Evaluating scaled windowed variance methods for estimating the Hurst coefficient of time series

Three-scaled windowed variance methods (standard, linear regression detrended, and brdge detrended) for estimating the Hurst coefficient (H) are evaluated. The Hurst coefficient, with 0 < H < 1, characterizes self-similar decay in the time-series autocorrelation function. The scaled windowed variance methods estimate H for fractional Brownian motion (fBm) signals which are cumulative sums of fractional Gaussian noise (fGn) signals. For all three methods both the bias and standard deviation of estimates are less than 0.05 for series having N ≥ 2(9) points. Estimates for short series (N < 2(8)) are unreliable. To have a 0.95 probability of distinguishing between two signals with true H differing by 0.1, more than 2(15) points are needed. All three methods proved more reliable (based on bias and variance of estimates) than Hurst's rescaled range analysis, periodogram analysis, and autocorrelation analysis, and as reliable as dispersional analysis. The latter methods can only be applied to fGn or differences of fBm, while the scaled windowed variance methods must be applied to fBm or cumulative sums of fGn.     

Use and misuse of some Hurst parameter estimators applied to stationary and non-stationary financial time series

The detection of long range dependence (LRD) is an important task in time series analysis. LRD is often summarized by the well-known Hurst parameter (or exponent) H∈[0,1], which can be estimated by a number of methods. Some of these techniques are designed to be applied to signals behaving as a stationary fractional Gaussian noise (fGn), whereas others imply that the analyzed time series behave as a non-stationary fractional Brownian motion (fBm). Moreover, some estimators do not yield the Hurst parameter but indexes related to H and ranging outside the unit interval. Therefore, the fGn or fBm nature of the studied time series has to be preliminarily analyzed before applying any estimation method, and the relationships between H and the indexes resulting from the analyses have to be taken into account to obtain coherent results. Since fGn-like series represent the increments of fBm-like processes and both the signals are characterized by the same H value by definition, estimators designed for fGn-like series can be applied to fBm-like sequences after preventive differentiation, and conversely estimators designed for fBm-like processes can be applied to fGn-like series after preventive integration. The signal characterization is particularly important when H is estimated on financial time series because the returns represent the first difference of price time series, which are often assumed to behave like self-affine sequences. The analysis of simulated fGn and fBm time series shows that all the considered methods yield comparable H values when properly applied. The reanalysis of several market price time series already studied in the literature points out that a correct application of the estimators (supported by a preventive signal classification) yields homogeneous H values allowing for a useful cross-validation of results reported in different works. Moreover, some conclusions reported in the literature about the anti-persistence of some financial series are shown to be incorrect because of the inappropriate application of the estimation methods.     

Inhibitory effect induced by fractional Gaussian noise in neuronal system

We discover a phenomenon of inhibition effect induced by fractional Gaussian noise in a neuronal system. Firstly, essential properties of fractional Brownian motion (fBm) and generation of fractional Gaussian noise (fGn) are presented, and representative sample paths of fBm and corresponding spectral density of fGn are discussed at different Hurst indexes. Next, we consider the effect of fGn on neuronal firing, and observe that neuronal firing decreases first and then increases with increasing noise intensity and Hurst index of fGn by studying the time series evolution. To further quantify the inhibitory effect of fGn, by introducing the average discharge rate, we investigate the effects of noise and external current on neuronal firing, and find the occurrence of inhibitory effect about noise intensity and Hurst index of fGn at a certain level of current. Moreover, the inhibition effect is not easy to occur when the noise intensity and Hurst index are too large or too small. In view of opposite action mechanism compared with stochastic resonance, this suppression phenomenon is called inverse stochastic resonance (ISR). Finally, the inhibitory effect induced by fGn is further verified based on the inter-spike intervals (ISIs) in the neuronal system. Our work lays a solid foundation for future study of non-Gaussian-type noise on neuronal systems.     

Analyzing exact fractal time series: evaluating dispersional analysis and rescaled range methods

Precise reference signals are required to evaluate methods for characterizing a fractal time series. Here we use fGp (fractional Gaussian process) to generate exact fractional Gaussian noise (fGn) reference signals for one-dimensional time series. The average autocorrelation of multiple realizations of fGn converges to the theoretically expected autocorrelation. Two methods commonly used to generate fractal time series, an approximate spectral synthesis (SSM) method and the successive random addition (SRA) method, do not give the correct correlation structures and should be abandoned. Time series from fGp were used to test how well several versions of rescaled range analysis (R/S) and dispersional analysis (Disp) estimate the Hurst coefficient (0 < H < 1.0). Disp is unbiased for H < 0.9 and series length N ≥ 1024, but underestimates H when H > 0.9. R/S-detrended overestimates H for time series with H < 0.7 and underestimates H for H > 0.7. Estimates of H(?) from all versions of Disp usually have lower bias and variance than those from R/S. All versions of dispersional analysis, Disp, now tested on fGp, are better than we previously thought and are recommended for evaluating time series as long-memory processes.     

