Effect of trends on detrended fluctuation analysis

Superstatistical model of bacterial DNA architecture

Understanding the physical principles that govern the complex DNA structural organization as well as its mechanical and thermodynamical properties is essential for the advancement in both life sciences and genetic engineering. Recently we have discovered that the complex DNA organization is explicitly reflected in the arrangement of nucleotides depicted by the universal power law tailed internucleotide interval distribution that is valid for complete genomes of various prokaryotic and eukaryotic organisms. Here we suggest a superstatistical model that represents a long DNA molecule by a series of consecutive ~150 bp DNA segments with the alternation of the local nucleotide composition between segments exhibiting long-range correlations. We show that the superstatistical model and the corresponding DNA generation algorithm explicitly reproduce the laws governing the empirical nucleotide arrangement properties of the DNA sequences for various global GC contents and optimal living temperatures. Finally, we discuss the relevance of our model in terms of the DNA mechanical properties. As an outlook, we focus on finding the DNA sequences that encode a given protein while simultaneously reproducing the nucleotide arrangement laws observed from empirical genomes, that may be of interest in the optimization of genetic engineering of long DNA molecules.

Evolution of scaling behaviors embedded in sentence series from A Story of the Stone

The novel entitled A Story of the Stone provides us precise details of life and social structure of the 18th century China. Its writing lasted a long duration of about 10 years, in which the author’s habit may change significantly. It had been published anonymously up to the beginning of the 20th century, which left a mystery of the author’s attribution. In the present work we focus our attention on scaling behavior embedded in the sentence series from this novel, hope to find how the ideas are organized from single sentences to the whole text. Especially we are interested in the evolution of scale invariance to monitor the changes of the author’s language habit and to find some clues on the author’s attribution. The sentence series are separated into a total of 69 non-overlapping segments with a length of 500 sentences each. The correlation dependent balanced estimation of diffusion entropy (cBEDE) is employed to evaluate the scaling behaviors embedded in the short segments. It is found that the total, the part attributed currently to Xueqin Cao (X-part), and the other part attributed to E Gao (E-part), display scale invariance in a large scale up to 10³ sentences, while their scaling exponents are almost identical. All the segments behave scale invariant in considerable wide scales, most of which reach one third of the length. In the curve of scaling exponent versus segment number, the X-part has rich patterns with averagely larger values, while the E-part has a U-shape with a significant low bottom. This finding is a new clue to support the attribution of the E-part to E Gao.

A Follow-Up Study on the Renal Protective Efficacy of Telbivudine for Hepatitis B Virus-Infected Taiwanese Patients After Living Donor Liver Transplant

OBJECTIVES

Preservation of renal function is an important issue after living donor liver transplant. We aimed to examine the renal protective efficacy of telbivudine in hepatitis B virus-infected patients after living donor liver transplant.

MATERIALS AND METHODS

In this retrospective study, we compared 18 patients who received telbivudine 600 mg once per day and 23 patients who received entecavir 1 mg once per day after living donor liver transplant. Clinical data were obtained through chart review and included Model for End-Stage Liver Disease score and pre- and postoperative aspartate aminotransferase, alanine aminotransferase, and creatinine levels and estimated glomerular filtration rate.

RESULTS

Posttransplant estimated glomerular filtration rates and creatinine levels were calculated, and improvement of renal function was found in the group of patients who received telbivudine. Significant improvements were shown in estimated glomerular filtration rates started after 9 months of administration and creatinine levels after 12 months compared with patients who received entecavir.

CONCLUSIONS

In our study, long-term telbivudine therapy is associated with a sustained improvement of renal function in patients with hepatitis B virus infection after living donor liver transplant.

Multifractals embedded in short time series: An unbiased estimation of probability moment

An exact estimation of probability moments is the base for several essential concepts, such as the multifractals, the Tsallis entropy, and the transfer entropy. By means of approximation theory we propose a new method called factorial-moment-based estimation of probability moments. Theoretical prediction and computational results show that it can provide us an unbiased estimation of the probability moments of continuous order. Calculations on probability redistribution model verify that it can extract exactly multifractal behaviors from several hundred recordings. Its powerfulness in monitoring evolution of scaling behaviors is exemplified by two empirical cases, i.e., the gait time series for fast, normal, and slow trials of a healthy volunteer, and the closing price series for Shanghai stock market. By using short time series with several hundred lengths, a comparison with the well-established tools displays significant advantages of its performance over the other methods. The factorial-moment-based estimation can evaluate correctly the scaling behaviors in a scale range about three generations wider than the multifractal detrended fluctuation analysis and the basic estimation. The estimation of partition function given by the wavelet transform modulus maxima has unacceptable fluctuations. Besides the scaling invariance focused in the present paper, the proposed factorial moment of continuous order can find its various uses, such as finding nonextensive behaviors of a complex system and reconstructing the causality relationship network between elements of a complex system.

Detrended fluctuation analysis of the Ornstein-Uhlenbeck process: Stationarity versus nonstationarity

The stationary/nonstationary regimes of time series generated by the discrete version of the Ornstein-Uhlenbeck equation are studied by using the detrended fluctuation analysis. Our findings point out to the prevalence of the drift parameter in determining the crossover time between the nonstationary and stationary regimes. The fluctuation functions coincide in the nonstationary regime for a constant diffusion parameter, and in the stationary regime for a constant ratio between the drift and diffusion stochastic forces. In the generalized Ornstein-Uhlenbeck equations, the Hurst exponent H influences the crossover time that increases with the decrease of H.

Order Statistics Concordance Coefficient With Applications to Multichannel Biosignal Analysis

In this paper, we propose a novel concordance coefficient, called order statistics concordance coefficient (OSCOC), to quantify the association among multichannel biosignals. To uncover its properties, we compare OSCOC with three other similar indexes, i.e., average Pearson's product moment correlation coefficient (APPMCC), Kendall's concordance coefficients (KCC), and average Kendall's tau (AKT), under a multivariate normal model (MNM), linear model (LM), and nonlinear model. To further demonstrate its usefulness, we present an example on atrial arrhythmia analysis based on real-world multichannel cardiac signals. Theoretical derivations as well as numerical results suggest that 1) under MNM and LM, OSCOC performs equally well with APPMCC, and outperforms the other two methods, 2) in nonlinear case, OSCOC even has better performance than KCC and AKT, which are well known to be robust under increasing nonlinear transformations, and 3) OSCOC performs the best in the case study of arrhythmia analysis in terms of the volume under the surface.

