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

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

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

Power-grid faults pose a great threat to the economy and to social stability. In this paper, the detrended fluctuation analysis method is used to investigate the scaling properties of power grid faults and the correlation between faults and meteorological factors that are closely related to power grid faults. Multifractal detrended fluctuation analysis showed that the fault time series were multifractal. Further investigations revealed the origins of multifractality in power-grid fault time series, and the results showed that the temporal correlations in the data represent the distribution of the returns, which are a significant source of multifractal scaling. Then, cross correlations between the fault and the four meteorological factors were investigated using the detrended cross-correlation analysis method. The results showed that maximum wind speed has a considerable impact on the number of transmission system line faults per day, whereas daily precipitation, daily mean air temperature, and maximum wind speed have a considerable impact on the number of distribution system line faults per day.

The internal mechanism of the social “earthquakes”- the wars burst in human society, is investigated by nonlinear theory. We explore the scaling behavior in the human activity and the historical evolution of ancient China. Based on the war chronology and history chronology from B.C.2000 to A.D.1840, using statistical analysis together with the detrended fluctuation analysis, we find that the social evolution system shows self-organized critical behavior and there exists self-similar scaling behavior during the evolution system. Based on time-delay reconstructed phase space, a modified radial basis function neural network (modified RBF-NN) is used to predict the number of wars and the transformation of the historical stage. It shows that the prediction result matches quite well with the real value for the following two historical stages. Furthermore, by singular value decomposition and the sparse identification algorithm in time-delay coordinates, the coefficients of the candidate functions in identified system can be solved from a sparse regression problem. We extracted the governing equations of big events burst in social system. The identified system fits well with the real data, and it can capture the same topology with the original system of wars. In this paper, we investigated the wars burst in social system by statistic analysis, neural network as well as sparse identification. It sheds light on the study of wars or conflicts from perspective of data science and dynamical theory.

Using detrended fluctuation analysis (DFA) we find that all continents are persistent in temperature. The scaling exponents of the southern hemisphere (SH) continents, i.e., South America (0.77) and Oceania (0.72) are somewhat higher than scaling exponents of Europe (0.70), Asia (0.69) and North America (0.64), but the scaling of Africa is by far the highest (0.86). The reason for this is the location of Africa near the equator. The scaling exponents of the precipitation are much smaller, i.e. between 0.55 (Europe) and 0.68 (North America). The scaling exponent of Europe is near that of the random noise (0.5), while the other continents are slightly persistent in precipitation. We also show that the persistence disappears in all time series when shuffling the data randomly, showing that persistence is not an intrinsic property of the estimator. We find that the monthly temperature is the more persistent the more wide area is analyzed. The persistence of precipitation is also increasing as a function of area, although not so clearly. Furthermore, our analysis shows that the persistence of the temperature decreases poleward in both hemispheres. The persistence of precipitation is also highest at equatorial zones, and decreases poleward, but is less dependent on the latitude than temperature. It seems that the persistence of the precipitation is more dependent on the corresponding climate type than the persistence of the temperature., Comment: 20 pages, 13 figures, 3 tables

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

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

As the earth's third pole, Qinghai-Tibet Plateau belongs to one of the most sensitive regions to climate change in the world. Based on the observed and the simulated daily precipitation from the Coupled Model Intercomparison Project Phase 5 (CMIP5), we evaluated the simulation performance of daily precipitation from selected CMIP5 models from 1975 to 2005 over the Qinghai-Tibet Plateau. We found that daily precipitation exhibited obvious long-range correlation characteristics using the detrended fluctuation analysis method. The scaling exponents of daily precipitation in summer and autumn are significantly larger than those in spring and winter. MIROC4H with the best performance can reproduce long-range correlation characteristic of daily precipitation series probably because of the higher resolution, which can capture small scale cloud convections. Besides there are seasonal differences in the simulation results among different regions of the Qinghai-Tibet Plateau, simulation effects of all climate models in summer and winter are better than those in spring and autumn. The performance of MIROC4H model works the best in spring. Overall, the scaling exponents of daily precipitation from BCC-CSM1-1-M, CMCC-CM and MIROC4H are close to the observations. CCSM4 and MIROC4H climate models could reproduce the internal dynamics characteristic of daily precipitation in autumn. But for winter, all climate models have exaggerated the scaling value in southeastern Qinghai-Tibet Plateau compared with the observed values.

