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We show that three prominent consumption-based asset pricing models - the Bansal-Yaron, Campbell-Cochrane and Cecchetti-Lam-Mark models - cannot explain the own-history predictability properties of stock market returns. We show this by estimating these models with GMM, deriving ex-ante expected returns from them and then testing whether the difference between realised and expected returns is a martingale difference sequence, which it is not. Furthermore, a semi-parametric test suggests that lagged returns have too much predictive power over current returns to be consistent with the state variables which explain market returns being the same as the state variables which explain market returns in any of the three models.

Some special sampling of discrete scale invariant (DSI) processes are presented to provide a multi-dimensional self-similar process in correspondence. By imposing Markov property we show that the covariance functions of such Markov DSI sequences are characterized by variance, and covariance of adjacent samples in the first scale interval. We also provide a theoretical method for estimating spectral density matrix of corresponding multi-dimensional self-similar Markov process. Some examples such as simple Brownian motion (sBm) with drift and scale invariant autoregressive model are presented and these properties are investigated. We present two new method to estimate Hurst parameter of DSI processes and apply them to some sBm and also to the SP500 indices for some period which has DSI property. We compare our estimates with the maximum-likelihood and rescaled range (R/S) method which are applied to the corresponding multi-dimensional self-similar processes.

Changes in the spatiotemporal patterns of precipitation have great impacts on drought/flood risk and utilization of water resources. In this study, we presented the results of a comparative analysis of spatial-temporal variability of precipitation in the Southwest China. The analysis investigated the trends of annual precipitation and explored the changes of two indices: the precipitation concentration index (PCI) and the concentration index (CI) which are designed for measuring seasonality and daily heterogeneity, respectively. The trends of annual precipitation and CI were tested by the Mann-Kendall method. The results show a significant seasonality of the rainfall distribution and very inhomogeneous temporal distribution of the daily rainfall in the study area. Positive trends in the CI were found at most stations, although most of them were not statistically significant. To detect the futures trends of precipitation in the study area, Hurst’s rescaled range (R/S) analysis was introduced and the corresponding Hurst Exponent was estimated. The results suggested that some drought hazards will happen in the intersection of Sichuan, Guizhou and Yunnan, and the west part of Sichuan, the north part of Chongqing and the middle part of Yunnan are under the risk of flood in the future.

The rescaled range ( R / S ) analysis was used for analyzing the statistical persistence of air pollutants in Mexico City. The air-pollution time series consisted of hourly observations of ozone, nitrogen dioxide, sulfur dioxide and particulate matter obtained at the Mexico City downtown monitoring station during 1999–2014. The results showed that long-range persistence is not a uniform property over a wide range of time scales, from days to months. In fact, although the air pollutant concentrations exhibit an average persistent behavior, environmental (e.g., daily and yearly) and socio-economic (e.g., daily and weekly) cycles are reflected in the dependence of the persistence strength as quantified in terms of the Hurst exponent. It was also found that the Hurst exponent exhibits time variations, with the ozone and nitrate oxide concentrations presenting some regularity, such as annual cycles. The persistence dynamics of the pollutant concentrations increased during the rainy season and decreased during the dry season. The time and scale dependences of the persistence properties provide some insights in the mechanisms involved in the internal dynamics of the Mexico City atmosphere for accumulating and dissipating dangerous air pollutants. While in the short-term individual pollutants dynamics seems to be governed by specific mechanisms, in the long-term (for monthly and higher scales) meteorological and seasonal mechanisms involved in atmospheric recirculation seem to dominate the dynamics of all air pollutant concentrations.

This paper considers the problem of estimation of the Hurst parameter H (1/2, 1) from longitudinal data with the error term of a fractional Brownian motion with Hurst parameter H that gives the amount of the long memory of its increment. We provide a new estimator of Hurst parameter H using a two scale sampling method based on -Sahalia and Jacod (2009). Asymptotic behaviors (consistent and central limit theorem) of the proposed estimator will be investigated. For the proof of a central limit theorem, we use recent results on necessary and sufficient conditions for multi-dimensional vectors of multiple stochastic integrals to converges in distribution to multivariate normal distribution studied by Nourdin et al. (2010), Nualart and Ortiz-Latorre (2008), and Peccati and Tudor (2005).

This work studies long-term autocorrelation asymmetries in the dynamics of crude oil markets for prices in the period from 1986 to 2014. Autocorrelations in crude oil price returns are quantified in terms of the Hurst exponent estimated with the rescaled range ( R / S ) method. The results obtained show that the autocorrelations can be different for price downturns and upturns, and the differences depend on the time scale. For time scales smaller than a quarter, the asymmetry is positive in the sense that negative price trends exhibit less persistence than positive price trends. The opposite behavior was detected for larger time scales. The herding behavior by uninformed (e.g., non-specialized) market participants is discussed as a possible explanation to the autocorrelations asymmetries in crude oil markets.

This study introduced an alternative method in long memory volatility financial time series evaluation using the autoregressive conditional heteroscedasticity models. Instead of direct measurement for long memory Hurst exponent parameter, this approach suggested a short memory filtering procedures using the conditional heteroscedastic specification. The stationary time series which is free from short memory is later evaluated using two heuristic long memory estimations. It is found that this method is able to eliminate the possible spurious long memory in the selected equity and energy markets. The findings of this study are important in long memory estimation and market efficiency analysis.