Detrended fluctuation analysis and the difference between external drifts and intrinsic diffusionlike nonstationarity

Detrended fluctuation analysis (DFA) has been shown to be an effective method to study long-range correlation of nonstationary series. In principle, DFA considers F_{DFA}^{2}(s), the mean of variance around the local polynomial fit in segments with length s, and then estimates the scaling exponent α_{DFA} in F_{DFA}(s)∼s^{α_{DFA}} with varying s. Usually, the methodological studies of DFA focus on its effect on removing the drift due to the external trends. Only few paid attention to nonstationary series without drift, such as fractional Brownian motion (FBM) with nonstationarity due to its intrinsic dynamics. Both of these types of nonstationarity can shift the local mean by drift or diffusion and can be treated as the additive nonstationarity eliminable by the additive decomposition. In this study, we limit our discussion to such additive nonstationarity and furthermore specifically distinguish these two types of nonstationarity, namely the drift and the intrinsic diffusionlike nonstationarity. To understand how DFA works for the intrinsic diffusionlike nonstationarity, we take FBM as the example and seek for the answers to two fundamental questions: (1) what DFA removes from FBM; and (2) why DFA can handle such intrinsic diffusionlike nonstationarity, in contrast to methods only applicable to stationary series such as the fluctuation analysis. A crucial condition, i.e., statistical equivalence among all segments, is proposed and checked in the fluctuation analysis and DFA. As shown, the crucial condition is a natural requirement for the connection between DFA and autocorrelation function. With the help of the crucial condition, our study analytically and numerically demonstrates for the intrinsic diffusionlike nonstationary series that (1) rather than the nonstationarity as thought, DFA actually removes the difference among all segments; (2) the detrended segments fulfill the crucial condition so that the average over segments becomes equivalent to the ensemble average over realizations. These answers are also true for series with a drift. Thus, we provide a unified perspective to refresh the understanding of how DFA works on nonstationarity and underpin the mathematical ground of DFA.

Network dynamics: quantitative analysis of complex behavior in metabolism, organelles, and cells, from experiments to models and back

Advancing from two core traits of biological systems: multilevel network organization and nonlinearity, we review a host of novel and readily available techniques to explore and analyze their complex dynamic behavior within the framework of experimental-computational synergy. In the context of concrete biological examples, analytical methods such as wavelet, power spectra, and metabolomics-fluxomics analyses, are presented, discussed, and their strengths and limitations highlighted. Further shown is how time series from stationary and nonstationary biological variables and signals, such as membrane potential, high-throughput metabolomics, O₂ and CO₂ levels, bird locomotion, at the molecular, (sub)cellular, tissue, and whole organ and animal levels, can reveal important information on the properties of the underlying biological networks. Systems biology-inspired computational methods start to pave the way for addressing the integrated functional dynamics of metabolic, organelle and organ networks. As our capacity to unravel the control and regulatory properties of these networks and their dynamics under normal or pathological conditions broadens, so is our ability to address endogenous rhythms and clocks to improve health-span in human aging, and to manage complex metabolic disorders, neurodegeneration, and cancer. WIREs Syst Biol Med 2017, 9:e1352. doi: 10.1002/wsbm.1352 For further resources related to this article, please visit the WIREs website.

Time and frequency domain characteristics of detrending-operation-based scaling analysis: Exact DFA and DMA frequency responses

We develop a general framework to study the time and frequency domain characteristics of detrending-operation-based scaling analysis methods, such as detrended fluctuation analysis (DFA) and detrending moving average (DMA) analysis. In this framework, using either the time or frequency domain approach, the frequency responses of detrending operations are calculated analytically. Although the frequency domain approach based on conventional linear analysis techniques is only applicable to linear detrending operations, the time domain approach presented here is applicable to both linear and nonlinear detrending operations. Furthermore, using the relationship between the time and frequency domain representations of the frequency responses, the frequency domain characteristics of nonlinear detrending operations can be obtained. Based on the calculated frequency responses, it is possible to establish a direct connection between the root-mean-square deviation of the detrending-operation-based scaling analysis and the power spectrum for linear stochastic processes. Here, by applying our methods to DFA and DMA, including higher-order cases, exact frequency responses are calculated. In addition, we analytically investigate the cutoff frequencies of DFA and DMA detrending operations and show that these frequencies are not optimally adjusted to coincide with the corresponding time scale.

Probing the Fractal Pattern of Heartbeats in Drosophila Pupae by Visible Optical Recording System

Judiciously tuning heart rates is critical for regular cardiovascular function. The fractal pattern of heartbeats — a multiscale regulation in instantaneous fluctuations — is well known for vertebrates. The most primitive heart system of the Drosophila provides a useful model to understand the evolutional origin of such a fractal pattern as well as the alterations of fractal pattern during diseased statuses. We developed a non-invasive visible optical heart rate recording system especially suitable for long-term recording by using principal component analysis (PCA) instead of fluorescence recording system to avoid the confounding effect from intense light irradiation. To deplete intracellular Ca²⁺ levels, the expression of sarco-endoplasmic reticulum Ca²⁺-ATPase (SERCA) was tissue-specifically knocked down. The SERCA group shows longer heart beat intervals (Mean ± SD: 1009.7 ± 151.6 ms) as compared to the control group (545.5 ± 45.4 ms, p < 0.001). The multiscale correlation of SERCA group (scaling exponent: 0.77 ± 0.07), on the other hand, is weaker than that of the control Drosophila (scaling exponent: 0.85 ± 0.03) (p = 0.016).

Self-Similar Random Process and Chaotic Behavior In Serrated Flow of High Entropy Alloys

The statistical and dynamic analyses of the serrated-flow behavior in the nanoindentation of a high-entropy alloy, Al0.5CoCrCuFeNi, at various holding times and temperatures, are performed to reveal the hidden order associated with the seemingly-irregular intermittent flow. Two distinct types of dynamics are identified in the high-entropy alloy, which are based on the chaotic time-series, approximate entropy, fractal dimension, and Hurst exponent. The dynamic plastic behavior at both room temperature and 200 °C exhibits a positive Lyapunov exponent, suggesting that the underlying dynamics is chaotic. The fractal dimension of the indentation depth increases with the increase of temperature, and there is an inflection at the holding time of 10 s at the same temperature. A large fractal dimension suggests the concurrent nucleation of a large number of slip bands. In particular, for the indentation with the holding time of 10 s at room temperature, the slip process evolves as a self-similar random process with a weak negative correlation similar to a random walk.

The role of environmental constraints in walking: Effects of steering and sharp turns on gait dynamics

Stride durations in gait exhibit long-range correlation (LRC) which tends to disappear with certain movement disorders. The loss of LRC has been hypothesized to result from a reduction of functional degrees of freedom of the neuromuscular apparatus. A consequence of this theory is that environmental constraints such as the ones induced during constant steering may also reduce LRC. Furthermore, obstacles may perturb control of the gait cycle and also reduce LRC. To test these predictions, seven healthy participants walked freely overground in three conditions: unconstrained, constrained (constant steering), and perturbed (frequent 90° turns). Both steering and sharp turning reduced LRC with the latter having a stronger effect. Competing theories explain LRC in gait by positing fractal CPGs or a biomechanical process of kinetic energy reuse. Mediation analysis showed that the effect of the experimental manipulation in the current experiment depends partly on a reduction in walking speed. This supports the biomechanical theory. We also found that the local Hurst exponent did not reflect the frequent changes of heading direction. This suggests that the recovery from the sharp turn perturbation, a kind of relaxation time, takes longer than the four to seven meters between successive turns in the present study.