In this study, we investigated the performance of nine CMIP5 models for global daily precipitation by comparing with NCEP data from 1960 to 2005 based on the detrended fluctuation analysis (DFA) method. We found that NCEP daily precipitation exhibits long-range correlation (LRC) characteristics in most regions of the world. The LRC of daily precipitation over the central of North American continent is the strongest in summer, while the LRC of precipitation is the weakest for the equatorial central Pacific Ocean. The zonal average scaling exponents of NCEP daily precipitation are smaller in middle and high latitudes than those in the tropics. The scaling exponents are above 0.9 over the tropical middle and east Pacific Ocean for the year and four seasons. Most CMIP5 models can capture the characteristic that zonal mean scaling exponents of daily precipitation reach the peak in the tropics, and then decrease rapidly with the latitude increasing. The zonal mean scaling exponents simulated by CMCC-CMS, GFDL-ESM2G and IPSL-CM5A-MR show consistencies with those of NCEP, while BCC_CSM1.1(m) and FGOALS-g2 cannot capture the seasonal variations of daily precipitation’s LRC. The biases of scaling exponents between CMIP5 models and NCEP are smaller in the high latitudes, and even less than the absolute value of 0.05 in some regions, including Arctic Ocean, Siberian, Southern Ocean and Antarctic. However, for Western Africa, Eastern Africa, Tropical Eastern Pacific and Northern South America, the simulated biases of scaling exponents are greater than the absolute value of 0.05 for the year and all four seasons. In general, the spatial biases of LRC simulated by GFDL-ESM2G, HadGEM2-AO and INM-CM4 are relatively small, which indicating that the LRC characteristics of daily precipitation are well simulated by these models.

The daily average land surface air temperature (SAT) on land simulated by 8 CMIP5 models historical experiments and that from NCEP data during 1960 - 2005, are used to evaluate the performance of the CMIP5 model based on detrended fluctuation analysis (DFA) method. The DFA results of NCEP data show that SAT in most regions of the world exhibit long-range correlation. The Scaling exponents of NCEP SAT show the zonal distribution characteristics of larg in tropics while small in medium and high latitudes. The distribution characteristics of the zonal average Scaling exponents of CMCC-CMS, GFDL-ESM2G, IPSL-CM5A-MR are similar to that of NCEP data. From the DFA errors of model-simulated SAT, the performance of IPSL-CM5A-MR is the best among the 8 models throughout the year, the performance of FGOALS-g2 is good in spring and summer, GFDL-ESM2G is the best in autumn, CNRM-CM5 and CMCC-CMS is good in winter. The Scaling exponents of model-simulated SAT are closer to that of NCEP data in most areas of the mid-high latitude on the northern hemisphere. However, simulations of SAT in East Asia and Central North American are generally less effective. In spring, most models have better performance in Siberian (SIB), Central Asia (CAS) and Tibetan (TIB). SAT in Northern Europe (NEU) area are well simulated by most models in summer. In autumn, areas with better performance of most models are Mediterranean (MED), SIB and TIB regions. In winter, SAT in Greenland (GRL), SIB and TIB areas are well simulated by most models. Generally speaking, the performance of CMIP5 models for SAT on global continents varies in different seasons and different regions.

The long-range correlations associated with the presence of persistence are investigated by applying the detrended fluctuation analysis (DFA) on three different proxies of long-term solar activity. The considered datasets are a sunspot number reconstruction (SNR04) obtained from the atmospheric activity of the cosmogenic isotope 14C derived from tree rings, a total solar irradiance reconstruction (TSIR12) obtained from several 10Be ice core records from Greenland and Antarctica in combination with the global record of 14C in tree rings and a new multi-proxy sunspot number reconstruction (SNR18), also derived from 10Be datasets and the global 14C production series. The DFA scaling exponents found for the three time series are similar (lying in the range between 0.70 and 0.77) and the scaling ranges are comparable. These results indicate the presence of long-range correlations with persistence, in substantial agreement with the findings of previous studies carried out on other solar activity indices and proxies.

