Detrended Fluctuation Analysis

In this paper, we investigate the multifractal properties of MASI and MADEX logarithmic returns using the multifractal detrended fluctuation analysis MF-DFA. We first calculate the fluctuation functions from which we estimate the generalized Hurst exponents. Then we deduce the Rényi exponents and the singularity spectrum of the two indices. Moreover, in order to detect the sources contributing to the multifractality, we perform the shuffling and phase-randomization (surrogate) techniques on the originate time series. Our results suggest that there are two principal sources of multifractality for MASI and MADEX indices: the long-range temporal correlations and the fat-tail distribution. By observing the curves representing the generalized Hurst exponents, the Rényi exponents as well as the singularity spectrum, we conclude that the long-range temporal correlations contributes mainly to the multifractality of MASI time series than do the fat-tail distribution. However, the two sources contribute almost equally to the multifractality of the MADEX time series with a slight dominance for the first source. Finally, the comparison of the multifractal behavior of MASI and MADEX data leads us to conclude that the first data reveals a richer multifractality feature than the second one. This result can be perhaps explained by the fact that the dynamics of the financial market is more complex by considering the MASI index which includes all the shares of the Casablanca Stock Exchange. This study leads to the principal conclusion that the Casablanca Stock Exchange is characterized by a multifractal behavior.

applied to process the images of the objects that are locally irregular [17]. To overcome this difficulty, several multifrac- tal analysis (MFA) methods were proposed [18-22]. For example, Backes et al. [18,19] used multi-scale fractal di- mensions to describe the texture property of leaf’ s surface to identify plants, which turned out to be very efficient. Note that the classical MFA is based on capacity measure- ment or probability measurement and thus describes only stationary measurements [17]. For a leaf image, the surface itself is hardly stationary. Therefore, the multifractal detrended fluctuation analysis (MF-DFA) method that can deal with non-stationary is a desirable method for leaf image analysis [23]. Though the MF-DFA method has been successfully applied in many fields for non-stationary series and surfaces [24-30], to the best of our knowledge, no work yet has applied the MF-DFA on leaf images for plant identi- fication and classification. In this paper, we attempt to iden- tify plant species via leaf images by using the MF-DFA. More precisely, we first adopt the MF-DFA to extract im- portant texture features from leaf images and obtain several key multifractal parameters, and then we apply the support vector machines and kernel methods (SVMKM) to distin- guish leaves from different plant species. The widely used Swedish leaf data set [31] containing leaves from fifteen

Most of sleep disorders are diagnosed based on the sleep scoring and assessments. The purpose of this study is to combine detrended fluctuation analysis features and spectral features of single electroencephalograph (EEG) channel for the purpose of building an automated sleep staging sys- tem based on the hybrid prediction engine model. The testing results of the model were promising as the classification accuracies were 98.85%, 92.26%, 94.4%, 95.16% and 93.68% for the wake, non-rapid eye movement S1, non-rapid eye movement S2, non-rapid eye movement S3 and rapid eye movement sleep stages, respectively. The overall classification accuracy was 85.18%. We con- cluded that it might be possible to employ this approach to build an industrial sleep assessment system that reduced the number of channels that affected the sleep quality and the effort excreted by sleep specialists through the process of the sleep scoring.

Epilepsy is a medical condition that produces seizures affecting a variety of mental and physical functions. Seizures can last from a few seconds to a few minutes. They can have many symptoms, from convulsions and loss of consciousness to blank staring, lip smacking, or jerking movements of arms and legs. If early warning signals of an upcoming seizure (diag- nosis of preictal period) are detected, proper treat- ment can be applied to the patient to help prevent the seizure. In this research, an epileptic disorder has been divided into three subsets: Normal, Preictal (just before the seizure), and Ictal (during seizure). By using Detrended Fluctuation Analysis (DFA), Bis- pectral Analysis (BIS), and Standard Deviation (SD) three features from single-channel EEG signals have been derived in the foresaid groups. A fuzzy classifier is used to separate the three groups which can suc- cessfully separate them with a separation degree of 100% and further a fuzzy inference engine is used to extract a Seizure Intensity Index (SII) from the Elec- troencephalogram (EEG) signals of the three differ- ent states. One can apparently see the distinction of SII amounts between the three states. It is more im- portant when one remembers that these results are just from single-channel EEG signal.