Progression of Dementia Assessed by Temporal Correlations of Physical Activity: Results From a 3.5-Year, Longitudinal Randomized Controlled Trial

Cross-sectional studies show that activity fluctuations in healthy young adults possess robust temporal correlations that become altered with aging, and in dementia and depression. This study was designed to test whether or not within-subject changes of activity correlations (i) track the clinical progression of dementia, (ii) reflect the alterations of depression symptoms in patients with dementia, and (iii) can be manipulated by clinical interventions aimed at stabilizing circadian rhythmicity and improving sleep in dementia, namely timed bright light therapy and melatonin supplementation. We examined 144 patients with dementia (70-96 years old) who were assigned to daily treatment with bright light, bedtime melatonin, both or placebos only in a 3.5-year double-blinded randomized clinical trial. We found that activity correlations at temporal scales <~2 hours significantly decreased over time and that light treatment attenuated the decrease by ~73%. Moreover, the decrease of temporal activity correlations positively correlated with the degrees of cognitive decline and worsening of mood though the associations were relatively weak. These results suggest a mechanistic link between multiscale activity regulation and circadian/sleep function in dementia patients. Whether temporal activity patterns allow unobtrusive, long-term monitoring of dementia progression and mood changes is worth further investigation.

Detrending moving average algorithm: Frequency response and scaling performances

The Detrending Moving Average (DMA) algorithm has been widely used in its several variants for characterizing long-range correlations of random signals and sets (one-dimensional sequences or high-dimensional arrays) over either time or space. In this paper, mainly based on analytical arguments, the scaling performances of the centered DMA, including higher-order ones, are investigated by means of a continuous time approximation and a frequency response approach. Our results are also confirmed by numerical tests. The study is carried out for higher-order DMA operating with moving average polynomials of different degree. In particular, detrending power degree, frequency response, asymptotic scaling, upper limit of the detectable scaling exponent, and finite scale range behavior will be discussed.

Irregular Liesegang-type patterns in gas phase revisited. I. Experimental setup, data processing, and test of the spacing law

Since the early work on Liesegang rings in gels, they have been a reference point for the study of pattern formation in chemical physics. Here we present a variant of the Liesegang experiment in gas phase, where ammonia and hydrochloric acid react within a glass tube producing a precipitate, which deposits along the tube wall producing a spatial pattern. With this apparently simple experiment a wide range of rich phenomenon can be observed due to the presence of convective flows and irregular dynamics reminiscent of turbulent behavior, for which precise measurements are scarce. In this first part of our work, we describe in detail the experimental setup, the method of data acquisition, the image processing, and the procedure used to obtain an intensity profile, which is representative of the amount of precipitate deposited at the tube walls. Special attention is devoted to the techniques rendering a data series reliable for statistical studies and model building, which may contribute to a characterization and understanding of the pattern formation phenomenon under consideration. As a first step in this direction, based on our data, we are able to show that the observed band pattern follows, with slight deviations, the spacing law encountered in common Liesegang rings, despite that the experimental conditions are very different. A further statistical correlation analysis of the data constitutes Paper II of this research.

Fast algorithm for scaling analysis with higher-order detrending moving average method

Among scaling analysis methods based on the root-mean-square deviation from the estimated trend, it has been demonstrated that centered detrending moving average (DMA) analysis with a simple moving average has good performance when characterizing long-range correlation or fractal scaling behavior. Furthermore, higher-order DMA has also been proposed; it is shown to have better detrending capabilities, removing higher-order polynomial trends than original DMA. However, a straightforward implementation of higher-order DMA requires a very high computational cost, which would prevent practical use of this method. To solve this issue, in this study, we introduce a fast algorithm for higher-order DMA, which consists of two techniques: (1) parallel translation of moving averaging windows by a fixed interval; (2) recurrence formulas for the calculation of summations. Our algorithm can significantly reduce computational cost. Monte Carlo experiments show that the computational time of our algorithm is approximately proportional to the data length, although that of the conventional algorithm is proportional to the square of the data length. The efficiency of our algorithm is also shown by a systematic study of the performance of higher-order DMA, such as the range of detectable scaling exponents and detrending capability for removing polynomial trends. In addition, through the analysis of heart-rate variability time series, we discuss possible applications of higher-order DMA.

Long-Range Correlations of Global Sea Surface Temperature

Scaling behaviors of the global monthly sea surface temperature (SST) derived from 1870–2009 average monthly data sets of Hadley Centre Sea Ice and SST (HadISST) are investigated employing detrended fluctuation analysis (DFA). The global SST fluctuations are found to be strong positively long-range correlated at all pertinent time-intervals. The value of scaling exponent is larger in the tropics than those in the intermediate latitudes of the northern and southern hemispheres. DFA leads to the scaling exponent α = 0.87 over the globe (60°S~60°N), northern hemisphere (0°N~60°N), and southern hemisphere (0°S~60°S), α = 0.84 over the intermediate latitude of southern hemisphere (30°S~60°S), α = 0.81 over the intermediate latitude of northern hemisphere (30°N~60°N) and α = 0.90 over the tropics 30°S~30°N [fluctuation F(s) ~ s^α], which the fluctuations of monthly SST anomaly display long-term correlated behaviors. Furthermore, the larger the standard deviation is, the smaller long-range correlations (LRCs) of SST in the corresponding regions, especially in three distinct upwelling areas. After the standard deviation is taken into account, an index χ = α * σ is introduced to obtain the spatial distributions of χ. There exists an obvious change of global SST in central east and northern Pacific and the northwest Atlantic. This may be as a clue on predictability of climate and ocean variabilities.

Magnitude and sign of long-range correlated time series: Decomposition and surrogate signal generation

We systematically study the scaling properties of the magnitude and sign of the fluctuations in correlated time series, which is a simple and useful approach to distinguish between systems with different dynamical properties but the same linear correlations. First, we decompose artificial long-range power-law linearly correlated time series into magnitude and sign series derived from the consecutive increments in the original series, and we study their correlation properties. We find analytical expressions for the correlation exponent of the sign series as a function of the exponent of the original series. Such expressions are necessary for modeling surrogate time series with desired scaling properties. Next, we study linear and nonlinear correlation properties of series composed as products of independent magnitude and sign series. These surrogate series can be considered as a zero-order approximation to the analysis of the coupling of magnitude and sign in real data, a problem still open in many fields. We find analytical results for the scaling behavior of the composed series as a function of the correlation exponents of the magnitude and sign series used in the composition, and we determine the ranges of magnitude and sign correlation exponents leading to either single scaling or to crossover behaviors. Finally, we obtain how the linear and nonlinear properties of the composed series depend on the correlation exponents of their magnitude and sign series. Based on this information we propose a method to generate surrogate series with controlled correlation exponent and multifractal spectrum.

