Surrogate analysis of volatility series from long-range correlated noise

Detrended fluctuation analysis (DFA) [C.-K. Peng, S.V. Buldyrev, A.L. Goldberger, S. Havlin, F. Sciortino, M. Simons, H.E. Stanley, Nature 356 (1992) 168] of volatility series has been proposed to identify possible nonlinear/multifractal signatures in the given empirical sample [Y. Ashkenazy, P.Ch. Ivanov, S. Havlin, C.-K. Peng, A.L. Goldberger, H.E. Stanley, Phys. Rev. Lett. 86 (2001) 1900; Y. Ashkenazy, S. Havlin, P. Ch. Ivanov, C.-K. Peng, V. Schulte-Frohlinde, H.E. Stanley, Physica A. 323 (2003) 19; T. Kalisky, Y. Ashkenazy, S. Havlin, Phys. Rev. E 72 (2005) 011913]. Long-range volatility correlation can be an outcome of static as well as dynamical nonlinearity. In order to argue in favor of dynamical nonlinearity, surrogate testing is used in conjunction with volatility analysis [Y. Ashkenazy, P.Ch. Ivanov, S. Havlin, C.-K. Peng, A.L. Goldberger, H.E. Stanley, Phys. Rev. Lett. 86 (2001) 1900; Y. Ashkenazy, S. Havlin, P. Ch. Ivanov, C.-K. Peng, V. Schulte-Frohlinde, H.E. Stanley, Physica A. 323 (2003) 19; T. Kalisky, Y. Ashkenazy, S. Havlin, Phys. Rev. E 72 (2005) 011913]. In this brief communication, surrogate testing of volatility series from long-range correlated monofractal noise and their static, invertible nonlinear transforms is investigated. Long-range correlated noise is generated from fractional auto regressive integrated moving average (FARIMA) (0, d, 0), with Gaussian and non-Gaussian innovations. We show significant deviation in the scaling behavior between the empirical sample and the surrogate counterpart at large time-scales in the case of FARIMA (0, d, 0) with non-Gaussian innovations whereas no such discrepancy was observed in the case of Gaussian innovations. The results encourage cautious interpretation of surrogate analysis of volatility series in the presence of non-Gaussian innovations.     

Fractional diffusion equation in half-space with Robin boundary condition

The initial and boundary value problem for the fractional diffusion equation in half-space with the Robin boundary condition is considered. The solution is comprised of two parts: the contribution of the initial value and the contribution of the boundary value, for which the respective fundamental solutions are given. Finally, the solution formula of the considered problem is obtained.     

Non-linear characteristics and long-range correlations in Asian stock markets

We test several non-linear characteristics of Asian stock markets, which indicates the failure of efficient market hypothesis and shows the essence of fractal of the financial markets. In addition, by using the method of detrended fluctuation analysis (DFA) to investigate the long range correlation of the volatility in the stock markets, we find that the crossover phenomena exist in the results of DFA. Further, in the region of small volatility, the scaling behavior is more complicated; in the region of large volatility, the scaling exponent is close to 0.5, which suggests the market is more efficient. All these results may indicate the possibility of characteristic multifractal scaling behaviors of the financial markets.     

Detrended fluctuation analysis on spot and futures markets of West Texas Intermediate crude oil

In this paper, we study the auto-correlations and cross-correlations of West Texas Intermediate (WTI) crude oil spot and futures return series employing detrended fluctuation analysis (DFA) and detrended cross-correlation analysis (DCCA). Scaling analysis shows that, for time scales smaller than a month, the auto-correlations and cross-correlations are persistent. For time scales larger than a month but smaller than a year, the correlations are anti-persistent, while, for time scales larger than a year, the series are neither auto-correlated nor cross-correlated, indicating the efficient operation of the crude oil markets. Moreover, for small time scales, the degree of short-term cross-correlations is higher than that of auto-correlations. Using the multifractal extension of DFA and DCCA, we find that, for small time scales, the correlations are strongly multifractal, while, for large time scales, the correlations are nearly monofractal. Analyzing the multifractality of shuffled and surrogated series, we find that both long-range correlations and fat-tail distributions make important contributions to the multifractality. Our results have important implications for market efficiency and asset pricing models.     

