A Monte Carlo simulation to the performance of the R/S and V/S methods—Statistical revisit and real world application

A correct or precise estimation of the Hurst exponent is one of the fundamentally important problems in the financial economics literature. There are three widely used tools to estimate the Hurst exponent, the canonical rescaled range (R/S), the variance rescaled statistic (V/S) and the Modified rescaled range (Modified R/S). To clarify their performance, we compare them by Monte Carlo simulations; we generate many time-series of a fractal Brownian motion, of a Weierstrass–Mandelbrot cosine fractal function and of a fractionally integrated process, whose theoretical Hurst exponents are known, to compare the Hurst exponents estimated by the three methods. To better understand their pragmatic performance, we further apply all of these methods empirically in real-world applications. Our results imply it is not appropriate to conclude simply which method is better as V/S performs better when the analyzed market is anti-persistent while R/S seems to be a reliable tool used in persistent market.     

Hurst exponent and prediction based on weak-form efficient market hypothesis of stock markets

We empirically investigated the relationships between the degree of efficiency and the predictability in financial time-series data. The Hurst exponent was used as the measurement of the degree of efficiency, and the hit rate calculated from the nearest-neighbor prediction method was used for the prediction of the directions of future price changes. We used 60 market indexes of various countries. We empirically discovered that the relationship between the degree of efficiency (the Hurst exponent) and the predictability (the hit rate) is strongly positive. That is, a market index with a higher Hurst exponent tends to have a higher hit rate. These results suggested that the Hurst exponent is useful for predicting future price changes. Furthermore, we also discovered that the Hurst exponent and the hit rate are useful as standards that can distinguish emerging capital markets from mature capital markets.     

Hurst exponent determination for digital speckle patterns in roughness control of metallic surfaces

In this paper, we present a study of metallic surface roughness using the Hurst exponent calculated from speckle pattern. A set of samples was prepared using polishing techniques and the roughness was directly measured by means of an optical profilometer. To study the H exponent, an experiment was performed by illuminating the samples using an expanded laser beam and the surface image was captured by a CCD camera. We applied techniques of the Hurst exponent calculation, traditionally calculated from surface profile, in the digitalized speckle patterns generated by the rough surfaces. We showed a clear dependence of the H exponent on roughness of the samples. We demonstrated that this tool is very sensitive to defects in the surfaces and can be used for roughness control.     

Relation between the Hurst Exponent and the Efficiency of Self-organization of a Deformable System

We have established the degree of self-organization of a system under plastic deformation at different scale levels. Using fractal analysis, we have determined the Hurst exponent and correlation lengths in the region of formation of a corrugated (wrinkled) structure in [111] nickel single crystals under compression. This has made it possible to single out two (micro-and meso-) levels of self-organization in the deformable system. A qualitative relation between the values of the Hurst exponent and the stages of the stress–strain curve has been established.     

Relationship between efficiency and predictability in stock price change

In this study, we evaluate the relationship between efficiency and predictability in the stock market. The efficiency, which is the issue addressed by the weak-form efficient market hypothesis, is calculated using the Hurst exponent and the approximate entropy (ApEn). The predictability corresponds to the hit-rate; this is the rate of consistency between the direction of the actual price change and that of the predicted price change, as calculated via the nearest neighbor prediction method. We determine that the Hurst exponent and the ApEn value are negatively correlated. However, predictability is positively correlated with the Hurst exponent.     