Quantitative analysis of heart rate variability

In the modern industrialized countries every year several hundred thousands of people die due to sudden cardiac death. The individual risk for this sudden cardiac death cannot be defined precisely by common available, noninvasive diagnostic tools like Holter monitoring, highly amplified ECG and traditional linear analysis of heart rate variability (HRV). Therefore, we apply some rather unconventional methods of nonlinear dynamics to analyze the HRV. Especially, some complexity measures that are based on symbolic dynamics as well as a new measure, the renormalized entropy, detect some abnormalities in the HRV of several patients who have been classified in the low risk group by traditional methods. A combination of these complexity measures with the parameters in the frequency domain seems to be a promising way to get a more precise definition of the individual risk. These findings have to be validated by a representative number of patients. (c) 1995 American Institute of Physics.     

The performance of methods based on the fractional Fourier transform for detecting marine mammal vocalizations

The analysis of cetacean vocalizations is considered using Fourier-based techniques that employ chirp functions in their decomposition. In particular, the paper considers a short-time methods based on the fractional Fourier transform for detecting frequency modulated narrow-band signals, such as dolphin whistles, and compares this to the classical short-time Fourier methods. The fractional Fourier technique explored computes transforms associated with a range of chirp rates and automatically selects the rate for the final analysis. This avoids the need for prior knowledge of signal's chirp rate. An analysis is presented that details the performance of both methods as signal detectors and allows one to determine their detection thresholds. These thresholds are then used to measure the detectability of synthetic signals. This principle is then extended to measure performance on a set of recordings of narrow-band vocalizations from a range of cetacean species.     

Investigation of determinism in heart rate variability

The article searches for the possible presence of determinism in heart rate variability (HRV) signals by using a new approach based on NARMA (nonlinear autoregressive moving average) modeling and free-run prediction. Thirty-three 256-point HRV time series obtained from Wistar rats submitted to different autonomic blockade protocols are considered, and a collection of surrogate data sets are generated from each one of them. These surrogate sequences are assumed to be nondeterministic and therefore they may not be predictable. The original HRV time series and related surrogates are submitted to NARMA modeling and prediction. Special attention has been paid to the problem of stationarity. The results consistently show that the surrogate data sets cannot be predicted better than the trivial predictor-the mean-while most of the HRV control sequences are predictable to a certain degree. This suggests that the normal HRV signals have a deterministic signature. The HRV time series derived from the autonomic blockade segments of the experimental protocols do not show the same predictability performance, albeit the physiological interpretation is not obvious. These results have important implications to the methodology of HRV analysis, indicating that techniques from nonlinear dynamics and deterministic chaos may be applied to elicit more information about the autonomic modulation of the cardiovascular activity. (c) 2000 American Institute of Physics.     

Effectiveness of measures of performance during speculative bubbles

Statistical analysis of financial data mostly focused on testing the validity of Brownian motion (Bm). Analyses performed on several time series have shown deviation from the Bm hypothesis, that is at the base of the evaluation of many financial derivatives. We analyze the behavior of performance measures based on maximum drawdown movements (MDD(T)), testing their stability when the underlying process deviates from the Bm hypothesis. In particular we consider the fractional Brownian motion (fBm), and fluctuations estimated empirically on raw market data. The case study of the rising part of speculative bubbles is reported.     

Bispectral Analysis of Heart Rate Variability to Characterize and Help Diagnose Pediatric Sleep Apnea

Pediatric obstructive sleep apnea (OSA) is a breathing disorder that alters heart rate variability (HRV) dynamics during sleep. HRV in children is commonly assessed through conventional spectral analysis. However, bispectral analysis provides both linearity and stationarity information and has not been applied to the assessment of HRV in pediatric OSA. Here, this work aimed to assess HRV using bispectral analysis in children with OSA for signal characterization and diagnostic purposes in two large pediatric databases (0–13 years). The first database (training set) was composed of 981 overnight ECG recordings obtained during polysomnography. The second database (test set) was a subset of the Childhood Adenotonsillectomy Trial database (757 children). We characterized three bispectral regions based on the classic HRV frequency ranges (very low frequency: 0–0.04 Hz; low frequency: 0.04–0.15 Hz; and high frequency: 0.15–0.40 Hz), as well as three OSA-specific frequency ranges obtained in recent studies (BW1: 0.001–0.005 Hz; BW2: 0.028–0.074 Hz; BWRes: a subject-adaptive respiratory region). In each region, up to 14 bispectral features were computed. The fast correlation-based filter was applied to the features obtained from the classic and OSA-specific regions, showing complementary information regarding OSA alterations in HRV. This information was then used to train multi-layer perceptron (MLP) neural networks aimed at automatically detecting pediatric OSA using three clinically defined severity classifiers. Both classic and OSA-specific MLP models showed high and similar accuracy (Acc) and areas under the receiver operating characteristic curve (AUCs) for moderate (classic regions: Acc = 81.0%, AUC = 0.774; OSA-specific regions: Acc = 81.0%, AUC = 0.791) and severe (classic regions: Acc = 91.7%, AUC = 0.847; OSA-specific regions: Acc = 89.3%, AUC = 0.841) OSA levels. Thus, the current findings highlight the usefulness of bispectral analysis on HRV to characterize and diagnose pediatric OSA.     