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

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

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

We have investigated dynamics of the Internet performance through the assessment of scaling features of a network ICMP echo mechanism or pinging. Time series of round-trip times (RTT) from the host computer to 5 destination hosts and back, recorded during three consecutive days and nights, have been used. To assess correlation and scaling features of network echo mechanism, we used method of detrended fluctuation analysis (DFA) for RTT data sets. It was shown that for different, 10 minute long periods of day and night observations, RTT data sets mostly fluctuate within a narrow range, though sometimes we observe strong sharp spikes. RTT variations mostly reveal persistent behavior. DFA fluctuation curves often are characterized by crossovers indication stronger or lesser changes in the dynamics of network performance. Distribution function of DFA scaling exponents of considered RTT time series mostly was asymmetric with long tail on the right hand side. Dynamical changes occurring in the scaling features of Internet network as assessed by RTT fluctuations do not depend on the location of the host and destination nodes. Larger delays in round-trip time responses make the scaling behavior of the RTT series complicated and strongly influence their long range correlation features.

We study the scaling features in the evolutionary dynamics of two coupled chaotic systems based on the sequences of return times into a Poincare section, contaminated with additive (measuring) noise. Using three models of chaotic systems: the Rossler oscillator, the Lorenz system, and the nephron model, and the detrended fluctuation analysis (DFA) as an approach for data processing, we demonstrate that the anti-correlated sequences of return times of synchronous motions show a higher sensitivity to measuring noise than the positively correlated series of return times of asynchronous oscillations. This conclusion is confirmed by the results for various oscillatory regimes in all models considered.

We study for the first time the correlation properties and the multifractality of downward longwave radiation (DLR) temporal fluctuations, using the multi-fractal detrended fluctuation analysis. Our DLR data are taken from 40 stations of the Baseline Surface Radiation Network and span more than 20 years on temporal resolutions of a few minutes. In almost all stations, two different scaling regions are defined through the emergence of a crossover at around 10 days, similarly to earlier studies of other meteorological quantities. The crossover marks the transition from a persistent, non-stationary signal on weather scales to a more weakly persistent (and sometimes anti-persistent) behavior extending in the macroweather region between 2 months and at least several years. We observe typical multifractal behavior for DLR through the investigation of the generalized Hurst exponents and singularity spectra. Generally, the multifractality strength is larger for the weather regime compared to the macroweather. We find evidence that the shape of the probability density function of the DLR anomalies contributes to the multifractality strength for macroweather. The three Western Equatorial Pacific stations exhibit markedly different characteristics, with fluctuation functions closely following 1/f noise between the scales of a few hours up to 2 months. These stations appear to be strongly affected by the ENSO. Also, their timeseries are barely long enough so that the ENSO shows as a semi-periodic signal on the large scales of their long-term memory. Their multifractality is not clearly defined and its strength is much smaller than that of the other stations.

In the recent past, coronal mass ejection (CME) has received much research attention for its geo-effectiveness. In this paper, an investigation has been made to identify the scaling pattern of the CME linear speed time series data (February 1999 to December 2007 of solar cycle 23) collected from the Solar and Heliospheric Observatory (SOHO) using Multi-Fractal Detrended Fluctuation Analysis (MFDFA) and Multi-Fractal Detrended Moving Average (MFDMA) method. The scaling exponent, generalized Hurst exponent, singularity strength and also the singularity spectrum have been computed to quantify the multifractality and to identify the singularities of the time series data. An effort has also been made to find out the possible sources which are responsible for the multifractality in the signal by studying the scaling patterns of the shuffled and surrogate version of the original data. It has been revealed in this paper that CME linear speed signal exhibit multifractal behaviour with long-term persistence. Both the long-range temporal correlation and the broad probability density function (pdf) are found to be the primary source of multifractality in the signal. The singularities or abruptness present in the signal are found to vary with time, and this fluctuation follows an AR (2) model.