We study long-range correlations and trends in time series extracted from the data of seismic events occurred from 1973 to 2011 in a rectangular region that contains mainly all the continental part of Colombia. The long-range correlations are detected by the calculation of the Hurst exponents for the time series of interevent intervals, separation distances, depth differences and magnitude differences. By using a modification of the classical R / S method that has been developed to detect short-range correlations in time series, we find the existence of persistence for all the time series considered except for magnitude differences. We find also, by using the D F A until the third order, that the studied time series are not influenced by trends. Additionally, an analysis of the Hurst exponent as a function of the number of events in the time and the maximum window size is presented.

We present a method for estimating a time-scale local Hurst exponent on time series.The method has proven to be sensitive to sudden behavior changes on time series.Lower scales evaluate short-range correlations.Larger scales evaluate long-range correlations.The analysis evaluates pattern changes regardless the amplitude and scale. This paper introduces a method to perform a Time-Scale Local Hurst Exponent (TS-LHE) analysis for time series. The traditional Hurst exponent methods usually analyze time series as a whole, providing a single value that characterizes their global behavior. In contrast, the methods based on the Local Hurst Exponent allow the evaluation of the fractal structure of a time series on local events. However, a critical parameter in these methods is the selection of scale. Here, a TS-LHE method is presented, based on a systematic implementation of the rescaled-range (R/S) method, in a set of sliding windows of different sizes. This method allows calculating instantaneous values of Local Hurst Exponents at different scales, associating them with individual samples of a time series. This paper is organized as follows: first, an overview of the TS-LHE is provided; then, a proof-of-concept of this analysis is presented, considering (a) different fractional Brownian motion series, (b) a synthetic seismic signal under different noise conditions, and (c) a group of real seismic traces. Finally, the obtained results show that the TS-LHE analysis is particularly sensitive to sudden behavior changes of the time series, such as frequency or phase variations. This sensitivity is independent of the amplitude of the data, and thus, it can be used to identify pattern changes as well as long- and short-range correlations within a time series.

The objective of this study was to illustrate the temporal variation of runoff and sediment of loess tilled slopes under successive rainfall conditions. Loess tilled slopes with four microtopography types (straight cultivated slope, artificial backhoe, artificial digging, and contour tillage) under five slope gradients (5°, 10°, 15°, 20°, 25°) were simulated and a rainfall intensity of 60 mm/h was adopted. The temporal trends of runoff and sediment yield were predicted based on the Rescaled Range (R/S) analysis method. The results indicate that the Hurst indices of runoff time series and sediment time series are higher than 0.5, and a long-term positive correlation exists between the future and the past. This means that runoff and sediment of loess tilled slopes in the future will have the same trends as in the past. The results obtained by the classical R/S analysis method were the same as those of the modified R/S analysis method. The rationality and reliability of the R/S analysis method were further identified and the method can be used for predicting the trend of runoff and sediment yield. The correlation between the microtopography and the Hurst indices of the runoff and sediment yield time series, as well as between the slopes and the Hurst indices, were tested, and the result was that there was no significant correlation between them. The microtopography and slopes cannot affect the correlation and continuity of runoff and sediment yield time series. This study provides an effective method for predicting variations in the trends of runoff and sediment yield on loess tilled slopes.

Identification with accuracy of prospective and dry zone from well log data is of prime importance in reservoir or hydrocarbon studies. This issue has greater stake, where in return many conventional methods have been established. The purpose of this study is to recognize the hydrocarbon and non-hydrocarbon bearing portion within a reservoir by using a non-conventional technique. Application of rescaled range (R/S) analysis and wavelet-based fractal analysis (WBFA) on the wire-line log data to obtain the pre-defined hydrocarbon (HC) and non-hydrocarbon (NHC) zones by their fractal nature is demonstrated in this paper. Among these two techniques, the WBFA tool has provided more prolific results. Applicability of the proposed approach is tested with the help of the most commonly used well log data like self-potential, gamma ray, and porosity log responses. These are used in the industry to distinguish several HC and NHC zones of all wells in the study region belonging to the Upper Assam Basin, India. The results are found to be of lower fractal dimension in this study for a particular log response having HC-bearing zones, which are mainly situated in a variety of sandstone lithology. On the other hand, NHC-bearing zones correspond to lithology with higher shale content categorized with higher fractal dimension. Hence, the WBFA technique can overcome the chance of misinterpretation, which is quite possible in the case of conventional reservoir characterization.

The report considers the modified Peters algorithm to calculate the Hurst exponent (HE), the essential properties of HE are proved, an example, analysis of a random time series is given.

International audience; This paper extends the fractional Brownian motion to the complex-valued case. The model is defined as the centered, zero at zero, self-similar complex-valued stochastic process with stationary increments. We present a few properties of this new model and propose an estimation of its main index, the Hurst exponent characterizing the self-similarity property.

