range correlations. Finally, a binomial multiplicative model is investigated showing how the analyzed signal can be described by a concise multifractal model with only two parameters.

Relative Prices of Ethanol-Gasoline in the Major Brazilian Capitals: An Analysis to Support Public Policies

The use of biomass as an energy source has advanced in recent decades, given the scientific evidence that it is a solution to the environmental problems faced globally. In this context, biofuels derived from biomass have a prominent role. Among the countries where this alternative is the most promising, Brazil stands out, just behind the USA. It is, therefore, necessary to assess whether such a replacement is economically viable. For such an assessment, the behavior of the relative price of bioethanol/gasoline is crucial. In the present work, the degree of temporal persistence of relative prices, considering the existence of shocks to which they are exposed, is evaluated, considering 15 important Brazilian capitals, via the detrended fluctuation analysis (DFA). The degree of correlation is also evaluated through the detrended cross-correlation analysis (DCCA) between fuel prices in São Paulo, the capital of the most populous state and main producer of bioethanol, with the capitals of the 14 states selected for the analysis. The period of analysis takes place between 2004 and 2020. The use of DCCA with sliding windows was recently proposed and we also evaluate DFA dynamically in this way, and this, together with an extended sample in the context of Brazilian fuel prices, represents the main innovations of the present work. We found that the degree of persistence varies significantly depending on the capitals analyzed, which means that price variations are localized and demand regional stimulus policies. Furthermore, it was found that the correlation with São Paulo is less intense in the most geographically distant capitals. Such evidence is important and complementary to infer how integrated the national bioethanol market is, in order to support public policies aimed at its consolidation.

Ensemble FARIMA Prediction with Stable Infinite Variance Innovations for Supermarket Energy Consumption

This paper concerns a fractional modeling and prediction method directly oriented toward an industrial time series with obvious non-Gaussian features. The hidden long-range dependence and the multifractal property are extracted to determine the fractional order. A fractional autoregressive integrated moving average model (FARIMA) is then proposed considering innovations with stable infinite variance. The existence and convergence of the model solutions are discussed in depth. Ensemble learning with an autoregressive moving average model (ARMA) is used to further improve upon accuracy and generalization. The proposed method is used to predict the energy consumption in a real cooling system, and superior prediction results are obtained.

Fractal Dimension Analysis of Earth Magnetic Field during 26 August 2018 Geomagnetic Storm

We analyse the fractal nature of geomagnetic field northward and eastward horizontal components with 1 min resolution measured by the four stations Belsk, Hel, Sodankylä and Hornsund during the period of 22 August–1 September, when the 26 August 2018 geomagnetic storm appeared. To reveal and to quantitatively describe the fractal scaling of the considered data, three selected methods, structure function scaling, Higuchi, and detrended fluctuation analysis are applied. The obtained results show temporal variation of the fractal dimension of geomagnetic field components, revealing differences between their irregularity (complexity). The values of fractal dimension seem to be sensitive to the physical conditions connected with the interplanetary shock, the coronal mass ejection, the corotating interaction region, and the high-speed stream passage during the storm development. Especially, just after interplanetary shock occurrence, a decrease in the fractal dimension for all stations is observed, not straightforwardly visible in the geomagnetic field components data.

On the Autocorrelation Function of 1/f Noises

The outputs of many real-world complex dynamical systems are time series characterized by power-law correlations and fractal properties. The first proposed model for such time series comprised fractional Gaussian noise (fGn), defined by an autocorrelation function C(k) with asymptotic power-law behavior, and a complicated power spectrum S(f) with power-law behavior in the small frequency region linked to the power-law behavior of C(k). This connection suggested the use of simpler models for power-law correlated time series: time series with power spectra of the form S(f)∼1/fβ, i.e., with power-law behavior in the entire frequency range and not only near f=0 as fGn. This type of time series, known as 1/fβ noises or simply 1/f noises, can be simulated using the Fourier filtering method and has become a standard model for power-law correlated time series with a wide range of applications. However, despite the simplicity of the power spectrum of 1/fβ noises and of the known relationship between the power-law exponents of S(f) and C(k), to our knowledge, an explicit expression of C(k) for 1/fβ noises has not been previously published. In this work, we provide an analytical derivation of C(k) for 1/fβ noises, and we show the validity of our results by comparing them with the numerical results obtained from synthetically generated 1/fβ time series. We also present two applications of our results: First, we compare the autocorrelation functions of fGn and 1/fβ noises that, despite exhibiting similar power-law behavior, present some clear differences for anticorrelated cases. Secondly, we obtain the exact analytical expression of the Fluctuation Analysis algorithm when applied to 1/fβ noises.