A hybrid detrending method for fractional Gaussian noise

Determining trend and implementing detrending operations are important steps in data analysis. Yet there is neither precise definition of “trend” nor any logical algorithm for extracting it. In this paper, we propose a Hybrid Detrending Method (HDM) which is based on the Empirical Mode Decomposition (EMD) and the Detrended Fluctuation Analysis (DFA). Our method can remove the polynomial and sinusoidal trends from the fractional Gaussian noise (fGn) background. We illustrate the method by selected examples from artificial time series and climate data. In comparison with existing frequency domain based detrending methods, our method is a posteriori, the trend defined by our method is only derived from the data. Further, our method also preserves the scaling behavior of the original signals.     

Modified diffusion with memory for cyclone track fluctuations

Fluctuations in a time series for tropical cyclone tracks are investigated based on an exponentially modified Brownian motion. The mean square displacement (MSD) is evaluated and compared to a recent work on cyclone tracks based on fractional Brownian motion (fBm). Unlike the work based on fBm, the present approach is found to capture the behavior of MSD versus time graphs for cyclones even for large values of time.     

Dynamical complexity changes during two forms of meditation

Detection of dynamical complexity changes in natural and man-made systems has deep scientific and practical meaning. We use the base-scale entropy method to analyze dynamical complexity changes for heart rate variability (HRV) series during specific traditional forms of Chinese Chi and Kundalini Yoga meditation techniques in healthy young adults. The results show that dynamical complexity decreases in meditation states for two forms of meditation. Meanwhile, we detected changes in probability distribution of m-words during meditation and explained this changes using probability distribution of sine function. The base-scale entropy method may be used on a wider range of physiologic signals.     

Multiscale Entropy Analysis of Short Signals: The Robustness of Fuzzy Entropy-Based Variants Compared to Full-Length Long Signals

Multiscale entropy (MSE) analysis is a fundamental approach to access the complexity of a time series by estimating its information creation over a range of temporal scales. However, MSE may not be accurate or valid for short time series. This is why previous studies applied different kinds of algorithm derivations to short-term time series. However, no study has systematically analyzed and compared their reliabilities. This study compares the MSE algorithm variations adapted to short time series on both human and rat heart rate variability (HRV) time series using long-term MSE as reference. The most used variations of MSE are studied: composite MSE (CMSE), refined composite MSE (RCMSE), modified MSE (MMSE), and their fuzzy versions. We also analyze the errors in MSE estimations for a range of incorporated fuzzy exponents. The results show that fuzzy MSE versions—as a function of time series length—present minimal errors compared to the non-fuzzy algorithms. The traditional multiscale entropy algorithm with fuzzy counting (MFE) has similar accuracy to alternative algorithms with better computing performance. For the best accuracy, the findings suggest different fuzzy exponents according to the time series length.     

Scaling pseudo-Zernike expansion coefficients to different pupil sizes

Orthogonal polynomials are routinely used to represent complex surfaces over a specified domain. In optics, Zernike polynomials have found wide application in optical testing, wavefront sensing, and aberration theory. This set is orthogonal over the continuous unit circle matching the typical shape of optical components and pupils. A variety of techniques has been developed to scale Zernike expansion coefficients to concentric circular subregions to mimic, for example, stopping down the aperture size of an optical system. Here, similar techniques are used to rescale the expansion coefficients to new pupil sizes for a related orthogonal set: the pseudo-Zernike polynomials.     

Multifractal estimates of monofractality in RR-heart series in power spectrum ranges

Two popular estimators of multifractal properties: the Wavelet Transform Modulus Maxima method and Multifractal Detrended Fluctuation Analysis are applied to investigate signals consisting of normal RR-series in 39 healthy subjects and 90 patients suffering from systolic dysfunction of the left ventricle. However, differently from standards for multifractal analysis the scaling is performed separately in intervals corresponding to standard power spectral bands: low (LF), very low (VLF) and ultra low frequencies (ULF). Tests on fractional Brownian motions (fBm) are done to quantify properties of the estimators as detectors of monofractality in LF, VLF and ULF bands. Arguments are given that multifractal analysis of RR-series performed in these bands has a physiological meaning. The increased activation of the sympathetic nervous system caused by heart disease is detected evidently only by analysis in LF. The transition in multifractal characteristics between diurnal and nocturnal activity takes place when the analysis moves from LF and VLF to ULF. Only in ULF, the diurnal heart rate variability can be approximated by fBm with a self-similarity index of H=0.20.     