International audience; Temporal and spatial analysis of seismicity is performed via the Rescaled Range (R/S) analysis for revealing the hidden characteristics of long memory dependence and clustering between earthquakes. The analysis is applied in two seismogenic units belonging to the extensional Aegean back-arc region, namely the Corinth rift and the Mygdonian graben. The Hurst exponent estimations were used for the interpretation of earthquake collective properties, regarding magnitude, interevent time and interevent epicentral distance for consecutive events. Additional stochastic tools were then engaged for the validation of the results. Τhe analysis outcome is a significant long memory content in the seismic process of both areas, especially for the interevent time of recent micro seismicity and moderate earthquakes in the last decades. This property is not ascertained for the strong (M ≥ 6.0) historical earthquakes indicating that stronger events are rather independent, whereas the weaker ones may be primary carriers of persistence in the seismogenesis process.

(ProQuest: ... denotes formulae omitted.)I. IntroductionIn the last two decades, borrowing and firm value issues receded into the background due to the structured financial instruments and the exotic options coming into prominence. During this period, some models have been improved by financial engineers in order to evaluate these instruments and to model the path of financial time series. However, these models are generally based on the assumptions of conventional financial theory, such as normal distribution and random walk, which are deeply criticized in different studies (Ramasamy and Helmi, 2011).As the existing conventional theory oversimplifies the complicated structure of financial markets and explains the problems under ideal and normal conditions, we need accurate and credible models in order to build up an effective finance theory (Velasquez, 2009). Conventional theory based on the Bachelier's (1900) study reduced the tail probabilities to an insignificant level using normal distribution. However, this insignificance forms the basis of models such as Markowitz (1952), Sharpe (1964), Fama (1965, 1970) and Black and Scholes (1973), which are the basic studies of conventional theory. The scandals and crises of the last 50 years, including Enron (2001), Parmalat (2003), Black Monday (1987), the collapse of Long Term Capital Management (LTCM) (1998) and the mortgage crisis (2008), demonstrated that the random walk assumption of the Efficient Market Hypothesis is not valid. This is because, if the distribution of price changes were normal, five standard deviation observations would be seen every 7,000 years. However, in this case changes are seen almost every 4 to 5 years. As stated by Mandelbrot and Hudson (2004), the chain is only as strong as its weakest link and the weakest links in conventional theory are the Efficient Market Hypothesis, random walk and normal distribution assumptions.The first and the most important criticism of conventional theory was made by Mandelbrot regarding the ignoring of self-similarity and long memory properties of the financial time series. It should be noted that while the long memory issue was associated with Mandelbrot, the first person who adressed this topic was Hurst (1951). That is why Mandelbrot (1972) used the H notation as the long memory parameter in honor of Hurst1. In that period, Hurst showed that there were significant correlations between the floods. According to his studies, a good flood year had a tendency to be followed by another good flood year, and similarly a bad flood year was followed by another bad flood year. That is, the movements of good and bad years are not in a random process (Grabbe, 2001). Hurst explained this correlation via the power law. One of the most crucial observations of Hurst can be explained as follows. Natural events generally appear to follow a normal distribution but only when the occurrence order of the events is not taken into account (Mandelbrot and Hudson, 2004). The statistic introduced by Hurst measures the persistency between the events and it is known as the Rescaled Range (R/S). The Rescaled Range is the range of the partial sums of the deviations from the mean of any time series (Chaitip et al., 2011). Inspired by the Bible, Mandelbrot and Wallis (1968) named the persistency situations the "Joseph Effect" and adapted them into the financial time series theory as the long memory issue (Grabbe, 2001).2 Mandelbrot (1963, 1983) also contributed to the emergence of self-similarity and the fractal dimension as new topics in finance theory. In fact, all of these models paved the way for the evolution of conventional finance theory.According to conventional finance theory based on the random walk assumption of Fama (1965), there is no correlation between asset returns, and therefore asset prices develop with the geometric Brownian motion. Almost all models of conventional finance theory are formed under this assumption and probability distributions of the returns are accepted as normal. …

The Wei River Basin, a typical arid and semi-arid region, is frequently attacked by droughts, which heavily limits its sustainable development of society and economy. Therefore, the main motivation of this study is to investigate the drought evolution characteristics and risk evaluation in the basin, which is crucial for drought warning, water resources management as well as agricultural development. The modified Mann-Kendall test trend method was used to capture the trend of the severity and duration of historical drought events, and the persistence of droughts was analyzed based on the Rescaled Range (R/S) analysis. Additionally, the Morlet wavelet analysis was utilized to calculate the period of dry and wet condition, and copulas were applied to calculate the joint return period of two typical scenarios. The main results are as follows: (1) the whole basin has an obviously dry trend, especially in autumn and winter; furthermore, the months with a wet trend are primarily concentrated in July and August; (2) the persistence of droughts is strong, indicating that droughts in the near future may be persistent; (3) the average “or” and “and” joint return periods under scenario (1) where drought duration is 6 months and drought severity is 4 are approximately 15.6 and 28.9 years, respectively, while those under scenario (2) where drought duration is 8 months and drought severity is 6 are approximately 34.9 and 53.1 years, respectively. In general, the drought risk of the northwestern basin is larger than that of the southeastern basin.

In this paper, we study long-range dependence of hydrological records with high frequent and massive data set. For detecting breakpoints, we apply the Evolutionary Wavelet Spectrum (EWS) to provide a segmentation of the original time series. And rescaled range analysis (R/S) for estimating the Hurst exponent that describe the long-range dependence phenomenon are used. The results affirm that the hydrological records have long-range dependent (LRD) behaviors.

