Hurst Exponent Estimation Research Articles

Comments on "PCA Based Hurst Exponent Estimator for fBm Signals Under Disturbances

In this note we indicate a mistake in the paper mentioned in the title (Li et al. (2009)). It is observed that the cited paper uses an expansion of the fractional Brownian motion which is not the Karhunen-Loeve expansion, and hence the asymptotic behaviour of the constants appearing in the expansion is not related to the asymptotic behaviour of the Karhunen-Loeve eigenvalues.

Dlouhá paměť a její vývoj ve výnosech burzovního indexu PX v letech 1997-2009

Long-term memory processes have been extensively examined in recent literature as they provide simple way to test for predictabilty in the underlying process. However, most of the literature interprets the results of estimated Hurst exponent simply by its comparison to its asymptotic limit of 0.5. Therefore, we use moving block bootstrap method for rescaled range and periodogram method. In our analysis of evolution of Hurst exponent between 1997 and 2009, we show that PX experienced persistent behavior which weakened in time. Nevertheless, the returns of PX remain close to confidence interval separating independent and persistent behavior.

On Hurst exponent estimation under heavy-tailed distributions

In this paper, we show how the sampling properties of the Hurst exponent methods of estimation change with the presence of heavy tails. We run extensive Monte Carlo simulations to find out how rescaled range analysis ( R / S ), multifractal detrended fluctuation analysis ( M F - D F A ), detrending moving average ( D M A ) and generalized Hurst exponent approach ( G H E ) estimate Hurst exponent on independent series with different heavy tails. For this purpose, we generate independent random series from stable distribution with stability exponent α changing from 1.1 (heaviest tails) to 2 (Gaussian normal distribution) and we estimate the Hurst exponent using the different methods. R / S and G H E prove to be robust to heavy tails in the underlying process. G H E provides the lowest variance and bias in comparison to the other methods regardless the presence of heavy tails in data and sample size. Utilizing this result, we apply a novel approach of the intraday time-dependent Hurst exponent and we estimate the Hurst exponent on high frequency data for each trading day separately. We obtain Hurst exponents for S&P500 index for the period beginning with year 1983 and ending by November 2009 and we discuss the surprising result which uncovers how the market’s behavior changed over this long period.

تطبيق تحليل الانتظام على بيانات صوتية مسجلة في بئر: دراسة حالة من بئر KTB

Borehole logs provide in situ information about the fluctuations of petrophysical properties with depth and thus allow the characterization of the crustal heterogeneities. A detailed investigation of these measurements may lead to extract features of the geological media. In this study, we suggest a regularity analysis based on the continuous wavelet transform to examine sonic logs data. The description of the local behavior of the logs at each depth is carried out using the local Hurst exponent estimated by two (02) approaches: the local wavelet approach and the average–local wavelet approach. Firstly, a synthetic log, generated using the random midpoints displacement algorithm, is processed by the regularity analysis. The obtained Hurst curves allowed the discernment of the different layers composing the simulated geological model. Next, this analysis is extended to real sonic logs data recorded at the Kontinentales Tiefbohrprogramm (KTB) pilot borehole (Continental Deep Drilling Program, Germany). The results show a significant correlation between the estimated Hurst exponents and the lithological discontinuities crossed by the well. Hence, the Hurst exponent can be used as a tool to characterize underground heterogeneities.

The use of fractal geometry to determine the impact of inlet migration on the dynamics of a seagrass landscape

The use of fractal geometry to evaluate seagrass scaling behavior and the persistence of seagrass landscape patterns in relation to a disturbance is presented in this paper. Ria Formosa is a dynamic barrier-island system with a migrating inlet that creates a cyclic disturbance in a seagrass landscape. Seagrass patches which develop in the intertidal and shallow subtidal areas of Ria Formosa were digitized from a temporal sequence of aerial photographs, from 1980 to 1998. The methodology used to evaluate seagrass scaling behavior was proposed by Meltzer and Hastings (1992), and relates the frequency distribution of patch size with the existence of patch size-related patterns. The Hurst exponent was calculated to assess the temporal persistence of the seagrass landscape. Univariate regression was used to investigate relations between temporal persistence and disturbance. The existence of patch size-related patterns was identified for all years suggesting shifts in generating processes occurring at different domains of scales in the seagrass landscape. The results enforces the idea that it is important to recognize the existence of diverse processes occurring at different domains of scales and, emphasizes the importance of evaluating issues of temporal and spatial scale while trying to understand changes in seagrass landscapes. The Hurst exponent estimates show that although the migration and relocation of the inlet affected this system the evolutionary trajectory of the seagrass landscape is persistent, i.e., the patch dynamics observed is stable. Furthermore, persistence values were different for differently sized patches, small patches having lower persistence then larger patches.

