quantifying signals with power law correlations a comparative study of detrended fluctuation analysis and detrended moving average techniques detrended fluctuation analysis dfa and detrended moving average dma are two scaling analysis methods designed to quantify correlations in noisy nonstationary signals we systematically study the performance of different variants of the dma method when applied to artificially generated long range power law correlated signals with an a priori known scaling exponent alpha 0 and compare them with the dfa method we find that the scaling results obtained from different variants of the dma method strongly depend on the type of the moving average filter further we investigate the optimal scaling regime where the dfa and dma methods accurately quantify the scaling exponent alpha 0 and how this regime depends on the correlations in the signal finally we develop a three dimensional representation to determine how the stability of the scaling curves obtained from the dfa and dma methods depends on the scale of analysis the order of detrending and the order of the moving average we use as well as on the type of correlations in the signal

multifractal diffusion entropy analysis on stock volatility in financial markets abstract this paper introduces a generalized diffusion entropy analysis method to analyze long range correlation then applies this method to stock volatility series the method uses the techniques of the diffusion process and renyi entropy to focus on the scaling behaviors of regular volatility and extreme volatility respectively in developed and emerging markets it successfully distinguishes their differences where regular volatility exhibits long range persistence while extreme volatility reveals anti persistence

empirical mode decomposition and correlation properties of long daily ozone records sreceived 16 november 2004 revised manuscript received 17 march 2005 published 27 may 2005d correlations for daily data of total ozone column are investigated by detrended fluctuation analysis sdfad the removal of annual periodicity does not result in a background free signal for the tropical station mauna loa in order to identify the remaining quasiperiodic constituent the relatively new method of empirical mode decomposition semdd is tested we found that the so called intrinsic mode functions do not represent real signal components of the ozone time series their amplitude modulation is very sensitive to local changes such as random data removal or smoothing tests on synthetic data further corroborate the limitations of decomposing quasiperiodic signals from noise with emd nevertheless the emd algorithm helps to identify dominating frequencies in the time series which allows to separate fluctuations from the remaining background we demonstrate that dfa analysis for the cleaned mauna loa record yields scaling comparable to a mid latitude station

a high order nodal discontinuous galerkin method for a linearized fractional cahnhilliard equation in this paper we develop and analyze a nodal discontinuous galerkin method for the linearized fractional cahnhilliard equation containing derivatives of fractional order in space the caputo derivative is chosen as the representation of spatial derivative the linearized fractional cahnhilliard problem has been expressed as a system of low order differential integral equations we adopt the nodal discontinuous galerkin methods for the full spatial discretization using a high order nodal basis set of orthonormal lagrangelegendre polynomials of arbitrary order in space on each element of computational domain moreover we prove the stability and optimal order of convergence n 1 for the linearized fractional cahnhilliard problem when polynomials of degree n are used numerical experiments are displayed to verify the theoretical results

fractional sturm liouville eigen problems we first consider a regular fractional sturm liouville problem of two kinds rfslp i and rfslp ii of order 0 2 the corresponding fractional differential operators in these problems are both of riemann liouville and caputo type of the same fractional order 2 0 1 we obtain the analytical eigensolutions to rfslp i ii as non polynomial functions which we define as jacobi poly fractonomials these eigenfunctions are orthogonal with respect to the weight function associated with rfslp i ii subsequently we extend the fractional operators to a new family of singular fractional sturm liouville problems of two kinds sfslp i and sfslp ii we show that the primary regular boundary value problems rfslp i ii are indeed asymptotic cases for the singular counterparts sfslp i ii furthermore we prove that the eigenvalues of the singular problems are real valued and the corresponding eigenfunctions are orthogonal in addition we obtain the eigen solutions to sfslp i ii analytically also as non polynomial functions hence completing the whole family of the jacobi poly fractonomials in numerical examples we employ the new poly fractonomial bases to demonstrate the exponential convergence of the approximation in agreement with the theoretical results

application of the detrended fluctuation analysis dfa method for describing cloud breaking abstract a method to sort out correlations and decorrelations in stratus cloud formation persistence and breakup is introduced the detrended fluctuation analysis dfa statistical method is applied to microwave radiometer data the existence of long range power law correlations in stratus cloud liquid water content fluctuations is demonstrated over a 2 h period moreover using a finite size time interval window a change from brownian to non brownian fluctuation regimes is clearly shown to define the cloud structure changes such findings are similar to those found in dna and financial data sequences when mosaics of persistent and antipersistent patches are present the occurrence of these statistics in stratus cloud liquid water content suggests the usefulness of similar studies on cloud resolving model output and for better meteorological predictability