In this study, the scaling properties of the oil and gold return volatilities have been analyzed in the context of bull and bear periods. In the determination of bull and bear turning points, we used the Modified Bry-Boschan Quarterly (MBBQ) algorithm. Results showed that the business cycle phase shapes of the bear periods in the oil market are almost linear, whereas the bull and bear periods of the gold and bull period of the oil market are convex. This means that there are sharper declines in the bear period of the oil market. Following the detection of bull and bear periods, scaling exponent H analysis was performed via the aggregated variance, Higuchi’s statistic, Peng’s statistic, rescaled range, boxed periodogram and wavelet fit models, which are from the time, frequency and wavelet domains. As there are conflicts about the credibility of these methods in the literature, we have used the shuffling procedure in order to determine the most robust methods. According to the results, bear periods have higher volatility persistency than bull periods. © 2014 by the author; licensee MDPI, Basel, Switzerland.

In this paper, we introduce a new approach which generalizes the GM2 algorithm (introduced in Sanchez-Granero et al. (2008) [52] ) as well as fractal dimension algorithms (FD1, FD2 and FD3) (first appeared in Sanchez-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 Levy stable motions are considered. We conclude the paper with a dynamic study of the Hurst exponent evolution in the S&P500 index stocks.

Daily price movements of seventeen commodities are tested for the possible presence of nonlinear speculative bubbles during 1991-2012. A VAR model for logarithmic first differences of each is estimated with one-year Treasury bill rates, U.S. dollar value, a world stock market index, and an overall commodities price index using Hamilton regime switching and Hurst rescaled range tests. Residuals after removing ARCH for all seventeen commodity price series are tested for remaining nonlinearity using the BDS test. These tests fail to reject the presence of bubble-like trends and nonlinearity beyond ARCH for all seventeen commodity series. However, we note that we are unable to overcome the misspecified fundamentals problem, which means we cannot argue that we have definitely found speculative bubbles. At most we can argue that our results indicate that these markets appear to exhibit excess volatility and unexplained trends.

It seems reasonable to expect financial market efficiency to be related to the economic development level. We study a 16 year sample, covering 22 countries. The Hurst–Mandelbrot–Wallis rescaled range is our efficiency measure, which we apply to returns and volatility. We find strong evidence of long memory persistence in volatility over time, which is unsurprising. However, unlike previous researchers, we could not find evidence of rescaled ranges trending down over time. However, we introduce an alternative measure of economic development, namely, whether FTSE (2011) classify an emerging market as ‘advanced’ or ‘secondary’. This measure shows greater efficiency in returns and volatility for ‘advanced’ emerging markets.

Hurst’s memory that roots in early work of the British hydrologist H.E. Hurst remains an open problem in stochastic hydrology. Today, the Hurst analysis is widely used for the hydrological studies for the memory and characteristics of time series and many methodologies have been developed for the analysis. So, there are many different techniques for the estimation of the Hurst exponent (H). 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, rescaled range (RR) analysis, modified rescaled 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. We found that DFA is the most appropriate technique for the Hurst exponent estimation for both the short term memory and long term memory. We analyze the SOI (Southern Oscillations Index) and 6 tree-ring series for USA sites by means of DFA and the BDS statistic is used for nonlinearity test of the series. From the results, we found that SOI series is nonlinear time series which has a long term memory of H = 0.92. Contrary to earlier work, all the tree ring series are not random from our analysis. A certain tree ring series show a long term memory of H = 0.97 and nonlinear property. Therefore, we can say that the SOI series has the properties of long memory and nonlinearity and tree ring series could also show long memory and non-linearity.

Rill erosion is one of the most serious problems in sloping farmland of purple soil. The formation and evolution of runoff and sediment associated with rill erosion are complex processes that are influenced by many factors. The aim of the present study was to investigate the temporal variation characteristics of runoff and sediment associated with rill erosion during the maize-growing season. Simulated rainfall experiments (1.0, 1.5 and 2.0 mm min−1) were carried out in three experimental croplands on a typical slope gradient of 15°. Multi-scale temporal variation characteristics of runoff and sediment associated with rill erosion were identified by rescaled range (R/S) and wavelet analyses. The results showed that the Hurst index of the runoff ranged between 0.572 and 0.875 during the maize-growing season, while those of the sediment ranged between 0.575 and 0.855, indicating a long-range positive correlation. Therefore, the time series of runoff and sediment were persistent time series, and the future trends of runoff and sediment would be consistent with the present. The principal cycles of runoff and sediment presented an initial increasing and then a decreasing trend with maize growth, and the runoff and sediment yield varied periodically every 8–15 minutes. The changes in the primary cycles of runoff and sediment were basically consistent with the changes in vegetation coverage. The results provided a new insight for describing multi-time scale characteristics of runoff and sediment associated with rill erosion.