Dynamical generalized Hurst exponent as a tool to monitor unstable periods in financial time series

We investigate the use of the Hurst exponent, dynamically computed over a weighted moving time-window, to evaluate the level of stability/instability of financial firms. Financial firms bailed-out as a consequence of the 2007–2008 credit crisis show a neat increase with time of the generalized Hurst exponent in the period preceding the unfolding of the crisis. Conversely, firms belonging to other market sectors, which suffered the least throughout the crisis, show opposite behaviors. We find that the multifractality of the bailed-out firms increase at the crisis suggesting that the multi fractal properties of the time series are changing. These findings suggest the possibility of using the scaling behavior as a tool to track the level of stability of a firm. In this paper, we introduce a method to compute the generalized Hurst exponent which assigns larger weights to more recent events with respect to older ones. In this way large fluctuations in the remote past are less likely to influence the recent past. We also investigate the scaling associated with the tails of the log-returns distributions and compare this scaling with the scaling associated with the Hurst exponent, observing that the processes underlying the price dynamics of these firms are truly multi-scaling.     

A note on geometric method-based procedures to calculate the Hurst exponent

Geometric method-based procedures, which we will call GM algorithms hereafter, were introduced in M.A. Sánchez-Granero, J.E. Trinidad Segovia, J. García Pérez, Some comments on Hurst exponent and the long memory processes on capital markets, Phys. A 387 (2008) 5543-5551, to calculate the Hurst exponent of a time series. The authors proved that GM algorithms, based on a geometrical approach, are more accurate than classical algorithms, especially with short length time series. The main contribution of this paper is to provide a mathematical background for the validity of these two algorithms to calculate the Hurst exponent H of random processes with stationary and self-affine increments. In particular, we show that these procedures are valid not only for exploring long memory in classical processes such as (fractional) Brownian motions, but also for estimating the Hurst exponent of (fractional) Lévy stable motions.     

Hurst exponent analysis of moving metallic surfaces

We report on the application of Hurst exponent analysis to digital speckle patterns for investigating moving rough surfaces in the presence of defects. Digital speckle patterns were generated by recording the scattered light from moving surfaces illuminated by a laser beam. It was found that it is possible to identify the presence of the defects by means of the variation of the Hurst exponent along the sample.     

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 (MF-DFA), detrending moving average (DMA) and generalized Hurst exponent approach (GHE) 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 GHE prove to be robust to heavy tails in the underlying process. GHE 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.     

Scaling behavior of earthquakes’ inter-events time series

In this paper, we investigate the statistical and scaling properties of the California earthquakes’ inter-events over a period of the recent 40 years. To detect long-term correlations behavior, we apply detrended fluctuation analysis (DFA), which can systematically detect and overcome nonstationarities in the data set at all time scales. We calculate for various earthquakes with magnitudes larger than a given M. The results indicate that the Hurst exponent decreases with increasing M; characterized by a Hurst exponent, which is given by, H = 0:34 + 1:53/M, indicating that for events with very large magnitudes M, the Hurst exponent decreases to 0:50, which is for independent events.      

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.     

Cantor型人工DNA序列的关联属性及输运性质

对按膨胀规律A→ABA和B→BBB生成的有限长度Cantor型人工DNA序列,采用统计方法研究了序列的净位移及其标准偏差、重标极差函数及其Hurst指数,并将结果与一维随机二元序列进行了对比,直接论证了Cantor序列具有关联、标度不变及自相似等性质.从Anderson紧束缚模型出发,采用重整化群方法研究了该序列的电子输运特性.研究表明具有好的输运效率的扩展态能在较宽的能量区间上存在,随着序列长度的增加,扩展态的能量区间变得更为细碎,但具有好的透射性的电子态数量只是略有减少,共振能态可以在较长的序列中存在 关键词： Cantor型人工DNA序列 关联属性 电荷输运效率 Lyapunov指数     

Effects of Different Connectivity Topologies in Small World Networks on EEG-Like Activities

Based on our previously pulse-coupled integrate-and-fire neuron model in small world networks, we investigate the effects of different connectivity topologies on complex behavior of electroencephalographic-like signals produced by this model. We show that several times series analysis methods that are often used for analyzing complex behavior of electroencephalographic-like signals, such as reconstruction of the phase space, correlation dimension, fractal dimension, and the Hurst exponent within the rescaled range analysis (R/S). We find that the different connectivity topologies lead to different dynamical behaviors in models of integrate-and-fire neurons.     