小波变换的信号分形分析及其在心电信号处理中的应用研究

介绍了一种基于小波变换的分形分析方法,并将它成功地应用于心房纤颤的检测中.该方法提取了一个特征值r,结果表明在心房纤颤开始时,特征值是上升的,心房纤颤结束后,特征值又慢慢恢复到原来的数值,由此可以成功地检测出心房纤颤的开始和结束.此外,由于特征值同时反映了heartratevariability(HRV)信号的复杂程度,所以同时它也表明:心房纤颤开始后,HRV信号的复杂度是下降的 关键词： HRV 小波变换 分形     

Long memory in patterns of mobile phone usage

In this article we show that usage of a mobile phone, i.e. daily series of number of calls made by a customer, exhibits long memory. We use a sample of 4502 postpaid users from a Polish mobile operator and study their two-year billing history. We estimate Hurst exponent by nine estimators: aggregated variance method, differencing the variance, absolute values of the aggregated series, Higuchi’s method, residuals of regression, the R/S method, periodogram method, modified periodogram method and Whittle estimator. We also analyze empirically relations between estimators. Long memory implies an inertial effect in clients’ behavior which may be used by mobile operators to accelerate usage and gain additional profit.     

Estimation of the effective and functional human cortical connectivity with structural equation modeling and directed transfer function applied to high-resolution EEG

Different brain imaging devices are presently available to provide images of the human functional cortical activity, based on hemodynamic, metabolic or electromagnetic measurements. However, static images of brain regions activated during particular tasks do not convey the information of how these regions are interconnected. The concept of brain connectivity plays a central role in the neuroscience, and different definitions of connectivity, functional and effective, have been adopted in literature. While the functional connectivity is defined as the temporal coherence among the activities of different brain areas, the effective connectivity is defined as the simplest brain circuit that would produce the same temporal relationship as observed experimentally among cortical sites. The structural equation modeling (SEM) is the most used method to estimate effective connectivity in neuroscience, and its typical application is on data related to brain hemodynamic behavior tested by functional magnetic resonance imaging (fMRI), whereas the directed transfer function (DTF) method is a frequency-domain approach based on both a multivariate autoregressive (MVAR) modeling of time series and on the concept of Granger causality.

This study presents advanced methods for the estimation of cortical connectivity by applying SEM and DTF on the cortical signals estimated from high-resolution electroencephalography (EEG) recordings, since these signals exhibit a higher spatial resolution than conventional cerebral electromagnetic measures. To estimate correctly the cortical signals, we used a subject's multicompartment head model (scalp, skull, dura mater, cortex) constructed from individual MRI, a distributed source model and a regularized linear inverse source estimates of cortical current density. Before the application of SEM and DTF methodology to the cortical waveforms estimated from high-resolution EEG data, we performed a simulation study, in which different main factors (signal-to-noise ratio, SNR, and simulated cortical activity duration, LENGTH) were systematically manipulated in the generation of test signals, and the errors in the estimated connectivity were evaluated by the analysis of variance (ANOVA). The statistical analysis returned that during simulations, both SEM and DTF estimators were able to correctly estimate the imposed connectivity patterns under reasonable operative conditions, that is, when data exhibit an SNR of at least 3 and a LENGTH of at least 75 s of nonconsecutive EEG recordings at 64 Hz of sampling rate.

Hence, effective and functional connectivity patterns of cortical activity can be effectively estimated under general conditions met in any practical EEG recordings, by combining high-resolution EEG techniques and linear inverse estimation with SEM or DTF methods. We conclude that the estimation of cortical connectivity can be performed not only with hemodynamic measurements, but also with EEG signals treated with advanced computational techniques.     

Inverse statistics and information content

Inverse statistics analysis studies the distribution of investment horizons to achieve a predefined level of return. This distribution provides a maximum investment horizon which determines the most likely horizon for gaining a specific return. There exists a significant difference between inverse statistics of financial market data and a fractional Brownian motion (fBm) as an uncorrelated time-series, which is a suitable criteria to measure information content in financial data. In this paper we perform this analysis for the DJIA and S&P500 as two developed markets and Tehran price index (TEPIX) as an emerging market. We also compare these probability distributions with fBm probability, to detect when the behavior of the stocks are the same as fBm.