Detrended Fluctuation Analysis of Heart Rate Dynamics Is an Important Prognostic Factor in Patients with End-Stage Renal Disease Receiving Peritoneal Dialysis

BACKGROUND AND OBJECTIVES

Patients with severe kidney function impairment often have autonomic dysfunction, which could be evaluated noninvasively by heart rate variability (HRV) analysis. Nonlinear HRV parameters such as detrended fluctuation analysis (DFA) has been demonstrated to be an important outcome predictor in patients with cardiovascular diseases. Whether cardiac autonomic dysfunction measured by DFA is also a useful prognostic factor in patients with end-stage renal disease (ESRD) receiving peritoneal dialysis (PD) remains unclear. The purpose of the present study was designed to test the hypothesis.

MATERIALS AND METHODS

Patients with ESRD receiving PD were included for the study. Twenty-four hour Holter monitor was obtained from each patient together with other important traditional prognostic makers such as underlying diseases, left ventricular ejection fraction (LVEF) and serum biochemistry profiles. Short-term (DFAα1) and long-term (DFAα2) DFA as well as other linear HRV parameters were calculated.

RESULTS

A total of 132 patients (62 men, 72 women) with a mean age of 53.7±12.5 years were recruited from July 2007 to March 2009. During a median follow-up period of around 34 months, eight cardiac and six non-cardiac deaths were observed. Competing risk analysis demonstrated that decreased DFAα1 was a strong prognostic predictor for increased cardiac and total mortality. ROC analysis showed that the AUC of DFAα1 (<0.95) to predict mortality was 0.761 (95% confidence interval (CI). = 0.617-0.905). DFAα1≧ 0.95 was associated with lower cardiac mortality (Hazard ratio (HR) 0.062, 95% CI = 0.007-0.571, P = 0.014) and total mortality (HR = 0.109, 95% CI = 0.033-0.362, P = 0.0003).

CONCLUSION

Cardiac autonomic dysfunction evaluated by DFAα1 is an independent predictor for cardiac and total mortality in patients with ESRD receiving PD.

The effect of footwear and footfall pattern on running stride interval long-range correlations and distributional variability

The presence of long-range correlations (self-similarity) in the stride-to-stride fluctuations in running stride interval has been used as an indicator of a healthy adaptable system. Changes to footfall patterns when running with minimalist shoes could cause a less adaptable running gait. The purpose of this study was to investigate stride interval variability and the degree of self-similarity of stride interval in runners wearing minimalist and conventional footwear. Twenty-six trained habitual rearfoot footfall runners, unaccustomed to running in minimalist footwear, performed 6-min sub-maximal treadmill running bouts at 11, 13 and 15 km·h(-1) in minimalist and conventional shoes. Force sensitive resistors were placed in the shoes to quantify stride interval (time between successive foot contacts). Footfall position, stride interval mean and coefficient of variation (CV), were used to assess performance as a function of shoe type. Long-range correlations of stride interval were assessed using detrended fluctuation analysis (α). Mean stride interval was 1-1.3% shorter (P=0.02) and 27% of runners adopted a midfoot footfall (MFF) in the minimalist shoe. There was a significant shoe effect on α and shoe*speed*footfall interaction effect on CV (P<0.05). Runners that adopted a MFF in minimalist shoes, displayed reduced long-range correlations (P<0.05) and CV (P<0.06) in their running stride interval at the 15 km·h(-1) speed. The reduced variability and self-similarity observed for runners that changed to a MFF in the minimalist shoe may be suggestive of a system that is less flexible and more prone to injury.

Model selection for identifying power-law scaling

Long-range temporal and spatial correlations have been reported in a remarkable number of studies. In particular power-law scaling in neural activity raised considerable interest. We here provide a straightforward algorithm not only to quantify power-law scaling but to test it against alternatives using (Bayesian) model comparison. Our algorithm builds on the well-established detrended fluctuation analysis (DFA). After removing trends of a signal, we determine its mean squared fluctuations in consecutive intervals. In contrast to DFA we use the values per interval to approximate the distribution of these mean squared fluctuations. This allows for estimating the corresponding log-likelihood as a function of interval size without presuming the fluctuations to be normally distributed, as is the case in conventional DFA. We demonstrate the validity and robustness of our algorithm using a variety of simulated signals, ranging from scale-free fluctuations with known Hurst exponents, via more conventional dynamical systems resembling exponentially correlated fluctuations, to a toy model of neural mass activity. We also illustrate its use for encephalographic signals. We further discuss confounding factors like the finite signal size. Our model comparison provides a proper means to identify power-law scaling including the range over which it is present.

Influence of Sub-Daily Variation on Multi-Fractal Detrended Fluctuation Analysis of Wind Speed Time Series

Using multi-fractal detrended fluctuation analysis (MF-DFA), the scaling features of wind speed time series (WSTS) could be explored. In this paper, we discuss the influence of sub-daily variation, which is a natural feature of wind, in MF-DFA of WSTS. First, the choice of the lower bound of the segment length, a significant parameter of MF-DFA, was studied. The results of expanding the lower bound into sub-daily scope shows that an abrupt declination and discrepancy of scaling exponents is caused by the inability to keep the whole diel process of wind in one single segment. Additionally, the specific value, which is effected by the sub-daily feature of local meteo-climatic, might be different. Second, the intra-day temporal order of wind was shuffled to determine the impact of diel variation on scaling exponents of MF-DFA. The results illustrate that disregarding diel variation leads to errors in scaling. We propose that during the MF-DFA of WSTS, the segment length should be longer than 1 day and the diel variation of wind should be maintained to avoid abnormal phenomena and discrepancy in scaling exponents.

The fractal structure of cardiovascular beat-to-beat series described over a broad range of scales: Differences between blood pressure and heart rate, and the effect of gender

The fractal characteristics of heart rate variability are usually assessed by estimating short- and long-term scale coefficients, α1 and α2, by detrended fluctuation analysis. Recently we extended this approach introducing a temporal spectrum of scale coefficients, α(τ), that describes the deviations of self-similarity from the bi-fractal model at each scale τ. Until now relatively short recordings were considered and α(τ) was characterized only for scales τ<;100 s. Aim of this work is to describe α(τ) of cardiovascular signals extending the range τ by an order of magnitude with respect to previous studies. We considered 2-hour recordings of systolic and diastolic blood pressure (SBP and DBP) and of pulse interval (PI) in 68 volunteers (26 males, 42 females) sitting at rest. The α(τ) spectra were estimated for 5s ≤τ ≤1000s and compared. We found important differences between α(τ) of SBP, DBP and PI. In particular, α(τ) of PI was lower than α(τ) of SBP at all the scales τ, with a relative maximum at τ =26 s and a minimum at τ =300 s that were completely missing in α(τ) of DBP. Significant differences were also found between α(τ) of males and females, probably linked to gender differences in the cardiovascular autonomic tone.