PCA Based Hurst Exponent Estimator for fBm Signals Under Disturbances

In this paper, the validity of PCA eigenspectrum based Hurst exponent estimator proposed in[J. B. Gao, Y. Cao, and J.-M. Lee, ldquoPrincipal Component Analysis of 1/ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">alpha</sup> noise,rdquo <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Phys.</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Lett.</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</i> , vol. 314, no. 5-6, pp. 392-400, 2003] for single fBm signal is proved. Moreover, how to apply this estimator for fBm signals corrupted with some other signals are discussed. Theoretical analysis and experiments show that it can also be used for 1) mixed fBm signals with different Hurst exponents, 2) fBm signals corrupted with additive Gaussian white noise when the signal-to-noise ratio (SNR) is not too small, and 3) fBm signals corrupted with additive deterministic sine/cosine signals. However, the estimation accuracy depends on the SNR value for the first two situations.

Statistical persistence analysis of hourly ground level ozone concentrations in Delhi

Persistence in hourly ground level ozone concentration observed during 2006 at two sites in Delhi is analyzed using rescaled range analysis based on Hurst exponent estimation. The analysis is carried out separately for whole data series and monthly time series to investigate the seasonal effects on the results of the analysis. High intermittence in the original ozone concentrations at traffic site and in the differenced ozone concentration at both traffic and mixed-use site is observed. The results showed break in the scaling behavior with persistence up to 5 days in the ozone concentration over 2006. The differencing however purged the persistence property. For the monthly observations, persistence is observed up to 2 days. In the monthly time series also, differencing resulted in low persistence as compared to original one during all the months at two sites except during July and August at traffic site, where random behavior is observed. This is mainly attributed to high and intermittent concentration of ozone due to high VOC emissions. Except these months, the behavior of ozone pollution in the observed period is observed to be causal and governed by the temporal dependence on the previous ozone levels and atmospheric processes. The results indicated the differencing has significant effect on the persistence property of the time series.

Fewer jumps, less memory: Homogenized temperature records and long memory

Air temperature records are commonly subjected to inhomogeneities, e.g., sudden jumps caused by a relocation of the measurement station or by installing a new type of shelter. We study the effect of these inhomogeneities on the estimation of the Hurst exponent and show that they bias the estimates toward larger values. The Hurst exponent is a parameter to measure long‐range dependence (LRD), which is a characteristic frequently used to describe the natural variability of temperature records. Analyzing a set of temperature time series before and after homogenization with respect to LRD, we find that the average Hurst exponent is clearly reduced for the homogenized series. To test whether (1) jumps cause this positive bias and (2) the homogenization does not artificially reduce the Hurst exponent estimates, we perform a simulation study. This test shows that inhomogeneities in the form of jumps bias the Hurst exponent estimation and that the homogenization procedure is able to remove this bias, leaving the Hurst exponent unchanged. This result holds for fractional autoregressive integrated moving average (FARIMA)‐based as well as for detrended fluctuation analysis‐based estimation. We conclude that the use of homogenized series is necessary to prevent misleading conclusions about the dependence structure and thus about subsequent analysis such as trend tests.

Blocks adjustment—reduction of bias and variance of detrended fluctuation analysis using Monte Carlo simulation

The length of minimal and maximal blocks equally distant on log–log scale versus fluctuation function considerably influences bias and variance of DFA. Through a number of extensive Monte Carlo simulations and different fractional Brownian motion/fractional Gaussian noise generators, we found the pair of minimal and maximal blocks that minimizes the sum of mean-squared error of estimated Hurst exponents for the series of length N = 2 p , p = 7 , … , 15 . Sensitivity of DFA to sort-range correlations was examined using ARFIMA( p , d , q ) generator. Due to the bias of the estimator for anti-persistent processes, we narrowed down the range of Hurst exponent to 1 2 ⩽ H < 1 .

The sampling properties of Hurst exponent estimates

The classical rescaled adjusted range ( R/ S) statistic is a popular and robust tool for identifying fractal structures and long-term dependence in time-series data. Subsequent to Mandelbrot and Wallis [Water Resour. Res. 4 (1968) 909] who proposed the statistic be measured over several subseries contained within the whole series length, the use overlapping vs. contiguous subseries has been a source of some debate amongst R/S theorists. This study examines the distributional characteristics of rescaled adjusted range and Hurst exponent estimates derived using overlapping vs. contiguous subseries, henceforth closing debate on the issue of relative bias due to either technique. Confidence intervals for the statistical significance of the Hurst exponent are also determined.

Multifractal character of surface latent heat flux

Surface latent heat flux (SLHF) has been associated with the study of natural hazards, such as earthquakes and hurricanes and has been proposed as a useful quantity in the prediction and monitoring of their evolution. In the present study the Mediterranean Sea, parts of Europe, Africa and Turkey are mapped with regard to the multifractal characteristics of SLHF time series during a period of eight years (1997–2004). The estimated Hurst exponents are markedly larger over land than over sea. In the case of land, SLHF has the characteristics of a mean-averting process, while its records are over sea noticeably uncorrelated. In contrast to the rather monofractal and weak multifractal character observed in most regions, with the application of a detrended fluctuation analysis, intense multifractality is seen mainly in North Africa. Crossover segments are present in the scaling of negative moments, implying that the small SLHF fluctuations are affected by seasonal components. Identification of anomalous SLHF deviations from their long-term multifractal behavior may serve as a precursor of extreme atmospheric phenomena.