Long-Range Correlations and Natural Time Series Analyses from Acoustic Emission Signals

This work focuses on analyzing acoustic emission (AE) signals as a means to predict failure in structures. There are two main approaches that are considered: (i) long-range correlation analysis using both the Hurst (H) and the detrended fluctuation analysis (DFA) exponents, and (ii) natural time domain (NT) analysis. These methodologies are applied to the data that were collected from two application examples: a glass fiber-reinforced polymeric plate and a spaghetti bridge model, where both structures were subjected to increasing loads until collapse. A traditional (AE) signal analysis was also performed to reference the study of the other methods. The results indicate that the proposed methods yield reliable indication of failure in the studied structures.

On the Validity of Detrended Fluctuation Analysis at Short Scales

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.

Bitcoin's price efficiency and safe haven properties during the COVID-19 pandemic: A comparison

As the COVID-19 outbreak became a global pandemic, traditional financial market indicators were significantly affected. We examine the price efficiency and net cross-correlations among Bitcoin, gold, a US dollar index, and the Morgan Stanley Capital International World Index (MSCI World) during the four months after the World Health Organization officially designated COVID-19 as a global pandemic. Using intraday data, we find that Bitcoin prices were more efficient than the US dollar and MSCI World indices. Using a detrended partial-cross-correlation analysis, our results show that net cross-correlations vary across time scales. Our results suggest that when the time scale is greater than two months, gold can be considered as a safe haven for investors holding the MSCI World and US dollar indices and when the time scale exceeds three months, Bitcoin can be considered a safe haven for the MSCI World index.

Memory in Ion Channel Kinetics

Ion channels are transport proteins present in the lipid bilayers of biological membranes. They are involved in many physiological processes, such as the generation of nerve impulses, hormonal secretion, and heartbeat. Conformational changes in the ion channel-forming protein allow the opening or closing of pores to control the ionic flux through the cell membranes. The opening and closing of the ion channel have been classically treated as a random kinetic process, known as a Markov process. Here the time the channel remains in a given state is assumed to be independent of the condition it had in the previous state. More recently, however, several studies have shown that this process is not random but a deterministic one, where both the open and closed dwell-times and the ionic current flowing through the channel are history-dependent. This property is called long memory or long-range correlation. However, there is still much controversy regarding how this memory originates, which region of the channel is responsible for this property, and which models could best reproduce the memory effect. In this article, we provide a review of what is, where it is, its possible origin, and the mathematical methods used to analyze the long-term memory present in the kinetic process of ion channels.

Broadband Dynamics Rather than Frequency-Specific Rhythms Underlie Prediction Error in the Primate Auditory Cortex

Detection of statistical irregularities, measured as a prediction error response, is fundamental to the perceptual monitoring of the environment. We studied whether prediction error response is associated with neural oscillations or asynchronous broadband activity. Electrocorticography was conducted in three male monkeys, who passively listened to the auditory roving oddball stimuli. Local field potentials (LFPs) recorded over the auditory cortex underwent spectral principal component analysis, which decoupled broadband and rhythmic components of the LFP signal. We found that the broadband component captured the prediction error response, whereas none of the rhythmic components were associated with statistical irregularities of sounds. The broadband component displayed more stochastic, asymmetrical multifractal properties than the rhythmic components, which revealed more self-similar dynamics. We thus conclude that the prediction error response is captured by neuronal populations generating asynchronous broadband activity, defined by irregular dynamic states, which, unlike oscillatory rhythms, appear to enable the neural representation of auditory prediction error response.SIGNIFICANCE STATEMENT This study aimed to examine the contribution of oscillatory and asynchronous components of auditory local field potentials in the generation of prediction error responses to sensory irregularities, as this has not been directly addressed in the previous studies. Here, we show that mismatch negativity-an auditory prediction error response-is driven by the asynchronous broadband component of potentials recorded in the auditory cortex. This finding highlights the importance of nonoscillatory neural processes in the predictive monitoring of the environment. At a more general level, the study demonstrates that stochastic neural processes, which are often disregarded as neural noise, do have a functional role in the processing of sensory information.