retarded and instantaneous evolution equations of macroobservables in non equilibrium statistical mechanics abstract the relations between evolution equations ee for macroobservables of the retarded and instantaneous time convolutionless type are discussed from a general point of view far from equilibrium processes are included in using linear operator representations of nonlinear ee conditions under which a given retarded ee can be memory renormalized are formulated and the renormalized versions of nonequilibrium theories of robertson and grabert are derived memory kernels containing long time tails which cannot be renormalized are considered explicitly the corresponding instantaneous ee exhibit the existence of two different regimes of decay the transition between them taking place at a critical time t k ln where characterizes the strength of the long time tail contribution if using a logarithmic time scale at very large times the relaxation is described by a markovian equation with a universal transport coefficient which is independent of the microscopic properties of the system

blocks adjustment reduction of bias and variance of detrended fluctuation analysis using monte carlo simulation abstract the length of minimal and maximal blocks equally distant on log log scale versus fluctuation function considerably influences bias and variance of dfa through a number of extensive monte carlo simulations and different fractional brownian motion fractional gaussian noise generators we found the pair of minimal and maximal blocks that minimizes the sum of mean squared error of estimated hurst exponents for the series of length n 2 p p 7 15 sensitivity of dfa to sort range correlations was examined using arfima p d q generator due to the bias of the estimator for anti persistent processes we narrowed down the range of hurst exponent to 1 2 h 1

finite difference methods for two dimensional fractional dispersion equation fractional order partial differential equations as generalizations of classical integer order partial differential equations are increasingly used to model problems in fluid flow finance and other areas of application in this paper we discuss a practical alternating directions implicit method to solve a class of two dimensional initial boundary value fractional partial differential equations with variable coefficients on a finite domain first order consistency unconditional stability and therefore first order convergence of the method are proven using a novel shifted version of the classical grunwald finite difference approximation for the fractional derivatives a numerical example with known exact solution is also presented and the behavior of the error is examined to verify the order of convergence

finite element simulation and efficient algorithm for fractional cahn hilliard equation abstract this article is devoted to designing efficient linear finite element algorithm for the fractional cahn hilliard equation an important and newly proposed phase field model combining the advantages of the classic bicg algorithm and the toeplitz like structure of the coefficient matrix we develop a fast bicg fbicg algorithm for the linearized scheme to compute numerically the fractional cahn hilliard equation our theoretical analysis and numerical experiments demonstrate that the proposed fbicg reduces the computation cost and the storage to o m log m and o m possesses the same convergence rates as newton s algorithm does in space and time and preserves the energy dissipation and equality laws the numerical experiments also demonstrate that the fbicg is almost mass conserved recognizes accurately the phase separation by a very clear coarse graining process and the influences of different indices r and s of fractional laplacian and different coefficients k and a on the width of the interfaces

stochastic modeling of daily temperature fluctuations classical spectral hurst and detrended fluctuation analysis have been revealed asymptotic power law correlations for daily average temperature data for short time intervals however strong correlations characterize the dynamics that permits a satisfactory description of temperature changes as a low order linear autoregressive process dominating the texts on climate research here we propose a unifying stochastic model reproducing correlations for all time scales the concept is an extension of a first order autoregressive model with power law correlated noise the inclusion of a nonlinear atmospheric response function conveys the observed skew for the amplitude distribution of temperature fluctuations while stochastic models cannot help to understand the physics behind atmospheric processes they are capable to extract useful features promoting to benchmark physical models an example is shown possible applications for other systems of strong short range and asymptotic power law correlations are discussed

comparison of the tokuyama mori type projection operator method to that of mori type near the glass transition abstract two types of projection operator methods mori type and tokuyama mori type are tested for studying the slow dynamics of a single particle in complex systems near the glass transition from a unified point of view based on the mean field theory recently proposed by the present author the molecular and brownian dynamics simulations and the corresponding solutions of the mode coupling theory developed by employing mori type formalism are compared consistently disagreement is found for the mean square displacements in stage thus validity of both types is fully investigated near the glass transition from a new standpoint