Of the characteristics of tight sandstone reservoirs, heterogeneity is one of the most difficult characteristics to analyze. Heterogeneity is mainly manifested in three properties: lithology, porosity and electricity. The Chang 7 reservoir in the Dingbian oilfield is a tight sandstone reservoir controlled by delta front deposits. In this sedimentary environment, the underwater distributary channel oscillates frequently, resulting in considerable heterogeneity of the Chang 7 formation in the study area. The rescaled range method can calculate the fractal dimension D of logging curves, thus quantitatively characterizing reservoir heterogeneity. After analyzing the relationship between average wave number and IMF number by EMD method, the reservoir heterogeneity can be quantitatively characterized by ρ value. By comparing D value, ρ value and F value of different heterogeneous types, it is found that F value can distinguish different heterogeneous types. The order of F value also satisfies the actual situation of formation heterogeneity. The new heterogeneity coefficient is used to study the heterogeneity of Chang 7 reservoir, it is found that the relationship between the lithological heterogeneity represented by the natural gamma (GR) log and oil production is a quadratic function that initially increases and then decreases, that the relationship between the pore heterogeneity represented by the interval transit time (AC) log and oil production is linear, and that the relationship between the reservoir electrical heterogeneity represented by the deep induction log (LLD) and oil production is linear too.

During coal oxidation and temperature rise, a significantly large amount of electromagnetic radiation (EMR) signals are generated as a result of thermal deformation and thermal cracking. The generation of EMR signal is the comprehensive embodiment of the physical and chemical changes in coal during its oxidation and subsequent heating. Therefore, the generated signals contain complex and rich messages that can reflect the changes in the internal structure of coal. In this work, the characteristics of EMR signal were analyzed by multifractal theory. Multifractal analysis was used to deconstruct EMR signals. Our preliminary study indicated that the multifractal spectrum of the EMR signal had a feature of small probability event. Further analysis demonstrated that the characteristic parameters of EMR signal are quite different at the later heating stage from those at the early heating stage. At the initial stage of coal spontaneous combustion, the multifractal spectrum of the signal is wider, but when coal combustion is reached, the scale range of EMR signal increases significantly, and its structure changes at different temperature ranges. The research results presented provide a basis for monitoring and presenting an early warning of coal spontaneous combustion risk.

This study conducts a systematic survey on whether the pricing behavior of cryptocurrencies is predictable. Thus, the Efficient Market Hypothesis is rejected and speculation is feasible via trading. We center interest on the Rescaled Range (R/S) and Detrended Fluctuation Analysis (DFA) as well as other relevant methodologies of testing long memory in returns and volatility. It is found that the majority of academic papers provides evidence for inefficiency of Bitcoin and other digital currencies of primary importance. Nevertheless, large steps towards efficiency in cryptocurrencies have been traced during the last years. This can lead to less profitable trading strategies for speculators.

We studied the predictability of the 10.7 cm solar radio flux by using stationary and non-stationary time-series analysis techniques of fractal theory to find the correlation exponent, the spectral exponent, the Hurst exponent, and the fluctuation exponent of the time series. The Hurst exponent was determined, from which the fractal dimension and consequently the predictability was evaluated. The results suggest that stationary methods of analysis yield inconsistent result, that is, amongst the four techniques used, the values of the exponents show great disparity. While two of the techniques, namely the auto-correlation function analysis and the spectral analysis, indicate long-term positive correlation, the other two methods, specifically the Hurst rescaled range-analysis and the fluctuation analysis, clearly exhibit the anti-correlated nature of the time series. The two non-stationary methods, that is, the discrete wavelet transform and the centered moving-average analysis, yielded values of the Hurst exponent that are indicative of positive correlation, of persistent behavior, and also showed that the time series is predictable to a certain extent.

This paper addresses issues of self-affinity, long-memory and self-organisation in variations of radon in soil recorded in Lesvos Island, Greece. Several techniques were employed, namely (a) power-law wavelet spectral fractal analysis, (b) estimation of Hurst exponents through (b1) rescaled-range, (b2) roughness-length, (b3) variogram and (a), (c) detrended fluctuation analysis, (d) investigation of fractal dimensions and (e) analysis of five block entropies: (e1) Shannon entropy, (e2) Shannon entropy per letter, (e3) conditional entropy, (e4) Tsallis entropy, and (e5) normalised Tsallis entropy. Long-lasting antipersistency was identified during a period of anomalous radon variations following fractional Brownian modelling. Remaining variations did not exhibit analogous behaviour and followed fractional Gaussian modelling. Antipersistent power-law-beta-exponent-values between 1.5 and 2.0 were detected during anomalies. Persistent values were also found. Hurst exponents were mainly within 0

Commonly, the interpretation of starch XRD patterns is hampered by the lack of sharp intensity peaks and the presence of a broad background band. This work proposes a fractal analysis approach for the analysis of XRD patterns obtained from starches. The idea behind the fractal analysis approach is that the broad background pattern contains information pertaining the complexity of the material structure that presumably is arranged within fractal geometries. Thus, the quest is to detect regularities reflected as auto-correlations of the complex intensity signal. The fractal rescaled range (R/S) analysis is proposed in this work to obtain an index for the regularity of the complex intensity signal. In turn, such regularity index should reflect the regularity of the complex starch structure. The approach is illustrated with native starches from maize, banana, and potato, showing that the former is more regular (i.e. more ordered starch structure) than the latter two. Gelatinization is used to contrast the fractal analysis results. In turn, it is shown that gelatinized starches exhibit a nearly amorphous structure reflected as non-correlated XRD sequences.