Multifractal nature of particulate matters (PMs) in Hong Kong urban air

In this study, we investigate the persistent variation and the multifractal nature of particulate matter (PM) concentrations from vehicle emissions at a typical traffic intersection of street canyon in Hong Kong. Six size groups of PMs are measured and collected during rush hour sessions on different dates respectively. A recently developed model, namely multifractal detrended fluctuation analysis (MF-DFA), is employed to decompose and analyze the collected database. Through estimating the scaling exponent, it is found that the PM levels from vehicular emissions display long-term correlation characters. By employing MF-DFA method to calculate the generalized Hurst exponent and discuss the multifractal spectrums of all size groups, it is noticed that the fine particulate matters in grain diameter of 0.3-0.499 μm present strong multifractal nature, intensive oscillations of concentration variations, and long-term persistence. For fine particulate matters in the grain diameter ranges from 0.5 μm to 4.99 μm, their similar and weak multifractal natures reflect the self-similarity behaviors among these groups and the gradual decreases of the lasting effects. For large size particulate matters, i.e., grain diameter above 5 μm, certain mono-fractal nature and sharp decay of long-term persistence are obtained, even for intermittent effects. It can therefore be concluded that the fine particulate matter diffuses anomaly and persists for a long time.

Assessing complexity of heart rate variability in people with spinal cord injury using local scale exponents

Detrended fluctuation analysis (DFA) has been widely used to study dynamics of heart rate variability (HRV), which provides a quantitative parameter, the scaling exponent a, to represent the correlation properties of RR interval series. However, it has been demonstrated that HRV exhibits complex behavior that cannot be fully described by a single exponent. This study aimed to investigate whether local scale exponent α(t) with t being the time scale can reveal new features of HRV that cannot be reflected by DFA coefficients. To accurately estimate α(t), we developed an approach for correcting a(t) at small scales and verified the approach using simulated signals. We studied HRV in 12 subjects with spinal cord injury and 14 able-bodied controls during sitting and prone postures. The results showed that α(t) provides complementary views of HRV, suggesting that it may be used to evaluate the effects of SCI-induced autonomic damage on HRV.

Establishing a direct connection between detrended fluctuation analysis and Fourier analysis

To understand methodological features of the detrended fluctuation analysis (DFA) using a higher-order polynomial fitting, we establish the direct connection between DFA and Fourier analysis. Based on an exact calculation of the single-frequency response of the DFA, the following facts are shown analytically: (1) in the analysis of stochastic processes exhibiting a power-law scaling of the power spectral density (PSD), S(f)∼f(-β), a higher-order detrending in the DFA has no adverse effect in the estimation of the DFA scaling exponent α, which satisfies the scaling relation α=(β+1)/2; (2) the upper limit of the scaling exponents detectable by the DFA depends on the order of polynomial fit used in the DFA, and is bounded by m+1, where m is the order of the polynomial fit; (3) the relation between the time scale in the DFA and the corresponding frequency in the PSD are distorted depending on both the order of the DFA and the frequency dependence of the PSD. We can improve the scale distortion by introducing the corrected time scale in the DFA corresponding to the inverse of the frequency scale in the PSD. In addition, our analytical approach makes it possible to characterize variants of the DFA using different types of detrending. As an application, properties of the detrending moving average algorithm are discussed.

Scaling behavior and complexity of plastic deformation for a bulk metallic glass at cryogenic temperatures

We explore the scaling behavior and complexity in the shear-branching process during the compressive deformation of a bulk metallic glass Zr(64.13)Cu(15.75)Al(10)Ni(10.12) (at. %) at cryogenic temperatures. The fractal dimension of the stress rate signal ranges from 1.22 to 1.72 with decreasing temperature and a larger shear-branching rate occurs at lower temperature. A stochastic model is introduced for the shear-branching process. In particular, at a temperature of 213K, the shear-branching process evolves as a self-similar random process. In addition, the complexity of the stress rate signal conforms to the larger activation energy of the shear transformation zone at lower temperatures.

Resting state MEG oscillations show long-range temporal correlations of phase synchrony that break down during finger movement

The capacity of the human brain to interpret and respond to multiple temporal scales in its surroundings suggests that its internal interactions must also be able to operate over a broad temporal range. In this paper, we utilize a recently introduced method for characterizing the rate of change of the phase difference between MEG signals and use it to study the temporal structure of the phase interactions between MEG recordings from the left and right motor cortices during rest and during a finger-tapping task. We use the Hilbert transform to estimate moment-to-moment fluctuations of the phase difference between signals. After confirming the presence of scale-invariance we estimate the Hurst exponent using detrended fluctuation analysis (DFA). An exponent of >0.5 is indicative of long-range temporal correlations (LRTCs) in the signal. We find that LRTCs are present in the α/μ and β frequency bands of resting state MEG data. We demonstrate that finger movement disrupts LRTCs correlations, producing a phase relationship with a structure similar to that of Gaussian white noise. The results are validated by applying the same analysis to data with Gaussian white noise phase difference, recordings from an empty scanner and phase-shuffled time series. We interpret the findings through comparison of the results with those we obtained from an earlier study during which we adopted this method to characterize phase relationships within a Kuramoto model of oscillators in its sub-critical, critical, and super-critical synchronization states. We find that the resting state MEG from left and right motor cortices shows moment-to-moment fluctuations of phase difference with a similar temporal structure to that of a system of Kuramoto oscillators just prior to its critical level of coupling, and that finger tapping moves the system away from this pre-critical state toward a more random state.

A comparison of heart rate variability in women at the third trimester of pregnancy and during low-risk labour

Heart rate variability (HRV) has been recognised as a non-invasive method for assessing cardiac autonomic regulation. Aiming to characterize HRV changes at labour in women, we studied 10 minute ECG recordings from young mothers (n=30) at the third trimester of pregnancy (P) or during augmentation of labour (L) (n=30). Data of the L group were collected when no-contractions (L-NC) or the contractile activity (L-C) was manifested. Accordingly, the inter-beat interval (IBI) time series were processed to estimate relevant parameters of HRV such as the mean IBI (IBI¯), the mean heart rate HR¯, the root mean square of successive differences (RMSSD) in IBIs, the natural logarithm of high-frequency component (LnHF), the short-term scaling parameters from detrended fluctuation and magnitude and sign analyses such as (α1, α1(MAG), α1(SIGN)), and the sample entropy (SampEn). We found statistical differences (p<0.05) for RMSSD among P and L-NC/L-C groups (25 ± 13 vs. 36 ± 14/34 ± 16 ms) and for LnHF between P and L-NC (5.37 ± 1.15 vs. 6.05 ± 0.86 ms(2)). Likewise, we identified statistical differences (p<0.05) for α1(SIGN) among P and L-NC/L-C groups (0.19 ± 0.20 vs. 0.32 ± 0.17/0.39 ± 0.13). By contrast, L-NC and L-C groups showed statistical differences (p<0.05) in α1(MAG) (0.67 ± 0.12 vs. 0.79 ± 0.12), and SampEn (1.62 ± 0.26 vs. 1.20 ± 0.44). These results suggest that during labour, despite preserving a concomitant non-linear influence, the maternal short-term cardiac autonomic regulation becomes weakly anticorrelated (as indicated by α1(SIGN)); furthermore, an increased vagally mediated activity is observed (as indicated by RMSSD and LnHF), which may reflect a cholinergic pathway activation owing to the use of oxytocin or the anti-inflammatory cholinergic response triggered during labour.