The relationship between ozone and its main precursors exhibits complex non-linear feature in the atmospheric photochemical system. Therefore, sophisticated mathematical and nonlinear methods are very important for studying the O3 sensitivity to its precursors and improving the O3 prediction model. In this study, the coupling detrended fluctuation analysis (CDFA) method has been used to investigate the multi-temporal scaling pattern of coupling correlations between O3 and its precursors. Hourly concentrations of ozone (O3), nitrogen oxide (NOx) and non-methane hydrocarbons (NMHCs) from 1 March 2016 to 28 February 2017 of two traffic monitoring sites in Taiwan are analyzed. Coupling multifractality features are confirmed in the temporal distribution of coupling correlations between O3 and its precursors. And the sources of coupling multifractal features are investigated by the phase randomized surrogates and the shuffling procedures. Moreover, in order to confirm the ozone sensitivity to its precursors at different time scales, a novel dynamical index, namely the coupling sensitivity degree (CSD), is introduced based on CDFA method. High temporal scale resolution of the O3-NMHCs-NOx sensitivity is presented based on the concept of sliding window. And the O3-NMHCs-NOx sensitivity regime can be shown visually by the temporal evolutions of the CSD on the whole year. Furthermore, statistics of seasonal CSD values are obtained by averaging the hourly sliding values. Result shows that the CSD values for both NMHCs and NOx are greatest in summer due to the more active photochemical reaction, compared to the other seasons. Meanwhile, the photochemical system is more sensitive to NMHCs for the two sites, which is consistent with the characteristics of pollutant emissions in the urban traffic area. At last, the diurnal pattern of the CSD for NMHCs and NOx are investigated. It is found that the greatest CSD for both NMHCs and NOx series are synchronized to the maximum O3 generation. This study provides a feasible and effective dynamical index to describe O3-NOx-NMHCs sensitivity, and suggests a new approach to improve the ozone prediction models in the photochemical system.

Heart rate asymmetry is a phenomenon by which the accelerations and decelerations of heart rate behave differently, and this difference is consistent and unidirectional, i.e. in most of the analyzed recordings the inequalities have the same directions. So far, it has been established for variance and runs based types of descriptors of R R intervals time series. In this paper we apply the newly developed method of Asymmetric Detrended Fluctuation Analysis, which so far has mainly been used with economic time series, to the set of 420 stationary 30 min time series of R R intervals from young, healthy individuals aged between 20 and 40. This asymmetric approach introduces separate scaling exponents for rising and falling trends. We systematically study the presence of asymmetry in both global and local versions of this method. In this study global means “applying to the whole time series” and local means “applying to windows jumping along the recording”. It is found that the correlation structure of the fluctuations left over after detrending in physiological time series shows strong asymmetric features in both magnitude, with α + α − , where α + is related to heart rate decelerations and α − to heart rate accelerations, and the proportion of the signal in which the above inequality holds. A very similar effect is observed if asymmetric noise is added to a symmetric self-affine function. No such phenomena are observed in the same physiological data after shuffling or with a group of symmetric synthetic time series.

In this paper, a generalized method of detrended fluctuation analysis (DFA) is proposed as a new measure to assess the complexity of a complex dynamical system such as stock market. We extend DFA and local scaling DFA to higher moments such as skewness and kurtosis (labeled SMDFA and KMDFA), so as to investigate the volatility scaling property of financial time series. Simulations are conducted over synthetic and financial data for providing the comparative study. We further report the results of volatility behaviors in three American countries, three Chinese and three European stock markets by using DFA and LSDFA method based on higher moments. They demonstrate the dynamics behaviors of time series in different aspects, which can quantify the changes of complexity for stock market data and provide us with more meaningful information than single exponent. And the results reveal some higher moments volatility and higher moments multiscale volatility details that cannot be obtained using the traditional DFA method.

Defining the morphological disorders causing neurodegenerative diseases is an unresolved problem. In this study, we propose a statistical-physical approach to quantify neurite morphology and evaluate the pathological states induced by Alzheimer's disease (AD). We analyzed the two-dimensional morphologies of neurites of in vitro-cultured human induced-pluripotent stem cell-derived neurons, reprogrammed from both a healthy person and a patient with AD, using discrete chordal Loewner evolution. For the numerically calculated Loewner driving forces, detrended fluctuation analysis was performed, and the morphological characteristics of the neurites were quantified using short-range and long-range scaling exponents. The day in vitro (DIV)-dependent behaviors of the scaling exponents and the associated neurite-type categorizations suggested that differences between healthy and AD neurites can be observed from the early stage (DIV3) of their development. Notably, AD neurites have less long-range autocorrelations than healthy neurites, particularly in the earlier stages (DIV3-10). Immunofluorescence-staining results suggested that these differences precede significant expressions of β-amyloid and phosphorylated tau, which are known as biological factors causing AD. We expect that these results will lead to a theoretical interpretation of the neurogenerative disease, providing the physical properties of individual neurites with different morphologies.