Long-Range Dependence and Multifractality of Ship Flow Sequences in Container Ports: A Comparison of Shanghai, Singapore, and Rotterdam

The prediction of ship traffic flow is an important fundamental preparation for layout and design of ports as well as management of ship navigation. However, until now, the temporal characteristics and accurate prediction of ship flow sequence in port are rarely studied. Therefore, in this study, we investigated the presence of long-range dependence in container ship flow sequences using the Multifractal Detrended Fluctuation Analysis (MF-DFA). We considered three representative container ports in the world—including Shanghai, Singapore, and Rotterdam container ports—as the study sample, from 1 January 2013 to 31 December 2017. Empirical results suggested that the ship flow sequences are deviated from normal distribution, and the sequences with different time scales exhibited varying degrees of long-range dependence. Furthermore, the ship flow sequences possessed a multifractal nature, where the larger the time scale of ship flow time series, the stronger the multifractal characteristics are. The weekly ship flow sequence in the port of Singapore owned the highest degree of multifractality. Furthermore, the multifractality presented in the ship flow sequences of container ports are due to the correlation properties as well as the probability density function of the ship flow sequences. The study outlines the importance of adopting these features for an accurate modeling and prediction for maritime ship flow series.

Inferring long memory using extreme events

Many natural and physical processes display long memory and extreme events. In these systems, the measured time series is invariably contaminated by noise and/or missing data. As the extreme events display a large deviation from the mean behavior, noise and/or missing data do not affect the extreme events as much as it affects the typical values. Since the extreme events also carry the information about correlations in the full-time series, we can use them to infer the correlation properties of the latter. In this work, we construct three modified time series using only the extreme events from a given time series. We show that the correlations in the original time series and in the modified time series are related, as measured by the exponent obtained from the detrended fluctuation analysis technique. Hence, the correlation exponents for a long memory time series can be inferred from its extreme events alone. We demonstrate this approach for several empirical time series.

Persistent homology of fractional Gaussian noise

In this paper, we employ the persistent homology (PH) technique to examine the topological properties of fractional Gaussian noise (fGn). We develop the weighted natural visibility graph algorithm, and the associated simplicial complexes through the filtration process are quantified by PH. The evolution of the homology group dimension represented by Betti numbers demonstrates a strong dependency on the Hurst exponent (H). The coefficients of the birth and death curves of the k-dimensional topological holes (k-holes) at a given threshold depend on H which is almost not affected by finite sample size. We show that the distribution function of a lifetime for k-holes decays exponentially and the corresponding slope is an increasing function versus H and, more interestingly, the sample size effect completely disappears in this quantity. The persistence entropy logarithmically grows with the size of the visibility graph of a system with almost H-dependent prefactors. On the contrary, the local statistical features are not able to determine the corresponding Hurst exponent of fGn data, while the moments of eigenvalue distribution (M_{n}) for n≥1 reveal a dependency on H, containing the sample size effect. Finally, the PH shows the correlated behavior of electroencephalography for both healthy and schizophrenic samples.

The futility of long-term predictions in bipolar disorder: mood fluctuations are the result of deterministic chaotic processes

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 < 0.005]. A post-hoc Tukey test showed that Lyapunov exponent in HC was significantly higher than both the BD (p = 0.003) and FDR groups (p = 0.03). Similarly, in the energy series, the largest Lyapunov exponent was found in HC (1.85), compared to BD (1.76) and FDR (1.67) [F (2, 87) = 11.02; p < 0.005]. There were no significant differences between groups for the detrended fluctuation analysis or fractal dimension.

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.