Detrended partial cross-correlation analysis of two nonstationary time series influenced by common external forces

When common factors strongly influence two power-law cross-correlated time series recorded in complex natural or social systems, using detrended cross-correlation analysis (DCCA) without considering these common factors will bias the results. We use detrended partial cross-correlation analysis (DPXA) to uncover the intrinsic power-law cross correlations between two simultaneously recorded time series in the presence of nonstationarity after removing the effects of other time series acting as common forces. The DPXA method is a generalization of the detrended cross-correlation analysis that takes into account partial correlation analysis. We demonstrate the method by using bivariate fractional Brownian motions contaminated with a fractional Brownian motion. We find that the DPXA is able to recover the analytical cross Hurst indices, and thus the multiscale DPXA coefficients are a viable alternative to the conventional cross-correlation coefficient. We demonstrate the advantage of the DPXA coefficients over the DCCA coefficients by analyzing contaminated bivariate fractional Brownian motions. We calculate the DPXA coefficients and use them to extract the intrinsic cross correlation between crude oil and gold futures by taking into consideration the impact of the U.S. dollar index. We develop the multifractal DPXA (MF-DPXA) method in order to generalize the DPXA method and investigate multifractal time series. We analyze multifractal binomial measures masked with strong white noises and find that the MF-DPXA method quantifies the hidden multifractal nature while the multifractal DCCA method fails.

Objective differentiation of neonatal EEG background grades using detrended fluctuation analysis

A quantitative and objective assessment of background electroencephalograph (EEG) in sick neonates remains an everyday clinical challenge. We studied whether long range temporal correlations quantified by detrended fluctuation analysis (DFA) could be used in the neonatal EEG to distinguish different grades of abnormality in the background EEG activity. Long-term EEG records of 34 neonates were collected after perinatal asphyxia, and their background was scored in 1 h epochs (8 h in each neonate) as mild, moderate or severe. We applied DFA on 15 min long, non-overlapping EEG epochs (n = 1088) filtered from 3 to 8 Hz. Our formal feasibility study suggested that DFA exponent can be reliably assessed in only part of the EEG epochs, and in only relatively short time scales (10-60 s), while it becomes ambiguous if longer time scales are considered. This prompted further exploration whether paradigm used for quantifying multifractal DFA (MF-DFA) could be applied in a more efficient way, and whether metrics from MF-DFA paradigm could yield useful benchmark with existing clinical EEG gradings. Comparison of MF-DFA metrics showed a significant difference between three visually assessed background EEG grades. MF-DFA parameters were also significantly correlated to interburst intervals quantified with our previously developed automated detector. Finally, we piloted a monitoring application of MF-DFA metrics and showed their evolution during patient recovery from asphyxia. Our exploratory study showed that neonatal EEG can be quantified using multifractal metrics, which might offer a suitable parameter to quantify the grade of EEG background, or to monitor changes in brain state that take place during long-term brain monitoring.

Correlation between cortical and subcortical neural dynamics on multiple time scales in Parkinson's disease

Complex amplitude dynamics of dominant alpha oscillations (8-13 Hz) in the cortex can be captured with long-range temporal correlations (LRTC) in healthy subjects and in various diseases. In patients with Parkinson's disease (PD), intra-nuclear coherence was demonstrated in dominant beta rhythms (10-30 Hz) in the basal ganglia. However, so far the relation between cortical LRTC (across tens of seconds) and subcortical coherence (millisecond scale) is unknown. We addressed these "multiscale interactions" by simultaneous recordings of surface electroencephalography (EEG) and deep local field potentials (LFP) from the bilateral subthalamic nucleus (STN) in eight patients with severe PD eligible for deep brain stimulation, who performed a lexical decision task on medication. In the continuous data set LRTC up to 20s were calculated in the amplitude envelope of 8-13-Hz EEG oscillations (across whole scalp), and subcortical coherence was assessed with measures being insensitive to volume conduction artifacts (imaginary part of coherency; iCOH) in 10-20 and 21-30-Hz oscillations in STN-LFP. We showed a significant positive correlation across patients between cortical LRTC (8-13Hz) and subcortical iCOH selectively in 10-20-Hz oscillations in the left STN. Our results suggest a relation between neural dynamics in the most dominant rhythms in the cortex and basal ganglia in PD, extending across multiple time scales (milliseconds vs. tens of seconds). Furthermore, the investigation of multiscale interactions might contribute to our understanding of cortical-subcortical neural coupling in PD.

Outlier-resilient complexity analysis of heartbeat dynamics

Complexity in physiological outputs is believed to be a hallmark of healthy physiological control. How to accurately quantify the degree of complexity in physiological signals with outliers remains a major barrier for translating this novel concept of nonlinear dynamic theory to clinical practice. Here we propose a new approach to estimate the complexity in a signal by analyzing the irregularity of the sign time series of its coarse-grained time series at different time scales. Using surrogate data, we show that the method can reliably assess the complexity in noisy data while being highly resilient to outliers. We further apply this method to the analysis of human heartbeat recordings. Without removing any outliers due to ectopic beats, the method is able to detect a degradation of cardiac control in patients with congestive heart failure and a more degradation in critically ill patients whose life continuation relies on extracorporeal membrane oxygenator (ECMO). Moreover, the derived complexity measures can predict the mortality of ECMO patients. These results indicate that the proposed method may serve as a promising tool for monitoring cardiac function of patients in clinical settings.

Analyzing long-term correlated stochastic processes by means of recurrence networks: potentials and pitfalls

Long-range correlated processes are ubiquitous, ranging from climate variables to financial time series. One paradigmatic example for such processes is fractional Brownian motion (fBm). In this work, we highlight the potentials and conceptual as well as practical limitations when applying the recently proposed recurrence network (RN) approach to fBm and related stochastic processes. In particular, we demonstrate that the results of a previous application of RN analysis to fBm [Liu et al. Phys. Rev. E 89, 032814 (2014)] are mainly due to an inappropriate treatment disregarding the intrinsic nonstationarity of such processes. Complementarily, we analyze some RN properties of the closely related stationary fractional Gaussian noise (fGn) processes and find that the resulting network properties are well-defined and behave as one would expect from basic conceptual considerations. Our results demonstrate that RN analysis can indeed provide meaningful results for stationary stochastic processes, given a proper selection of its intrinsic methodological parameters, whereas it is prone to fail to uniquely retrieve RN properties for nonstationary stochastic processes like fBm.