Abstract: The vibration signal of heavy gearbox presents non-stationary and nonlinear characteristics, which increases the difficulty to extract the fault feature. When the gear has a subtle fault, it may cause a perceptible change of local fluctuation rather than the large scale fluctuation. Therefore, the feature parameters extracted from local fluctuation can effectively improve the recognition performance of the gear fault. In this paper, a novel signal processing method based on variational mode decomposition (VMD) and detrended fluctuation analysis (DFA) is proposed to identify the gear fault of heavy gearbox. Firstly, the raw vibration signal is decomposed several mode components by VMD, which is an adaptive and non-recursive signal decomposition method. Next, the sensitive mode component is selected by a maximal indicator, which is composed of kurtosis and correlation coefficient of relative higher frequency mode components corresponding to local fluctuation of raw vibration signal. Finally, the characteristics of the double-scales feature parameters of selected sensitive mode are extracted by DFA. In addition, the position of turning point of double scales is estimated by sliding windowing algorithm. The proposed method is evaluated through its application to gear fault classification using vibration signal. The results demonstrates that the recognization rate of gear faults condition have marked improvement by proposed method than the DFA of Small Time Scale (STS-DFA) method.

Backgrounds: Patients with rheumatoid arthritis (RA) have increased risk of sudden cardiac death (SCD), which is two-fold higher than general population. The driving cause of SCD was considered due to lift-threatening arrhythmia where systemic inflammation acts as the pathophysiological basis linking RA to autonomicdysfunction. Methods: To assess the sympathetic over-activity of “ inflammatory reflex ” , we measured heart rate variability (HRV) in a rat collagen-induced arthritis (CIA) model, whose arthritis is induced in Lewis rats by intradermal injection of emulsion of type II collagen. Single-lead electrocardiogram (ECG) was recorded for 30 min every two days. Time and frequency-domain parameters, detrended fluctuation analysis (DFA), deceleration (DC) and acceleration capacity (AC) were analyzed.

Methods: PD participants fitted with an actigraph took either YXQN or placebo granules in a randomized manner for 12 weeks while maintaining other anti-parkinsonism medications (e.g., dopaminergic agent, dopamine agonist) unchanged. Additional participants without sleep disturbance or PD served as controls. The changes in detrended fluctuation analysis (DFA) of physical activity with respect to diurnal activity (DA), evening activity (EA), nocturnal activity (NA), Parkinson ’ s disease sleep scale (PDSS) score and unified Parkinson ’ s disease rating scale (UPDRS) score were evaluated every 4 weeks during the 12-week YXQN intervention period and again at week 16.

Abstract. Knowledge about the scaling properties of soil water storage is crucial in transferring locally measured fluc- tuations to larger scales and vice-versa. Studies based on remotely sensed data have shown that the variability in sur- face soil water has clear scaling properties (i.e., statistically self similar) over a wider range of spatial scales. However, the scaling property of soil water storage to a certain depth at a field scale is not well understood. The major challenges in scaling analysis for soil water are the presence of localized trends and nonstationarities in the spatial series. The objec- tive of this study was to characterize scaling properties of soil water storage variability through multifractal detrended fluctuation analysis (MFDFA). A field experiment was con- ducted in a sub-humid climate at Alvena, Saskatchewan, Canada. A north-south transect of 624-m long was estab- lished on a rolling landscape. Soil water storage was moni- tored weekly between 2002 and 2005 at 104 locations along the transect. The spatial scaling property of the surface 0 to 40 cm depth was characterized using the MFDFA technique for six of the soil water content series (all gravimetrically determined) representing soil water storage after snowmelt, rainfall, and evapotranspiration. For the studied transect, scaling properties of soil water storage are different between drier periods and wet periods. It also appears that local con- trols such as site topography and texture (that dominantly control the pattern during wet states) results in multiscaling property. The nonlocal controls such as evapotranspiration results in the reduction of the degree of multiscaling and im- provement in the simple scaling. Therefore, the scaling prop- erty of soil water storage is a function of both soil moisture status and the spatial extent considered.

statistical models. The degree of clustering can be mea- sured, and appropriate statistical models for clustered data must account for the degree of clustering/non-inde- pendence [34]. Generalized linear mixed models (GLMM) can account for lack of independence within clustered data by assigning each cluster of repeated mea- sures its own intercept, the whole of which are con- strained to be normally distributed. In this way, the proportion of correlation due to clustering can be mea- sured, allowing identification of reliable parameter esti- mates [35]. We implemented univariate, multilevel logistic regression with random intercepts, predicting the probability of a vasopressor up-titration being suc- cessful. Complexity measures were centered and scaled to facilitate convergence of regression models. Using bootstrapping to estimate power, we calculated that with a sample size of 95, we had 80 % power with two-tailed alpha = 0.05 to detect a 0.27 absolute difference in the ratio of detrended fluctuation analysis (DFA) exponents between successful and unsuccessful vasopressor up- titrations.