Fractal analysis of randomly rough surface is an interesting tool to establish relationships between surface geometry and properties. Nonetheless, the observation that different methods to determine the fractal dimension D yield different results has raised questions about its physical meaning. This work shows that such variations are caused by the mathematical details of the methods used, particularly by the effect of vertical scaling. For the triangular prism method (TPM), applied to fractional Brownian motion, the effect of vertical scaling on the numerical estimation of D can be addressed through analytic calculations. The analytic approach was compared to simulations of surface topography obtained by the random midpoint algorithm (RMA) using TPM, box count method (BCM), differential box count (DBC) and detrended fluctuation analysis (DFA). The effect of scaling for TPM is considerable and coincides with the mathematical predictions. BCM and DBC show no effect of scaling but provide poor estimates at high D . A small effect was found for DFA. It is concluded that TPM provides a precise estimate of D which is independent of vertical scaling for infinite image resolution. At finite resolutions, the estimation error on D can be minimised by choosing an optimal vertical scaling factor.

This paper discusses a major issue in computational linguistics; the automatic calculation of text coherence. Heretofore, only few methods have been proposed to automatically detect local coherence of texts. All of these methods need a lot of pre-processing tasks and computational efforts. Here we suggest a simple method to evaluate the coherence globally. First, we use a word ranking method to assign an importance value to each word-type in a text, then the importance time series associated with text is constructed. In the next step, Detrended Fluctuation Analysis(DFA) which is used for detecting inherent correlations in time series, is applied to texts importance time series. We found that the importance time series exhibits a bi-scale behavior; it is long-range correlated at large distances, while short-range correlations are observed in small distances. We also observed that for a shuffled text the scaling exponent decreases. This decrease becomes more and more significant when we reshuffle the chapters, paragraphs, sentences and words respectively. This fact leads us to consider the scaling exponent of text time series (or briefly STT) as a measure for quantifying the global coherence. We demonstrate our claim by carrying out an experiment on three sample texts and comparing our method by some entity grid based models.

The aim of this study is to investigate the scaling exponent properties of mountainous river flow fluctuations by detrended fluctuation analysis (DFA). Streamflow data were collected continuously using Fluvial Acoustic Tomography System (FATS), which is a novel system for measuring continuous streamflow at high-frequency scales. The results revealed that river discharge fluctuations have two scaling regimes and scaling break. In contrast to the Ranting Curve method (RC), the small-scale exponent detected by the FATS is estimated to be 1.02 ± 0.42% less than that estimated by RC. More importantly, the crossover times evaluated from the FATS delayed approximately by 42 ± 21 hr ≈ 2–3 days than their counterparts estimated by RC. The power spectral density analysis assists our findings. We found that scaling characteristics information evaluated for a river using flux data obtained by RC approach might not be accurately detected, because this classical method assumes that flow in river is steady and depends on constructing a relationship between discharge and water level, while the discharge obtained by the FATS decomposes velocity and depth into two ratings according to the continuity equation. Generally, this work highlights the performance of FATS as a powerful and effective approach for continuous streamflow measurements at high-frequency levels.

Stick-slip dynamics during nanoscratching is investigated for the Ni 62 Nb 38 metallic glass. Detrended fluctuation analysis is introduced to explore the influence of loading force on the temporal scaling and stick-slip behavior. The self-similar characteristics and complexity in the temporal scale of the lateral force signal are investigated. A modified Cauchy class model is used for the stochastic stick-slip process, which connects the fractal dimension and the Hurst exponent and features the positive correlation process. The confidence intervals of the differential friction coefficient at different loading forces elucidate the inhomogeneous (and homogeneous) shear-branching processes during the nanoscratching process.

Horizontal oil-water two-phase flows are frequently encountered in many industrial processes but the understanding of the dynamic behavior underlying the different flow patterns is still a challenge. In this study, we first conduct experiments of horizontal oil-water flows in a small diameter pipe, and collect the fluctuation signals from conductance probes. The multi-scale power-law correlations of the oil-water flow structures are investigated using detrended fluctuation analysis (DFA) based on the magnitude and sign decomposition of the raw signals. The analysis reveals the scaling behavior of different flow structures; five conductive flow patterns are indentified based on the magnitude and sign scaling exponents at different time scales. In addition, the transfer entropy (TE) in a state space is used to study the information transferring characteristics of the oil-water mixture flowing past a conductance cross-correlation velocity probe. The results of TE indicate that the transferring information depends on the flow conditions and can be used to show changes in the flow patterns.