Detrended partial-cross-correlation analysis: a new method for analyzing correlations in complex system

In this paper, a new method, detrended partial-cross-correlation analysis (DPCCA), is proposed. Based on detrended cross-correlation analysis (DCCA), this method is improved by including partial-correlation technique, which can be applied to quantify the relations of two non-stationary signals (with influences of other signals removed) on different time scales. We illustrate the advantages of this method by performing two numerical tests. Test I shows the advantages of DPCCA in handling non-stationary signals, while Test II reveals the "intrinsic" relations between two considered time series with potential influences of other unconsidered signals removed. To further show the utility of DPCCA in natural complex systems, we provide new evidence on the winter-time Pacific Decadal Oscillation (PDO) and the winter-time Nino3 Sea Surface Temperature Anomaly (Nino3-SSTA) affecting the Summer Rainfall over the middle-lower reaches of the Yangtze River (SRYR). By applying DPCCA, better significant correlations between SRYR and Nino3-SSTA on time scales of 6 ~ 8 years are found over the period 1951 ~ 2012, while significant correlations between SRYR and PDO on time scales of 35 years arise. With these physically explainable results, we have confidence that DPCCA is an useful method in addressing complex systems.

Synaptic plasticity enables adaptive self-tuning critical networks

During rest, the mammalian cortex displays spontaneous neural activity. Spiking of single neurons during rest has been described as irregular and asynchronous. In contrast, recent in vivo and in vitro population measures of spontaneous activity, using the LFP, EEG, MEG or fMRI suggest that the default state of the cortex is critical, manifested by spontaneous, scale-invariant, cascades of activity known as neuronal avalanches. Criticality keeps a network poised for optimal information processing, but this view seems to be difficult to reconcile with apparently irregular single neuron spiking. Here, we simulate a 10,000 neuron, deterministic, plastic network of spiking neurons. We show that a combination of short- and long-term synaptic plasticity enables these networks to exhibit criticality in the face of intrinsic, i.e. self-sustained, asynchronous spiking. Brief external perturbations lead to adaptive, long-term modification of intrinsic network connectivity through long-term excitatory plasticity, whereas long-term inhibitory plasticity enables rapid self-tuning of the network back to a critical state. The critical state is characterized by a branching parameter oscillating around unity, a critical exponent close to -3/2 and a long tail distribution of a self-similarity parameter between 0.5 and 1.

Existence of anticorrelations for local field potentials recorded from mice reared in standard condition and environmental enrichment

In the present paper, we analyze local field potentials (LFPs) recorded from the secondary motor cortex (M2) and primary visual cortex (V1) of freely moving mice reared in environmental enrichment (EE) and standard condition (SC). We focus on the scaling properties of the signals by using an integrated approach combining three different techniques: the Higuchi method, detrended fluctuation analysis, and power spectrum. Each technique provides direct or indirect estimations of the Hurst exponent H and this prevents spurious identification of scaling properties in time-series analysis. It is well known that the power spectrum of an LFP signal scales as 1/f(β) with β>0. Our results indicate the existence of a particular power spectrum scaling law 1/f(β) with β<0 for low frequencies (f<4 Hz) for both SC and EE rearing conditions. This type of scaling behavior is associated to the presence of anticorrelation in the corresponding LFP signals. Moreover, since EE is an experimental protocol based on the enhancement of sensorimotor stimulation, we study the possible effects of EE on the scaling properties of secondary motor cortex (M2) and primary visual cortex (V1). Notably, the difference between Hurst's exponents in EE and SC for individual cortical regions (M2) and (V1) is not statistically significant. On the other hand, using the detrended cross-correlation coefficient, we find that EE significantly reduces the functional coupling between secondary motor cortex (M2) and visual cortex (V1).

Evaluation of scaling invariance embedded in short time series

Scaling invariance of time series has been making great contributions in diverse research fields. But how to evaluate scaling exponent from a real-world series is still an open problem. Finite length of time series may induce unacceptable fluctuation and bias to statistical quantities and consequent invalidation of currently used standard methods. In this paper a new concept called correlation-dependent balanced estimation of diffusion entropy is developed to evaluate scale-invariance in very short time series with length ~10(2). Calculations with specified Hurst exponent values of 0.2,0.3,...,0.9 show that by using the standard central moving average de-trending procedure this method can evaluate the scaling exponents for short time series with ignorable bias (≤0.03) and sharp confidential interval (standard deviation ≤0.05). Considering the stride series from ten volunteers along an approximate oval path of a specified length, we observe that though the averages and deviations of scaling exponents are close, their evolutionary behaviors display rich patterns. It has potential use in analyzing physiological signals, detecting early warning signals, and so on. As an emphasis, the our core contribution is that by means of the proposed method one can estimate precisely shannon entropy from limited records.

Simulated shift work in rats perturbs multiscale regulation of locomotor activity

Motor activity possesses a multiscale regulation that is characterized by fractal activity fluctuations with similar structure across a wide range of timescales spanning minutes to hours. Fractal activity patterns are disturbed in animals after ablating the master circadian pacemaker (suprachiasmatic nucleus, SCN) and in humans with SCN dysfunction as occurs with aging and in dementia, suggesting the crucial role of the circadian system in the multiscale activity regulation. We hypothesized that the normal synchronization between behavioural cycles and the SCN-generated circadian rhythms is required for multiscale activity regulation. To test the hypothesis, we studied activity fluctuations of rats in a simulated shift work protocol that was designed to force animals to be active during the habitual resting phase of the circadian/daily cycle. We found that these animals had gradually decreased mean activity level and reduced 24-h activity rhythm amplitude, indicating disturbed circadian and behavioural cycles. Moreover, these animals had disrupted fractal activity patterns as characterized by more random activity fluctuations at multiple timescales from 4 to 12 h. Intriguingly, these activity disturbances exacerbated when the shift work schedule lasted longer and persisted even in the normal days (without forced activity) following the shift work. The disrupted circadian and fractal patterns resemble those of SCN-lesioned animals and of human patients with dementia, suggesting a detrimental impact of shift work on multiscale activity regulation.