Abstract—Using artificial intelligence algorithms, providers of news analytics calculate the sentiment score of almost every economic and financial news in real time. The sentiment score of negative, neutral, positive news are assigned to be -1, 0, 1, respectively. We constructed time series of news sentiments as follows: a nine-month period of 2015 was divided into non- overlapping consecutive intervals of equal length, and then we calculated the sum of sentiment scores of all news within each time interval. In this paper we examine long-range dependance and self-similarity of time series of sentiments of economic and financial news using the Detrended Fluctuation Analysis of order 1 (DFA), Rescaled Range Analysis (R/S), Average Wavelet Coefficient Method (AWC) and Fourier Transform Method (FTM). Empirical results obtained by this methods show that time series of news sentiments exhibit self-similarity (as well as a long memory property). The Hurst exponent (as well as the long-range correlation exponent) is greater than 0.55 over four orders of magnitude in time ranging from several minutes to dozen of days. DFA and AWC methods allowed us to reveal a strong scaling behavior as well as to detect a distinct crossover effect. On the other hand, it turns out that for the classic R/S analysis and Fourier transform techniques, the scaling regimes and/or positions of crossovers are hard to define.

The presence of human-made explosions in a seismic catalog leads to errors in statistical analyses of seismicity. We proposed the Hurst estimator by the Smoothed Detrended Fluctuation Analysis method (SDFA) like an additional classification tool to discriminate between natural tectonic earthquakes and human-made explosions.

Subcutaneous glucose profiles are characterized by a strong non-stationarity, which limits the application of correlation-spectral analysis. We derived an index of linear predictability by calculating the autocorrelation function of time series increments and applied detrended fluctuation analysis to assess the non-stationarity of the profiles. Time series from volunteers with both type 1 and type 2 diabetes and from control subjects were analysed. The results suggest that in control subjects, blood glucose variation is relatively uncorrelated, and this variation could be modelled as a random walk with no retention of ‘memory’ of previous values. In diabetes, variation is both greater and smoother, with retention of inter-dependence between neighboring values. Essential components for adequate longer term prediction were identified via a decomposition of time series into a slow trend and responses to external stimuli. Implications for diabetes management are discussed.

The traditional multifractal analysis has been developed for the multifractal characterisation of normalised, stationary time series. This standard formalism does not give cor- rect results for non-stationary time series which are affected by trends. Multifractal detrended fluctuation analysis (MF- DFA), which is a generalisation of the standard detrended fluctuation analysis (DFA), is based on the identification of the scaling of the q th-order moments of the time series, which may be non-stationary (Kantelhardt et al., 2002). DFA has been used to study the classification problem of protein secondary structures (Yu et al., 2006). Movahed et al. (2006) used the MF-DFA to study sunspot fluctuations.

nonlinearly combined; (2) to try to improve this performance with the inclusion of additional information and processing techniques. We analyzed a database of 727 consecutive cases of tilt test. Previously proposed features were measured from heart rate and systolic/diastolic pressure tachograms, in several representative signal segments. We aimed to improve the prediction performance: first, using new nonlinear features (detrended fluctuation analysis and sample entropy); second, using a multivariable nonlinear classifier (support vector machine); and finally, including additional physiological signals (stroke volume). The predictive performance of the nonlinearly combined previously proposed features was limited [area under receiver operating characteristic curve (ROC) 0.57 ± 0.12], especially at the beginning of the test, which is the most clinically relevant period. The improvement with additional available physiological information was limited too. We conclude that the use of a system for tilt test outcome prediction with current knowledge and processing should be considered with caution, and that further effort has to be devoted to understand the mechanisms of VVS.

However, the estimates for pure Gaussian process can strongly deviate from the limit of 0.5 (Weron, 2002; and Couillard & Davison, 2005). Moreover, the estimates are influenced by choice of minimum and maximum scale (Weron, 2002). There have been several papers dealing with finite sample properties of estimators of Hurst exponent (Peters, 1994; Couillard & Davison, 2005; Grech & Mazur, 2005; and Weron, 2002). However, none of the papers use the proposition for optimal scales presented elsewhere (Grech & Mazur, 2004; Matos et al., 2008; Alvarez-Ramirez, Rodriguez & Echeverria, 2005; and Einstein, Wu & Gil, 2001). This paper attempts to fill this gap and presents results of Monte Carlo simulations for two mostly used techniques – rescaled range analysis and detrended fluctuation analysis.