Testing for predictability in equity returns for European transition markets

This paper presents empirical evidence of short and long-run predictability in stock returns for European transition economies. We employ variance ratios with a bootstrap methodology to test for short-run predictability, which is present in most countries. We also estimate Hurst exponents to test for long-range dependence, and find evidence of such. Furthermore, we find evidence of strong time-varying long-range dependence in these economies stock returns, which is in line with evidence of multifractality of equity returns.

On fast generation of fractional Gaussian noise

A fast method of generating fractional Gaussian noise processes based on a power spectrum function is presented. Algorithms based on the fast Fourier transform reveal low computational complexity and good performance when compared to other methods of generating self-similar processes. These methods have good accuracy in that the results of estimation of Hurst exponent do not differ significantly from expected values. The aim is to improve the efficiency of Paxson's and other recent methods while keeping a high level of accuracy. The results of numerical simulations are presented and compared from the point of view of accuracy and running time. Furthermore, a new way of exact approximation of power spectrum based on the generalized Riemann zeta function is introduced. Subsequently, this method is used for accuracy evaluation of aforementioned generators.

허스트 지수 산정 방법에 대한 고찰

허스트 지수를 산정하기 위하여 기존에 여러 방법론들이 제안되어 왔다. 그러나, 이들 방법론들은 시계열들의 지속성에 대하여 각기 다른 특성들을 보이고 있음을 기존의 연구에서 알 수 있다 따라서 본 연구에서는 수문학에서 주로 이용하고 있는 보정용량, 조정용량, 수정조정용량 방법 이외에 생리학 분야와 전자 분야 등에서 이용되고 있는 1/f 파워 스펙트럼 밀도 분석, DFA, AVT 방법, 최우도법 등을 이용하여 허스트 지수를 산정하여 보았다. 즉, 단기간과 장기간 기억을 가진 카오스와 추계학적 시계열들에 대하여 각각의 방법들을 적용하여 비교 분석하고자 하였으며, 각 방법론들에 대한 장점 및 단점 그리고 한계에 대하여 논의하였다. There are many different techniques for the estimation of the Hurst exponent. However, the techniques can produce different characteristics for the persistence of a time series each other. This study uses several techniques such as adjusted range, resealed range(RR) analysis, modified restated range(MRR) analysis, 1/f power spectral density analysis, Maximum Likelihood Estimation(MLE), detrended fluctuations analysis(DFA), and aggregated variance time(AVT)method for the Hurst exponent estimation. The generated time series from chaos and stochastic systems are analyzed for the comparative study of the techniques. Then this study discusses the advantages and disadvantages of the techniques and also the limitations of them.

Evidence of Crossover Phenomena in Wind-Speed Data

In this paper, a systematic analysis of hourly wind-speed data obtained from three potential wind-generation sites (in North Dakota) is analyzed. The power spectra of the data exhibited a power-law decay characteristic of 1/f/sup /spl alpha// processes with possible long-range correlations. Conventional analysis using Hurst exponent estimators proved to be inconclusive. Subsequent analysis using detrended fluctuation analysis revealed a crossover in the scaling exponent (/spl alpha/). At short time scales, a scaling exponent of /spl alpha//spl sim/1.4 indicated that the data resembled Brownian noise, whereas for larger time scales the data exhibited long-range correlations (/spl alpha//spl sim/0.7). The scaling exponents obtained were similar across the three locations. Our findings suggest the possibility of multiple scaling exponents characteristic of multifractal signals.

Ranking efficiency for emerging equity markets II☆

Abstract This paper employs a “rolling sample” approach to estimate Hurst exponents for emerging markets squared and absolute returns. The findings suggests that these markets possess strong long-range dependence in volatility. Empirical results suggest that Asian equity markets are more efficient than those of Latin America and that the US is the most efficient country.

Rescaled-range and power spectrum analyses on well-logging data

SUMMARY Fractal theory and related developments have significantly enlightened our understanding of many natural phenomena and led to an upsurge in research efforts in many different disciplines. This has unfortunately caused some ambiguities, misunderstandings, and even misuses in the published literature concerning some new concepts in fractal theory and their actual implementations. As new applications and ideas arise almost daily, it is important to present the methods in a common framework to minimize further confusion. In this paper, we scrutinize and compare two of the most frequently applied methods for characterizing fractal time-series, namely the rescaled-range (R/S) and the power spectrum method, and describe their proper applications in wire line well-logging data. We outline the problems related to applications of these two methods. Based on numerical analysis on fractional Gaussian noise (fGn) and fractional Brownian motion (fBm), we demonstrate the clear necessity to distinguish between fGn-like and fBm-like data series. For fBm-like data series, the use of their successive increments is proposed against using various other alternatives in R/S analysis in order to make the estimated Hurst exponent (Hu) comparable with the global scaling exponent (H) from power spectrum analysis. We argue that well-logging data are generally analogous to fBm, and thus we use their incremental series rather than raw well data themselves to estimate Hu when applying R/S. We also reveal a connection between the transient zone length in the R/S technique and the shortest-wavelength that can be included for a linear regression in the power spectrum method.

Fractional Brownian Motions, Fractional Noises and Applications

Fractional Brownian Motions, Fractional Noises and Applications