Space-magnitude dependent scaling behaviour in seismic interevent series revealed by detrended fluctuation analysis

The scaling behaviour of the 1981-2007 seismicity data in central Italy, which is one of the most seismically active areas in Italy is investigated. In particular we examined the earthquakes located in a circular area centred on the epicentre of the strongest event, occurred in September 26, 1997 (duration magnitude M_D=5.8). On the base of the detrended fluctuation analysis (DFA), we found that in the magnitude range between 2.5 and 2.9 the scaling exponents fall into disjoint sets for events relatively close and far from the epicentre of the strongest event.     

Memory effect and multifractality of cross-correlations in financial markets

We introduce an instantaneous and an average instantaneous cross-correlation function to detect the temporal cross-correlations between individual stocks based on the daily data of the United States and the Chinese stock markets. The memory effect of the instantaneous cross-correlations is investigated by applying the detrended fluctuation analysis (DFA), where the DFA exponents can be partly explained by the correlation function from the common sense. Long-range memory is observed for the average instantaneous cross-correlations, and persists up to a month magnitude of timescale for the United States stock market and half a month magnitude of timescale for the Chinese stock market. In addition, multifractal nature is investigated by a multifractal detrended fluctuation analysis.     

Long-range dependence in tree-ring width time series of Austrocedrus Chilensis revealed by means of the detrended fluctuation analysis

By using the detrended fluctuation analysis and detrended moving average method, 823 time series of tree-ring widths in Austrocedrus Chilensis in Patagonia were analyzed. The tree-ring widths of A. Chilensis have been widely used for climatological studies. The results point out to the presence of significant scaling in the temporal fluctuations of tree-ring, which is not due to singular probability density function of the widths but due to the presence of long-range correlations. Such results are in good agreement with those concerning the evidence of long-range dependencies in weather time series.     

Long-range correlations in two-dimensional spatio-temporal seismic fluctuations

We analysed the scaling behaviour of the two-dimensional (2-D) sequence (Δs, Δt) of the 1981–1998 southern California seismicity, where Δs is the distance between two consecutive earthquakes (jump) and Δt is their interevent interval. The 2-D seismic spatio-temporal fluctuations were investigated by means of the detrended fluctuation analysis (DFA), well-known methodology used to detect scaling behaviour in observational time series possibly affected by nonstationarities. The estimated scaling exponents α_DFA, larger than 0.5, indicate the presence of persistent long-range correlations in the 2-D sequence analysed. The variation of the scaling exponent with the increase of threshold magnitude shows a two-fold behaviour: in the range between 1.5 (the completeness magnitude of the catalog) and 3.0, the scaling exponent is quite constant and denoting a flicker-noise dynamics; while for magnitudes larger than 3.0 it decreases with the increase of magnitude, indicating a tendency toward a 2-D space–time Poissonian process for large events.     

Different scaling behaviors in daily temperature records over China

Long-range correlations of five kinds of daily temperature records (i.e. daily average temperature records, daily maximum temperature records, daily minimum temperature records, diurnal temperature range and the sum of daily maximum and minimum temperature records) from 164 weather stations over China during 1951-2004 are analyzed by means of detrended fluctuation analysis (DFA). These five kinds of fluctuation series are found to be power-law correlated with scaling exponents larger than 0.5. Local changes of scaling exponents are examined and the spatial distributions of these different kinds of temperature records are similar except the diurnal temperature range (DTR for short) records. Furthermore, the differences of the scaling behavior among diurnal temperature records and other kinds of temperature records are discussed.     