We investigate the self-affinity of a fluid velocity time series in turbulent regime. We use the CFD OpenFoam library to simulate the flow of fluid in a duct with two obstacles injecting vorticity downstream. We calculated the fluid velocity in three sections in of the structure. The simulation presented more turbulences in sections further away from obstacles. We assess the self-affinity of properties of those sections by using the Detrended Fluctuation Analysis method. This scaling method presented two behaviors. First, a persistent behavior after passing through obstacles. Second, a sub-diffusive behavior in turbulent sections, presenting a long-range correlation α exponent ranging between 1.0 and 1.5. The findings suggest that chaotic states tend to follow a sub-diffusive process.

Based on the climatological-hydrological daily data recorded in the headwater region of the Kaidu River during 1972–2011, the multi-scale characteristics of runoff variability and four climatic factor fluctuations (i.e. temperature, precipitation, relative humidity and evaporation) are analyzed using detrended fluctuation analysis. Furthermore, multi-scale response of runoff to climate fluctuation is investigated using detrended cross-correlation analysis. Main findings are as follows: (1) The temporal scaling behaviors of runoff and four climate factor series all exhibit two different power laws. In shorter temporal scaling, all the series indicate the similar persistence corresponding to the annual cycle. However, in longer temporal scaling, their different trends reflect the different inherent dynamic nature of various hydro-climatic change. (2) In the double logarithm curve logF2(s) ∼ logs, the long-range correlation of runoff and temperature, long-range correlation of runoff and precipitation and long-range correlation of runoff and relative humidity (hereafter referred to as Lrc-R-T, Lrc-R-P and Lrc-R-H, respectively) show two scaling regimes with two different scale indexes and a critical time scale of about 1 year; the long-range correlation of runoff and evaporation (hereafter referred to as Lrc-R-E) presents three scaling regime with three different scale indexes and with two critical time scales of about 1 and 10 years. These results reflect the multi-scale response characteristics of the runoff to climate change on different time scales.

Previous studies in the literature show that the annual cycle of surface air temperature (SAT) is changing in both amplitude and phase, and the SAT departures from the annual cycle are long-term correlated. However, the classical definition of temperature anomalies is based on the assumption that the annual cycle is constant, which contradicts the fact of changing annual cycle. How to quantify the impact of the changing annual cycle on the long-term correlation of temperature anomaly variability still remains open. In this paper, a recently developed data adaptive analysis tool, the nonlinear mode decomposition (NMD), is used to extract and remove time-varying annual cycle to reach the new defined temperature anomalies in which time-dependent amplitude of annual cycle has been considered. By means of detrended fluctuation analysis, the impact induced by inter-annual variability from the time-dependent amplitude of annual cycle has been quantified on the estimation of long-term correlation of long historical temperature anomalies in Europe. The results show that the classical climatology annual cycle is supposed to lack inter-annual fluctuation which will lead to a maximum artificial deviation centering around 600 days. This maximum artificial deviation is crucial to defining the scaling range and estimating the long-term persistence exponent accurately. Selecting different scaling range could lead to an overestimation or underestimation of the long-term persistence exponent. By using NMD method to extract the inter-annual fluctuations of annual cycle, this artificial crossover can be weakened to extend a wider scaling range with fewer uncertainties.

We detect the asymmetric multi-fractality in the U.S. stock indices based on the asymmetric multi-fractal detrended fluctuation analysis (A-MFDFA). Instead using the conventional return-based approach, we propose the index-based model of A-MFDFA where the trend based on the evolution of stock index rather than stock price return plays a role for evaluating the asymmetric scaling behaviors. The results show that the multi-fractal behaviors of the U.S. stock indices are asymmetric and the index-based model detects the asymmetric multi-fractality better than return-based model. We also discuss the source of multi-fractality and its asymmetry and observe that the multi-fractal asymmetry in the U.S. stock indices has a time-varying feature where the degree of multi-fractality and asymmetry increase during the financial crisis.