In this study, we analyzed the multifractality and the source of multifractality of the returns of GBP/USD, EUR/USD, USD/JPY and USD/CHF currencies. In the examination of multifractality we performed the Multifractal Detrended Fluctuation Analysis (MF-DFA). Also, we used shuffled and surrogated data that was derived from the Statically Transformed Autoregressive Process (STAP) method to determine the source of multifractality. According to the results, GBP/USD returns have monofractal features, whereas EUR/USD, USD/JPY and USD/CHF returns have multifractal behaviors. The tests concerning the source of multifractality indicated that the reason of multifractality for EUR/USD and USD/JPY returns is fat-tails of the probability density function of returns, whereas the reason of multifractality of USD/CHF returns are both long memory and fat tails. Also we have seen that there is an ambiguous relationship between the liquidity of the currency market and multifractality.

A Modified Detrended Fluctuation Analysis (MDFA I) method is presented in Chapter 5. The DFA algorithm finds a trend on the integrated EEG signals. The integrations on EEG signals are divided in segments of length s. These segments are called box sizes. This trend is iteratively searched on different sizes of boxes over the signals. When the trend is found in each box, it is subtracted from the integrated signals. FFT and filter bank methods are designed to identify anaesthesia states and to separate EEG signals into five components. In MDFA I, the box sizes are reduced gradually from awake state, light anaesthesia, moderate anaesthesia, deep anaesthesia and very deep anaesthesia states. The simulation results demonstrate that the MDFA I can clearly discriminate five above anaesthesia states and the DoA values are close to the BIS values (Nguyen-Ky 2009b).

Abstract. Multifractal fluctuations in the time dynamics of geoelectrical data, recorded in a seismic area of southern Italy, have been revealed using the Multifractal Detrended Fluctuation Analysis (MF-DFA), which allows to detect mul- tifractality in nonstationary signals. Our findings show that the geoelectrical time series, recorded in the seismic area of southern Apennine Chain (Italy), is multifractal. The time evolution of the multifractality suggests that the multifractal degree increases prior the occurrence of earthquakes. This study aims to propose another approach to investigate the complex dynamics of earthquake-related geoelectrical sig- nals.

The measurement of blood volume is normally conducted in vivo using a technique called chamber- plethysmography. In order to conduct the experiment, the technique uses a chamber which makes the measurement need a plenty of space and complex measurement. However, in some cases, a simple technique that provides an immediate and advancement analysis is required. A non-invasive technique is used to obtain the blood volume changes called photoplethysmography (PPG). A PPG measurement device consists of an infra-red light source and a photo-detector. The infra-red light is illuminated through the skin. The penetration depth of light depends on the absorption and scattering coefficients of tissue. Therefore, the use of an infra-red light is preferred since it is more stable over time compared to red light [1]. The PPG instrument can be used in two modes, reflection mode and transmission mode. In the reflection mode, the light source and the photo-detector are placed side by side. The infra-red light is illuminated to the skin and the reflection light from tissue and bones is accepted by a photo-detector. Meanwhile in the transmission mode, the light is illuminated through the skin and detected by a photo- detector, which means that the light source and the photo-detector are facing each other. The light intensity accepted by the photo-detector represents the blood volume and the heart rate [2]. The PPG signal has a fundamental frequency of around 1 Hz. However, it has unequal periods. The PPG signal consists of pulsatile waveforms and DC components. The pulsatile waveforms related to the heart rate and the DC component represent the tissues and the average of blood volume [2]. A considerable number of studies on the processing of PPG signals using statistical and theory of non-linear dynamical analyses have been carried out [3, 4, 5, 6]. In addition, several time series analyses have increasingly been introduced to analyse the PPG signal, including fractal analysis. The fractal analysis provides an ability to measure the pattern behaviour of the time series and serial correlation over time windows [7]. Some of those fractal analyses introduce a Hurst exponent H which indicates the level of autocorrelated properties of the time series. One of these techniques is Detrended Fluctuation Analysis (DFA) [8].

The physical bases for the use of the techniques are the plasma features related to the MC processes. Physical– mathematical techniques have been selected for their ability to allow the investigation of MC occurrences. Those tech- niques have been developed in an original approach to char- acterize MC events in the SW. They consist of techniques of persistence exponents: Hurst, Hausdorff, the beta expo- nent from power-spectral density (Fourier), and the alpha exponent from detrended fluctuation analysis, respectively. Those numerical tools have a great advantage, because they are easy to implement with low computational cost and could be the creation of an automatic operation detection. In addi- tion, they characterize MC regions using (as input data) only the three components of the IMF measured by satellites at convenient space location, e.g., the Lagrangian point L1.