Investigating the temporal variation of the scaling behavior in rainfall data measured in central Argentina by means of detrended fluctuation analysis

Investigation of the time variant scaling behavior of six monthly rainfall series recorded in central Argentina from 1860 to 2006 by using detrending fluctuation analysis (DFA) was performed. Changes in precipitation extremes are analyzed for several regions of Argentina using long-term monthly rainfall data (back to 1860) recorded by rain gauges extended for more than 10 latitude-degrees, from subtropical regions until 39^° S. A moving window was employed in order to analyze statistical changes. Three different types of time patterns can be distinguished: (i) eastern stations show visible crossovers between persistent and random behavior; (ii) north-western stations are characterized by random behavior approximately at any time and do not present visible crossovers between different types of scaling behaviors; (iii) one station (i.e. Corrientes) shows a peculiar pattern, since it is characterized by several crossovers from persistent to random behavior, indicating a more fluctuating time dynamics without a well defined trend. The obtained results can be interpreted in the context of the climatological conditions of Central Argentina, which is characterized by the continental heat low in the northwest region, by the subtropical rainforest in the eastern provinces and by a transition zone (from maritime to continental regime) fluctuating with the South Atlantic high-pressure cell.     

Levels of complexity in turbulent time series for weakly and high Reynolds number

We use the detrended fluctuation analysis (DFA), the detrended cross correlation analysis (DCCA) and the magnitude and sign decomposition analysis to study the fluctuations in the turbulent time series and to probe long-term nonlinear levels of complexity in weakly and high turbulent flow. The DFA analysis indicate that there is a time scaling region in the fluctuation function, segregating regimes with different scaling exponents. We discuss that this time scaling region is related to inertial range in turbulent flows. The DCCA exponent implies the presence of power-law cross correlations. In addition, we conclude its multifractality for high Reynold’s number in inertial range. Further, we find that turbulent time series exhibit complex features by magnitude and sign scaling exponents.     

Scaling instability in self-potential earthquake-related signals

Scaling behavior of the temporal variability in self-potential data recorded in the frequency range during 1995 at Acapulco station (Mexico) was identified and analyzed. In this period, a strong earthquake (Ms=7.4) struck the monitored area on September 14, 1995. Using the detrended fluctuation analysis (DFA), which is a powerful method to detect scaling in nonstationary time series, deviations from stable power-law scaling were quantified. Our findings point to an evident unstable scaling behavior in self-potential data before the occurrence of the seismic event. These first results could be useful in the framework of earthquake prediction studies.     

Volatilities, traded volumes, and the hypothesis of price increments in derivative securities

A detrended fluctuation analysis (DFA) is applied to the statistics of Korean treasury bond (KTB) futures from which the logarithmic increments, volatilities, and traded volumes are estimated over a specific time lag. In this study, the logarithmic increment of futures prices has no long-memory property, while the volatility and the traded volume exhibit the existence of the long-memory property. To analyze and calculate whether the volatility clustering is due to a inherent higher-order correlation not detected by with the direct application of the DFA to logarithmic increments of KTB futures, it is of importance to shuffle the original tick data of future prices and to generate a geometric Brownian random walk with the same mean and standard deviation. It was found from a comparison of the three tick data that the higher-order correlation inherent in logarithmic increments leads to volatility clustering. Particularly, the result of the DFA on volatilities and traded volumes can be supported by the hypothesis of price changes.     

Cycles,scaling and crossover phenomenon in length of the day (LOD) time series

The dynamics of the temporal fluctuations of the length of the day (LOD) time series from January 1, 1962 to November 2, 2006 were investigated. The power spectrum of the whole time series has revealed annual, semi-annual, decadal and daily oscillatory behaviors, correlated with oceanic–atmospheric processes and interactions. The scaling behavior was analyzed by using the detrended fluctuation analysis (DFA), which has revealed two different scaling regimes, separated by a crossover timescale at approximately 23 days. Flicker-noise process can describe the dynamics of the LOD time regime involving intermediate and long timescales, while Brownian dynamics characterizes the LOD time series for small timescales.     