Purpose – This paper seeks to characterize the behavior of the naira/dollar foreign exchange rate series over the period 1999 through 2006 to determine if the process generating the series has long memory which is a special case of fractional Brownian motion. The existence of long memory contradicts the notion of market efficiency.Design/methodology/approach – The paper employs the modified rescaled range R/S test which is proposed by Lo to test the null hypothesis that daily and weekly NGN/USD exchange rates from 1999 through 2006 exhibit short‐memory process. The second test that was also employed is the Geweke‐Porter‐Hubak (GPH) test which was refined by Hurvich et al.Findings – The results show that long memory is present in daily and weekly foreign exchange level series of the Nigerian naira for the period sampled. This evidence implies that the Nigerian foreign exchange market may not be efficient. Thus, it is possible for investors to realize abnormal profit by taking an investment position based o...

For a sequence of i.i.d. zero mean random variables belonging to the domain of attraction of the normal law, two results concerning the rescaled range statistic are investigated in this paper. More specifically, we obtain precise asymptotics in the law of the logarithm related to complete convergence and a.s. convergence under some mild conditions.

In this study, we give a rescaled range analysis method employed to demonstrate long-range correlations and the presence of periodical features for a time series. An extension of the rescaled range method to estimate the Hurst exponent of high-dimensional fractals is proposed in this study. The two-dimensional rescaled range analysis is used to analyze traffic data, reveal interesting scaling behavior with physical grounds. The relation between the one-dimensional rescaled range method and two-dimensional rescaled range method is interpreted carefully.

[En]The present study aimed at investigating the existence of long memory properties in ten developed stock markets across the globe. When return series exhibit long memory, the series realizations are not independent over time and past returns can help predict future returns, thus violating the market efficiency hypothesis. It poses a serious challenge to the supporters of random walk behavior of the stock returns indicating a potentially predictable component in the series dynamics. We computed Hurst-Mandelbrot’s Classical R/S statistic, Lo’s statistic and semi parametric GPH statistic using spectral regression. The findings suggest existence of long memory in volatility and random walk for logarithmic return series in general for all the selected stock market indices. Findings are in line with the stylized facts of financial time series. [Es]El presente estudio pretende investigar la existencia de propiedades de memoria larga en diez mercados de valores de distintos países desarrollados. Cuando las series de rendimientos exhiben memoria larga, estas series no son independientes del tiempo y los rendimientos pasados pueden ayudar a predecir rendimientos futuros, violando por tanto la hipótesis de eficiencia de los mercados. Esto plantea un serio desafío a los que defienden que los rendimientos siguen un camino aleatorio, indicando un componente potencialmente predecible en la dinámica de las series. Hemos calculado el estadístico clásico de Hurst Mandelbrot (R/S), el estadístico de Lo y el estadístico semiparamétrico GPH utilizando un método de regresión espectral. Los resultados sugieren la existencia de memoria larga en la volatilidad de los rendimientos y un paseo aleatorio para los logaritmos de las series, en general para todos los índices de mercado seleccionados. Los resultados están en línea con hechos contrastados para series temporales financieras.

This paper presents Hurst exponent footprints from pseudo-dynamic measurements of significantly varied activities on a damaged bridge structure during rehabilitation through continuous monitoring. The system is interesting due to associated uncertainty in large-scale structures and significant presence of human intervention arising from fundamentally different processes. Investigations into the variation of computed Hurst exponents on time series of limited lengths are carried out in this regard. The Hurst exponents are compared with respect to specific events during the rehabilitation, as well as with the data collection locations. The variations of local Hurst exponents about the values computed for each activity are presented. The scaling of Hurst exponents for different activities is also investigated; these are representative of the extent of multifractality for each event. The extent of multifractality is assessed along with its source and time dependency. The National Roads Authority, Maudlins, Naas, Kildare, Ireland South Dublin County Council, Tallaght, Dublin, Ireland Coastway, Naas, Kildare, Ireland Datum Monitoring Services Limited, Moira, Armagh, Northern Ireland Structural Concrete Bonding Services Limited, Newbridge, Kildare, Ireland Complete Highway Maintenance Group, Walkinstown, Dublin, Ireland Testconsult Ireland Limited, Portlaoise, Laoise, Ireland

The Arab Spring which began on 17 December 2010 with the civil rebellions, revolutionary wave of demonstrations and protests in the Tunisia, Egypt, Libya, Yemen, Bahrain and Syria. The Arab Spring not only created a domino effect between Arabic countries but also it reflected a significant influence on the financial markets all over the world. The objective of this study is to analyze the impact of the Arab Spring in Turkey Financial Market in consideration of long memory. Long memory can be defined as the persistence of the unexpected shocks on the underlying has long lasting effects. Modeling long memory in stock returns and volatility has also attracted great deal of attention from finance literature recently. Existence of long memory is determined both for the returns and volatility of the time series by using different methods. Existence of long memory can be tested by Rescaled Range Statistics (R/S), Geweke and Porter-Hudak (GPH) Model and Gaussian Semi Parametric (GSP) Method. In consequence of these tests, if the stock returns have long memory affect then respectively Fractionally Integrated Autoregressive Moving Average Model (ARFIMA) and the Fractionally Integrated Generalized Autoregressive Conditional Heteroscedasticity (FIGARCH) model are used to detect the long memory in respectively return and volatility. In this study, the impact of the Arab Spring is investigated by modeled the long memory in Istanbul Stock Exchange using ISE 30 index prices in between December 17, 2010 and April 02, 2012.

