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 {\alpha} 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

Comparative study of nonlinear properties of EEG signals of a normal person and an epileptic patient

Background: Investigation of the functioning of the brain in living systems has been a major effort amongst scientists and medical practitioners. Amongst the various disorder of the brain, epilepsy has drawn the most attention because this disorder can affect the quality of life of a person. In this paper we have reinvestigated the EEGs for normal and epileptic patients using surrogate analysis, probability distribution function and Hurst exponent. Results: Using random shuffled surrogate analysis, we have obtained some of the nonlinear features that was obtained by Andrzejak \textit{et al.} [Phys Rev E 2001, 64:061907], for the epileptic patients during seizure. Probability distribution function shows that the activity of an epileptic brain is nongaussian in nature. Hurst exponent has been shown to be useful to characterize a normal and an epileptic brain and it shows that the epileptic brain is long term anticorrelated whereas, the normal brain is more or less stochastic. Among all the techniques, used here, Hurst exponent is found very useful for characterization different cases. Conclusions: In this article, differences in characteristics for normal subjects with eyes open and closed, epileptic subjects during seizure and seizure free intervals have been shown mainly using Hurst exponent. The H shows that the brain activity of a normal man is uncorrelated in nature whereas, epileptic brain activity shows long range anticorrelation.Comment: Keywords:EEG, epilepsy, Correlation dimension, Surrogate analysis, Hurst exponent. 9 page

Decomposing Intraday Dependence in Currency Markets: Evidence from the AUD/USD Spot Market

The local Hurst exponent, a measure employed to detect the presence of dependence in a time series, may also be used to investigate the source of intraday variation observed in the returns in foreign exchange markets. Given that changes in the local Hurst exponent may be due to either a time-varying range, or standard deviation, or both of these simultaneously, values for the range, standard deviation and local Hurst exponent are recorded and analyzed separately. To illustrate this approach, a high-frequency data set of the spot Australian dollar/U.S. dollar provides evidence of the returns distribution across the 24-hour trading day with time-varying dependence and volatility clearly aligning with the opening and closing of markets. This variation is attributed to the effects of liquidity and the price-discovery actions of dealers.Comment: 3 Figures, 3 Tables, 28 page

Fractal Heterogeneous Media

A method is proposed for generating compact fractal disordered media, by generalizing the random midpoint displacement algorithm. The obtained structures are invasive stochastic fractals, with the Hurst exponent varying as a continuous parameter, as opposed to lacunar deterministic fractals, such as the Menger sponge. By employing the Detrending Moving Average algorithm [Phys. Rev. E 76, 056703 (2007)], the Hurst exponent of the generated structure can be subsequently checked. The fractality of such a structure is referred to a property defined over a three dimensional topology rather than to the topology itself. Consequently, in this framework, the Hurst exponent should be intended as an estimator of compactness rather than of roughness. Applications can be envisaged for simulating and quantifying complex systems characterized by self-similar heterogeneity across space. For example, exploitation areas range from the design and control of multifunctional self-assembled artificial nano and micro structures, to the analysis and modelling of complex pattern formation in biology, environmental sciences, geomorphological sciences, etc

Long-range correlations and nonstationarity in the Brazilian stock market

We report an empirical study of the Ibovespa index of the Sao Paulo Stock Exchange in which we detect the existence of long-range correlations. To analyze our data we introduce a rescaled variant of the usual Detrended Fluctuation Analysis that allows us to obtain the Hurst exponent through a one-parameter fitting. We also compute a time-dependent Hurst exponent H(t) using three-year moving time windows. In particular, we find that before the launch of the Collor Plan in 1990 the curve H(t) remains, in general, well above 1/2, while afterwards it stays close to 1/2. We thus argue that the structural reforms set off by the Collor Plan has lead to a more efficient stock market in Brazil. We also suggest that the time dependence of the Ibovespa Hurst exponent could be described in terms of a multifractional Brownian motion.Comment: 19 pages with 11 figures, submitted to Physica

Relationship between degree of efficiency and prediction in stock price changes

This study investigates empirically whether the degree of stock market efficiency is related to the prediction power of future price change using the indices of twenty seven stock markets. Efficiency refers to weak-form efficient market hypothesis (EMH) in terms of the information of past price changes. The prediction power corresponds to the hit-rate, which is the rate of the consistency between the direction of actual price change and that of predicted one, calculated by the nearest neighbor prediction method (NN method) using the out-of-sample. In this manuscript, the Hurst exponent and the approximate entropy (ApEn) are used as the quantitative measurements of the degree of efficiency. The relationship between the Hurst exponent, reflecting the various time correlation property, and the ApEn value, reflecting the randomness in the time series, shows negative correlation. However, the average prediction power on the direction of future price change has the strongly positive correlation with the Hurst exponent, and the negative correlation with the ApEn. Therefore, the market index with less market efficiency has higher prediction power for future price change than one with higher market efficiency when we analyze the market using the past price change pattern. Furthermore, we show that the Hurst exponent, a measurement of the long-term memory property, provides more significant information in terms of prediction of future price changes than the ApEn and the NN method.Comment: 10 page

Distinguishing fractional and white noise in one and two dimensions

We discuss the link between uncorrelated noise and Hurst exponent for one and two-dimensional interfaces. We show that long range correlations cannot be observed using one-dimensional cuts through two-dimensional self-affine surfaces whose height distributions are characterized by a Hurst exponent lower than -1/2. In this domain, fractional and white noise are not distinguishable. A method analysing the correlations in two dimensions is necessary. For Hurst exponents larger than -1/2, a crossover regime leads to a systematic over estimate of the Hurst exponent.Comment: 3 pages RevTeX, 4 Postscript figure

Can One Make Any Crash Prediction in Finance Using the Local Hurst Exponent Idea?

We apply the Hurst exponent idea for investigation of DJIA index time-series data. The behavior of the local Hurst exponent prior to drastic changes in financial series signal is analyzed. The optimal length of the time-window over which this exponent can be calculated in order to make some meaningful predictions is discussed. Our prediction hypothesis is verified with examples of '29 and '87 crashes, as well as with more recent phenomena in stock market from the period 1995-2003.Some interesting agreements are found.Comment: LaTeX 2e, 7 figures (included), 17 page