Long-term correlations in hourly wind speed records in Pernambuco, Brazil

We study the statistical properties of hourly wind speed time series detected at four weather stations in the state of Pernambuco, Brazil, in the period 2008-2009. We find that the average and maximum hourly wind speeds deviate from a mutual linear relationship, and that they may be well explained individually by a Weibull distribution, however, with different shape parameter values. On the other hand, the long-term correlations of both of these observables obey the same power-law behavior, with two distinct scaling regimes. Our results agree with previous studies on wind speed series correlations in other regions of the world, which is suggestive of universal behavior.     

Fluctuation and synchronization of gait intervals and gait force profiles distinguish stages of Parkinson's disease

We study the effects of Parkinson's disease (PD) on the long-term fluctuation and phase synchronization properties of gait timing (series of interstride intervals) as well as gait force profiles (series characterizing the morphological changes between the steps). We find that the fluctuations in the gait timing are significantly larger for PD patients and early PD patients, who were not treated yet with medication, compared to age-matched healthy controls. Simultaneously, the long-term correlations and the phase synchronization of right and left leg are significantly reduced in both types of PD patients. Surprisingly, long-term correlations of the gait force profiles are relatively weak for treated PD patients and healthy controls, while they are significantly larger for early PD patients. The results support the idea that timing and morphology of recordings obtained from a complex system can contain complementary information.     

Detrended fluctuation analysis and Kolmogorov–Sinai entropy of electroencephalogram signals

We measured the electroencephalogram (EEG) of young students in the relaxed state and in the state of the mathematical activities. We applied the detrended fluctuation analysis and Kolmogorov–Sinai entropy (KSE) in the EEG signals. We found that the detrended fluctuation functions follow a power law with Hurst exponents larger than 1/2. The Hurst exponents enhanced at all EEG channels in the state of mathematical activities. The KSE in the relaxed state is larger than those in the state of the mathematical activities. These indicate that the entropy is enhanced in the disorder state of the brain.     

Strong anticipation: Sensitivity to long-range correlations in synchronization behavior

Strong anticipation has emerged as a new framework for studying prospective control. According to earlier theories of prediction, anticipatory behavior rests on temporally local predictions from internal models. Strong anticipation eschews internal models and draws on the embedding of an organism in its environment. In this formulation, behavior is sensitive to the non-local temporal structure of the environment. We present initial evidence for strong anticipation in a synchronization task with tapping as the behavior. Participants were instructed to synchronize, to the best of their abilities, with a (unpredictable) chaotic signal. Our data suggest a close relationship between the long-range correlations of the chaotic signal and the long-range correlations of the synchronization behavior.     

Temporal variations of long-term correlations in seismic activity fluctuations

The scaling behavior of the 1998-2009 seismicity in Guerrero, southern Mexico, was studied by means of the detrended fluctuation analysis (DFA). We found that inter-seismic periods are correlated with a transition in the scaling behavior at about 200 seismic events. Correlations are relatively weak for small time scales. However, for large time scales, correlations are associated with a 1/f fractional process, indicating that the seismicity pattern emerges from a self-organized critical state. Temporal variations of the scaling exponent along years computed from the DFA indicate the presence of a quasi-biennial cycle in the seismicity correlations. This cyclic behavior was apparently triggered by the large 2001-2002 slow slip event in the Guerrero seismic gap. Besides, the significant seismic events (M_w>5) originate, on the average, at deeper regions in each cycle.     

Extracting quantitative dynamics in Earth's apparent resistivity time series by using the detrended fluctuation analysis

The magnetotelluric method (MT) consists in the observation of the surface natural variations of the electromagnetic field in a broad range of periods. The apparent resistivity is proportional to the modulus of the surface impedance, which is the tensorial relation between the horizontal components of the electric with the magnetic field. The subsurface resistivity distribution is obtained by inverting the apparent resistivity.