We propose to estimate the Hurst parameter involved in fractional processes via a method based on the Karhunen–Loeve expansion of a Gaussian process. We specifically investigate the cases of the fractional Brownian motion, the fractional Ornstein–Uhlenbeck family and the fractional Brownian bridge. We numerically compare our results with the ones obtained by the maximum-likelihood method, which show the validity of our proposal.

Well-log (radioactivity, density and resistivity) analysis constitutes a stan- dard approach for inferring lithology from wells. However, due to inherent complex- ity of the signals (such as highly heterogeneous deep-water sedimentary sequences) lithology recognition is not straightforward. We used a rescaled range analysis, cali- brated with cores, to recognize lithological patterns from signal recorded along wire- line logs. The detected intervals coincide with zones of visual electro-facies associa- tions proposed by geologist well-log interpreters. In addition, we propose a rescaled range multifractal analysis to identify ranges of well-log signal complexities, which could be related to sedimentary process variations at specific stratigraphic order cy- cles.

Investigating long range dependence of river flows, especially in connection with various climate and storage related factors, is important in order to improve stochastic models for long range dependence and in order to understand deterministic and stochastic variability in long-term behaviour of streamflow. Long range dependence expressed by the Hurst coefficient H is estimated for 39 (deseasonalized) mean daily runoff time series in Europe of at least 59 years using five estimators (rescaled range, regression on periodogram, Whittle, aggregated variances, and least squares based on variance). All methods yield estimates of H > 0.5 for all data sets. The results from the different estimators are significantly positively correlated for all pairs of methods indicating consistency of the methods used. Correlations between H and various catchment attributes are also analysed. H is strongly positively correlated with catchment area. Apparently, increasing storage with catchment area translates into increasing long range dependence. H is also positively correlated with mean discharge and air temperature and negatively correlated with the mean specific discharge and the seasonality index (maximum Parde coefficient). No significant correlation is found between the Hurst coefficient and the length of the analyzed time series. The correlations are interpreted in terms of snow processes and catchment wetness. Copyright © 2012 John Wiley & Sons, Ltd.

In this paper, we evaluate the differences between Healthy subjects (HS), Parkinson Disease (PDS) subjects, and Epileptic subjects (EBS — Epileptic subject between seizures and ES — epileptic subjects in seizure) by computing some linear, statistic and non-linear features, such as: correlation dimension, maximum Lyapunov exponent and Hurst coefficient. A comparison is made between PDS, EBS, ES and HS groups using non-linear parameters. The EEG data are wavelet multiresolution decomposed into subbands of interest (alpha 8–12 Hz, beta 13–30 Hz, delta 0–4 Hz, theta 4–8 Hz, gamma 30–60 Hz). We applied rescaled range method to estimate the Hurst coefficient of the decomposed signals, with different types of wavelet transforms. The results obtained using the maximum Lyapunov exponent do not highlight very well differences between the 4 groups analyzed (HS, PDS, EBS and ES) but using Hurst coefficient we can make a very good differentiation of EEG signals. The results show that the Hurst coefficient is significant different for delta and theta rhythms extracted from the EEG signal in case of the healthy subjects compared with PDS or EBS subjects. The Hurst coefficient for healthy subjects has a value higher than 0.5 and for PD subjects or epileptic subject between seizures has a value lower than 0.5. Hence, the non-linear parameters such as Hurst coefficient can help in EEG interpretation and in neurological disorders diagnosis.

In this paper, we have utilized ANN (artificial neural network) modeling for the prediction of monthly rainfall in Mashhad synoptic station which is located in Iran. To achieve this black-box model, we have used monthly rainfall data from 1953 to 2003 for this synoptic station. First, the Hurst rescaled range statistical (R/S) analysis is used to evaluate the predictability of the collected data. Then, to extract the rainfall dynamic of this station using ANN modeling, a three-layer feed-forward perceptron network with back propagation algorithm is utilized. Using this ANN structure as a black-box model, we have realized the complex dynamics of rainfall through the past information of the system. The approach employs the gradient decent algorithm to train the network. Trying different parameters, two structures, M531 and M741, have been selected which give the best estimation performance. The performance statistical analysis of the obtained models shows with the best tuning of the developed monthly prediction model the correlation coefficient (R), root mean square error (RMSE), and mean absolute error (MAE) are 0.93, 0.99, and 6.02 mm, respectively, which confirms the effectiveness of the developed models.

This research paper focuses on the persistence level of earthquakes in and around Indonesia by using Rescaled Range Analysis. In the past, the long-term memories in many natural phenomena like floods, hurricanes, sunspots have been found using Rescaled Range Analysis. The Hurst exponent computed from the Rescaled Range determines the level of persistence. The test sample employed for conducting this analysis was taken from the seismic data of Indonesia for the period from 1943 to 2013 (70 years). The result analysis indicates that there is a fair level of persistence in the occurrences of earthquakes in Indonesia. It also suggests that this time series has long term memory and the intensity and recurrence of earthquakes in this area may increase in the coming years.