Mouse activity across time scales: fractal scenarios

In this work we devise a classification of mouse activity patterns based on accelerometer data using Detrended Fluctuation Analysis. We use two characteristic mouse behavioural states as benchmarks in this study: waking in free activity and slow-wave sleep (SWS). In both situations we find roughly the same pattern: for short time intervals we observe high correlation in activity - a typical 1/f complex pattern - while for large time intervals there is anti-correlation. High correlation of short intervals ( to : waking state and to : SWS) is related to highly coordinated muscle activity. In the waking state we associate high correlation both to muscle activity and to mouse stereotyped movements (grooming, waking, etc.). On the other side, the observed anti-correlation over large time scales ( to : waking state and to : SWS) during SWS appears related to a feedback autonomic response. The transition from correlated regime at short scales to an anti-correlated regime at large scales during SWS is given by the respiratory cycle interval, while during the waking state this transition occurs at the time scale corresponding to the duration of the stereotyped mouse movements. Furthermore, we find that the waking state is characterized by longer time scales than SWS and by a softer transition from correlation to anti-correlation. Moreover, this soft transition in the waking state encompass a behavioural time scale window that gives rise to a multifractal pattern. We believe that the observed multifractality in mouse activity is formed by the integration of several stereotyped movements each one with a characteristic time correlation. Finally, we compare scaling properties of body acceleration fluctuation time series during sleep and wake periods for healthy mice. Interestingly, differences between sleep and wake in the scaling exponents are comparable to previous works regarding human heartbeat. Complementarily, the nature of these sleep-wake dynamics could lead to a better understanding of neuroautonomic regulation mechanisms.

Markers of criticality in phase synchronization

The concept of the brain as a critical dynamical system is very attractive because systems close to criticality are thought to maximize their dynamic range of information processing and communication. To date, there have been two key experimental observations in support of this hypothesis: (i) neuronal avalanches with power law distribution of size and (ii) long-range temporal correlations (LRTCs) in the amplitude of neural oscillations. The case for how these maximize dynamic range of information processing and communication is still being made and because a significant substrate for information coding and transmission is neural synchrony it is of interest to link synchronization measures with those of criticality. We propose a framework for characterizing criticality in synchronization based on an analysis of the moment-to-moment fluctuations of phase synchrony in terms of the presence of LRTCs. This framework relies on an estimation of the rate of change of phase difference and a set of methods we have developed to detect LRTCs. We test this framework against two classical models of criticality (Ising and Kuramoto) and recently described variants of these models aimed to more closely represent human brain dynamics. From these simulations we determine the parameters at which these systems show evidence of LRTCs in phase synchronization. We demonstrate proof of principle by analysing pairs of human simultaneous EEG and EMG time series, suggesting that LRTCs of corticomuscular phase synchronization can be detected in the resting state and experimentally manipulated. The existence of LRTCs in fluctuations of phase synchronization suggests that these fluctuations are governed by non-local behavior, with all scales contributing to system behavior. This has important implications regarding the conditions under which one should expect to see LRTCs in phase synchronization. Specifically, brain resting states may exhibit LRTCs reflecting a state of readiness facilitating rapid task-dependent shifts toward and away from synchronous states that abolish LRTCs.

Dynamical glucometry: use of multiscale entropy analysis in diabetes

Diabetes mellitus (DM) is one of the world's most prevalent medical conditions. Contemporary management focuses on lowering mean blood glucose values toward a normal range, but largely ignores the dynamics of glucose fluctuations. We probed analyte time series obtained from continuous glucose monitor (CGM) sensors. We show that the fluctuations in CGM values sampled every 5 min are not uncorrelated noise. Next, using multiscale entropy analysis, we quantified the complexity of the temporal structure of the CGM time series from a group of elderly subjects with type 2 DM and age-matched controls. We further probed the structure of these CGM time series using detrended fluctuation analysis. Our findings indicate that the dynamics of glucose fluctuations from control subjects are more complex than those of subjects with type 2 DM over time scales ranging from about 5 min to 5 h. These findings support consideration of a new framework, dynamical glucometry, to guide mechanistic research and to help assess and compare therapeutic interventions, which should enhance complexity of glucose fluctuations and not just lower mean and variance of blood glucose levels.

Cross-evidence for hypnotic susceptibility through nonlinear measures on EEGs of non-hypnotized subjects

Assessment of hypnotic susceptibility is usually obtained through the application of psychological instruments. A satisfying classification obtained through quantitative measures is still missing, although it would be very useful for both diagnostic and clinical purposes. Aiming at investigating the relationship between the cortical brain activity and the hypnotic susceptibility level, we propose the combined use of two methodologies - Recurrence Quantification Analysis and Detrended Fluctuation Analysis - both inherited from nonlinear dynamics. Indicators obtained through the application of these techniques to EEG signals of individuals in their ordinary state of consciousness allowed us to obtain a clear discrimination between subjects with high and low susceptibility to hypnosis. Finally a neural network approach was used to perform classification analysis.

On the increase of the "non-uniform" scaling of the magnetic field variations before the M(w)9.0 earthquake in Japan in 2011

By applying Detrended Fluctuation Analysis (DFA) to the time series of the geomagnetic data recorded at three measuring stations in Japan, Rong et al. in 2012 recently reported that anomalous magnetic field variations were identified well before the occurrence of the disastrous Tohoku Mw9.0 earthquake that occurred on 11 March 2011 in Japan exhibiting increased "non-uniform" scaling behavior. Here, we provide an explanation for the appearance of this increase of "non-uniform" scaling on the following grounds: These magnetic field variations are the ones that accompany the electric field variations termed Seismic Electric Signals (SES) activity which have been repeatedly reported that precede major earthquakes. DFA as well as multifractal DFA reveal that the latter electric field variations exhibit scaling behavior as shown by analyzing SES activities observed before major earthquakes in Greece. Hence, when these variations are superimposed on a background of pseudosinusoidal trend, their long range correlation properties-quantified by DFA-are affected resulting in an increase of the "non-uniform" scaling behavior. The same is expected to hold for the former magnetic field variations. This explanation is strengthened by recent findings showing that the fluctuations of the order parameter of seismicity exhibited an unprecedented minimum almost two months before the Tohoku earthquake occurrence which is characteristic for an almost simultaneous emission of Seismic Electric Signals activity.

Modified multidimensional scaling approach to analyze financial markets

Detrended cross-correlation coefficient (σDCCA) and dynamic time warping (DTW) are introduced as the dissimilarity measures, respectively, while multidimensional scaling (MDS) is employed to translate the dissimilarities between daily price returns of 24 stock markets. We first propose MDS based on σDCCA dissimilarity and MDS based on DTW dissimilarity creatively, while MDS based on Euclidean dissimilarity is also employed to provide a reference for comparisons. We apply these methods in order to further visualize the clustering between stock markets. Moreover, we decide to confront MDS with an alternative visualization method, "Unweighed Average" clustering method, for comparison. The MDS analysis and "Unweighed Average" clustering method are employed based on the same dissimilarity. Through the results, we find that MDS gives us a more intuitive mapping for observing stable or emerging clusters of stock markets with similar behavior, while the MDS analysis based on σDCCA dissimilarity can provide more clear, detailed, and accurate information on the classification of the stock markets than the MDS analysis based on Euclidean dissimilarity. The MDS analysis based on DTW dissimilarity indicates more knowledge about the correlations between stock markets particularly and interestingly. Meanwhile, it reflects more abundant results on the clustering of stock markets and is much more intensive than the MDS analysis based on Euclidean dissimilarity. In addition, the graphs, originated from applying MDS methods based on σDCCA dissimilarity and DTW dissimilarity, may also guide the construction of multivariate econometric models.