Human breathing patterns contain both temporal scaling characteristics, and an innately random component. A stochastic and mathematically integrative model of breathing (SIMB) that simulated the natural random and fractal-like pattern of human breathing was designed using breath-to-breath interval (BBI) data recorded from 14 healthy subjects. Respiratory system memory was estimated with autocorrelation, and a probability density function (PDF) was created by fitting a polynomial curve to each normalized BBI sequence histogram. SIMB sequences were produced by randomly selecting BBI values using a PDF and imparting memory with an autocorrelation-based function. Temporal scaling was quantified with detrended fluctuation analysis. The SIMB BBI sequences were embedded with significant fractal scaling (p0.001) that was similar to the human data (p0.05), and increasing SIMB output length did not alter the temporal scaling (p0.05). This study demonstrated a new computational model that can reproduce the inherent stochastic and time scaling characteristics of human breathing.

Temporal-spatial cross-correlation analysis of non-stationary wind speed time series plays a crucial role in wind field reconstruction as well as in wind pattern recognition. Firstly, the near-surface wind speed time series recorded at different locations are studied using the detrended fluctuation analysis (DFA), and the corresponding scaling exponents are larger than 1. This indicates that all these wind speed time series have non-stationary characteristics. Secondly, concerning this special feature (i.e., non-stationarity) of wind signals, a cross-correlation analysis method, namely detrended cross-correlation analysis (DCCA) coefficient, is employed to evaluate the temporal-spatial cross-correlations between non-stationary time series of different anemometer pairs. Finally, experiments on ten wind speed data synchronously collected by the ten anemometers with equidistant arrangement illustrate that the method of DCCA cross-correlation coefficient can accurately analyze full-scale temporal-spatial cross-correlation between non-stationary time series and also can easily identify the seasonal component, while three traditional cross-correlation techniques (i.e., Pearson coefficient, cross-correlation function, and DCCA method) cannot give us these information directly.

We analyze European temperature variability from station data with the method of detrended fluctuation analysis. This method is known to give a scaling exponent indicating long range correlations in time for temperature anomalies. However, by a more careful look at the fluctuation function we are able to explain the emergent scaling behaviour by short time relaxation, the yearly cycle and one additional process. It turns out that for many stations this interannual variability is an oscillatory mode with a period length of approximately 7-8 years, which is consistent with results of other methods. We discuss the spatial patterns in all parameters and validate the finding of the 7-8 year period by comparing stations with and without this mode.

Using detrended fluctuation analysis (DFA) and rescaled range (R/S) analysis, we investigate the scaling properties of EUV intensity fluctuations of low-latitude coronal holes (CHs) and neighboring quiet-Sun (QS) regions in signals obtained with the Solar Dynamics Observatory/Atmospheric Imaging Assembly (SDO/AIA) instrument. Contemporaneous line-of-sight SDO/Helioseismic and Magnetic Imager (HMI) magnetic fields provide a context for the physical environment. We find that the intensity fluctuations in the time series of EUV images present at each spatial point a scaling symmetry over the range $\sim 20$ min to $\sim$ 1 hour. Thus we are able to calculate a generalized Hurst exponent and produce image maps, not of physical quantities like intensity or temperature, but of a single dynamical parameter that sums up the statistical nature of the intensity fluctuations at each pixel. In quiet-Sun (QS) regions and in coronal holes (CHs) with magnetic bipoles, the scaling exponent ($1.0 < \alpha \leq 1.5$) corresponds to anti-correlated turbulent-like processes. In coronal holes, and in quiet-Sun regions primarily associated with (open) magnetic field of dominant polarity, the generalized exponent (0.5 $< \alpha, Comment: 21 pages, 8 figures

We investigated the quality of smoothness during human unipedal quiet stance. Smoothness is quantified by the time rate of change of the accelerations, or jerks, associated with the motion of the foot and can be seen as an indicative of how controlled the balance process is. To become more acquainted with this as a quantity, we wanted to establish whether or not it can be modeled as a (stationary) stochastic process and, if so, explore its temporal scaling behavior. Specifically, our study focused on the jerk concerning the center-of-pressure (COP) for each foot. Data were collected via a force plate for individuals attempting to maintain upright posture using one leg (with eyes open). Positive tests for stochasticity allowed us to treat the time series as a stochastic process and, given this, we took the jerk to be proportional to the increment of the force realizations. Detrended fluctuation analysis was the primary tool used to explore the scaling behavior. Results suggest that both the medial-lateral and anterior-posterior components of the jerk display persistent and antipersistent correlations which can be modeled by fractional Gaussian noise over three different temporal scaling regions. Finally, we discussed certain possible implications of these features such as a jerk-based control over the force on the foot’s COP.