Time series and imaging using fractal analysis have been implemented in many areas of science and engineering, economics, biology and statistics among others, for their ability to detect auto-similarities or irregular behavior patterns. Such advantage has represented a significant advance in the monitoring and control of variables processes and research. But all this has been done in most cases with the help of software trademarks and sophisticated equipment that often do not realize the possibility of acquiring or present the results in a user-friendly graphical interface to analyze online or process variables seudoline their fractals descriptors. Therefore, it has been developed in graphic language (language-G) a program that performs the analysis of process variables crystallization of cane sugar for online monitoring of their acquired images and evolution crystallization with fractal descriptors.

The soil Rn-222 concentration non-linear patterns are investigated by the application of various chaos methodologies based on 70 272 measurement data from the East Anatolian Fault Zone, which is one of the world's most active faults. Among these methodologies are Lyapunov exponent, surrogate data, rescaled range (R/S) analysis, Fourier spectrum, phase space reconstruction, mutual information, false nearest neighbors, and correlation dimension. The results indicate that the nonlinear dynamical approach is convenient for characterization and prediction of the Rn-222 concentration dynamics, which are in turn usually used as an earthquake precursor. Behaviour of Rn-222 gas is important in earthquake prediction researches.

This paper reflects the studies undertaken to understand the complexity of the Electromyography (EMG) signal during biceps muscle contraction (flexion), by estimation of Fractal Dimension using Hurst's Rescaled Range (R/S) analysis. The measurements were done using surface electrodes with necessary signal acquisition circuitry and signal data is collected using sound card of PC through Matlab for data analysis purpose. The signal is also subjected to Frequency Domain techniques to estimate the power generated by muscle under different loading conditions. The Fractal Dimension, as obtained here, is found to vary under different loading conditions. The study of non-linear behavior of the EMG signal under varying loading conditions can lead to significant findings, pertaining to time based monitoring of muscle rehabilitation and endurance in a quantitative way.

Due to their potential volume and speed, large submarine landslides can generate destructive tsunamis or damage expensive seafloor structures. Understanding their timing is therefore important for hazard assessments; however, dating large numbers of landslides close to their origin is logistically difficult. Previous landslide studies are typically limited to fewer than ten observations of ages. To address this we analyse extensive, continuous and long-term turbidite records from four deep-sea basins which are interpreted to be the distal deposits of large, disintegrative landslides. Our records include sufficient numbers of turbidites (N = 151–1571) for robust statistical analysis of long-term controls on event timing and testing for relationships with triggering mechanisms such as earthquakes, sea level, climate change and volcanic activity. We explore statistical methods developed by medical, economic and biological disciplines and show how they can be applied to analysis of submarine landslide frequency and triggering. Frequency analysis of field data reveals two different distribution forms for landslide recurrence – exponential and log-normal. We discuss possible individual and combined effects of controlling factors that result in these distributions. Rescaled range and Gaussian finite mixture models determine whether and how landslides are clustered in time. Parametric Generalised Linear Models and non-parametric Proportional Hazards Models are used to test for the significance and influence of different variables and their rate of change. We demonstrate the value of unusually detailed long-term landslide records, and show how statistical analysis provides quantitative inputs for future hazard assessments, landslide-climate studies and understanding the tempo of deep-sea sediment flux.

This paper reports characteristic pre seismic electromagnetic disturbances in the MHz range detected prior to seven significant earthquakes of M L ≥ 5.0 which occurred in Greece between 2009 and 2015. The whole data set was investigated through Rescaled Range (R/S) analysis and wavelet based spectral fractal analysis. The long-memory trends hidden in the analysed signals were investigated via the Hurst exponent (H). The R/S method indicated that in the majority of the segments of the investigated signals, the H - exponents were in the range 0.7-0.9. Several exponents were above 0.9. This is associated with persistency. In several segments, the Hurst exponent exhibited small variance. Noteworthy fluctuations were also found. In many cases the H - time fluctuations were independent of the time evolution of the associated electromagnetic signals. The spectral fractal method, being more suited for this purpose, highlighted the characteristic signals’ epochs, governed by long-lasting fractal organisation. The calculated Hurst exponents in the characteristic long-lasting fractal epochs were associated with medium anti-persistent behaviour or with continuous switching between anti-persistency and persistency. The H - exponents of the random fractal organised segments were mainly persistent. From the data presented in this paper, it is deduced that the R/S method provides additional information on the estimation of the Hurst exponent when combined with the spectral fractal method. Both techniques should be employed in sequential steps to enhance the precursory value of the results.

We propose some class of statistics suitable for estimation of the Hurst index of the fractional Brownian motion based on the second order increments of an observed discrete trajectory., Comment: 4 pages

Countries across the globe have undergone financialization of their economies over the recent decades. Concomitantly, asset markets have exhibited high levels of volatility with sharp increases characteristic of speculative bubbles followed by even sharper crashes. This paper attempts to test the possible presence of nonlinear speculative bubbles in 23 international markets using daily data from January 1993-March 2015, and its possible link to the financialization phenomenon. To estimate fundamental values, we estimate VAR models for each market for stock market returns with world interest rates, exchange rates, and world stock indexes as the fundamental variables. Residuals from these VAR national market models are tested for significant movements away from the fundamentals using Hamilton regime switching and Hurst rescaled range tests. After removing ARCH effects from the residuals the remaining series is tested for nonlinearities using BDS statistics. Our results indicate the presence of speculative bubbles in all 23 of these markets with increasing incidence over time, which suggest a linkage with the phenomenon of financialization of the economies over the period.