Some comments on Hurst exponent and the long memory processes on capital markets

The analysis of long memory processes in capital markets has been one of the topics in finance, since the existence of the market memory could implicate the rejection of an efficient market hypothesis. The study of these processes in finance is realized through Hurst exponent and the most classical method applied is R/S analysis. In this paper we will discuss the efficiency of this methodology as well as some of its more important modifications to detect the long memory. We also propose the application of a classical geometrical method with short modifications and we compare both approaches.     

Self-affine Multiplicity Fluctuation of Slow Particles in 16O-AgBr Collisions at 3.7 A GeV

We study the slow particles in 160-AgBr collisions at 3.7A GeV in nuclear emulsion with the method of twodimensional factorial moments using the Hurst exponent. Our investigation reveals the power law behaviour, exhibited in self-affine analysis, better than that in self-similar analysis. This work shows a clear evidence of self-affine target fragmentation.     

Size matters: some stylized facts of the stock market revisited

We reanalyze high resolution data from the New York Stock Exchange and find a monotonic (but not power law) variation of the mean value per trade, the mean number of trades per minute and the mean trading activity with company capitalization. We show that the second moment of the traded value distribution is finite. Consequently, the Hurst exponents for the corresponding time series can be calculated. These are, however, non-universal: The persistence grows with larger capitalization and this results in a logarithmically increasing Hurst exponent. A similar trend is displayed by intertrade time intervals. Finally, we demonstrate that the distribution of the intertrade times is better described by a multiscaling ansatz than by simple gap scaling.     

Stock price fluctuations and the mimetic behaviors of traders

We give a stochastic microscopic modelling of stock markets driven by continuous double auction. If we take into account the mimetic behavior of traders, when they place limit order, our virtual market shows the power-law tail of the distribution of returns with the exponent outside the Levy stable region, the short memory of returns and the long memory of volatilities. The Hurst exponent of our model is asymptotically . An explanation is also given for the profile of the autocorrelation function, which is responsible for the value of the Hurst exponent.     

Interacting gaps model,dynamics of order book,and stock-market fluctuations

Inspired by order-book models of financial fluctuations, we investigate the Interacting gaps model, which is the schematic one-dimensional system mimicking the order-book dynamics. We find by simulations the power-law tail in return distribution, power-law decay of volatility autocorrelation with exponent 0.5 and Hurst exponent close to 1/2. Surprisingly, when we make a mean-field approximation, i.e. replace the one-dimensional system by effectively infinite-dimensional one, we obtain analytically the return exponent 5/2, in perfect accord with one-dimensional simulations.     

Time-varying Hurst exponent for US stock markets

In this work, the dynamical behavior of the US stock markets is characterized on the basis of the temporal variations of the Hurst exponent estimated with detrended fluctuation analysis (DFA) over moving windows for the historical Dow Jones (1928-2007) and the S&P-500 (1950-2007) daily indices. According to the results drawn: (i) the Hurst exponent displays an erratic dynamics with some episodes alternating low and high persistent behavior, (ii) the major breakthrough of the long-term trend of the scaling behavior occurred in 1972, at the end of the Bretton Woods system, when the Hurst exponent shifted form a positive to a negative long-term trend. Other effects, such as the 1987 crisis and the emergence of anti-correlated behavior in the recent two years, are also discussed.     

An accurate algorithm to calculate the Hurst exponent of self-similar processes

In this paper, we introduce a new approach which generalizes the GM2 algorithm (introduced in Sánchez-Granero et al. (2008) [52]) as well as fractal dimension algorithms (FD1, FD2 and FD3) (first appeared in Sánchez-Granero et al. (2012) [51]), providing an accurate algorithm to calculate the Hurst exponent of self-similar processes. We prove that this algorithm performs properly in the case of short time series when fractional Brownian motions and Lévy stable motions are considered.We conclude the paper with a dynamic study of the Hurst exponent evolution in the S&P500 index stocks.