the scaling properties of stock markets based on modified multiscale multifractal detrended fluctuation analysis abstract here we propose the new method dh mma based on multiscale multifractal detrended fluctuation analysis mma to investigate the scaling properties in stock markets it is demonstrated that our approach can provide a more stable and faithful description of the scaling properties in comprehensive range rather than fixing the window length and slide length it allows the assessment of more universal and subtle scaling characteristics we illustrate dh mma by selecting power law artificial data sets and six stock markets from us and china the us stocks exhibit very strong multifractality for positive values of q however the chinese stocks show stronger multifractality for negative q than positive q in general the us stock markets show similar behaviors but chinese stock markets display distinguishing characteristics

method for generating long range correlations for large systems we propose a new method to generate a sequence of random numbers with long range power law correlations that overcomes known difficulties associated with large systems the new method presents an improvement on the commonly used methods we apply the algorithm to generate enhanced diffusion isotropic and anisotropic self affine surfaces and isotropic and anisotropic correlated percolation

chaos based nonlinear analysis to study cardiovascular responses to changes in posture abstract cardiovascular responses to changes in posture have been the focus of numerous investigations in the past analysis of ecg and arterial blood pressure abp data for different postures obtained from physionet was done from a new perspective of analyzing nonlinear time series having multifractal features multifractal detrended fluctuation analysis mfdfa and also the cross correlation between ecg and bp with another technique multifractal detrended cross correlation analysis mfdxa reveals ecg and abp time series of different subjects to be multifractal in nature with different degree of complexity different degree of auto and cross correlation for both ecg and abp signals is also observed for all subjects further significant information obtained from the cross correlation parameters obtained from mfdxa analysis is very important and relevant particularly for understanding the dynamics of orthostatic stress

a second order accurate numerical method for the two dimensional fractional diffusion equation spatially fractional order diffusion equations are generalizations of classical diffusion equations which are used in modeling practical superdiffusive problems in fluid flow finance and others in this paper we present an accurate and efficient numerical method to solve a fractional superdiffusive differential equation this numerical method combines the alternating directions implicit adi approach with a crank nicolson discretization and a richardson extrapolation to obtain an unconditionally stable second order accurate finite difference method the stability and the consistency of the method are established numerical solutions for an example super diffusion equation with a known analytic solution are obtained and the behavior of the errors are analyzed to demonstrate the order of convergence of the method

numerical approximation of a time dependent nonlinear space fractional diffusion equation in this article we analyze a fully discrete numerical approximation to a time dependent fractional order diffusion equation which contains a nonlocal quadratic nonlinearity the analysis is performed for a general fractional order diffusion operator the nonlinear term studied is a product of the unknown function and a convolution operator of order 0 convergence of the approximation and a priori error estimates are given numerical computations are included which confirm the theoretical predictions

effect of nonlinear filters on detrended fluctuation analysis when investigating the dynamical properties of complex multiple component physical and physiological systems it is often the case that the measurable system s output does not directly represent the quantity we want to probe in order to understand the underlying mechanisms instead the output signal is often a linear or nonlinear function of the quantity of interest here we investigate how various linear and nonlinear transformations affect the correlation and scaling properties of a signal using the detrended fluctuation analysis dfa which has been shown to accurately quantify power law correlations in nonstationary signals specifically we study the effect of three types of transforms i linear y i a x i b ii nonlinear polynomial y i a x k i and iii nonlinear logarithmic y i log x i delta filters we compare the correlation and scaling properties of signals before and after the transform we find that linear filters do not change the correlation properties while the effect of nonlinear polynomial and logarithmic filters strongly depends on a the strength of correlations in the original signal b the power k of the polynomial filter and c the offset delta in the logarithmic filter we further apply the dfa method to investigate the apparent scaling of three analytic functions i exponential exp x a ii logarithmic log x a and iii power law x a lambda which are often encountered as trends in physical and biological processes while these three functions have different characteristics we find that there is a broad range of values for parameter a common for all three functions where the slope of the dfa curves is identical we further note that the dfa results obtained for a class of other analytic functions can be reduced to these three typical cases we systematically test the performance of the dfa method when estimating long range power law correlations in the output signals for different parameter values in the three types of filters and the three analytic functions we consider

multifractal detrended fluctuation analysis of nonstationary time series abstract we develop a method for the multifractal characterization of nonstationary time series which is based on a generalization of the detrended fluctuation analysis dfa we relate our multifractal dfa method to the standard partition function based multifractal formalism and prove that both approaches are equivalent for stationary signals with compact support by analyzing several examples we show that the new method can reliably determine the multifractal scaling behavior of time series by comparing the multifractal dfa results for original series with those for shuffled series we can distinguish multifractality due to long range correlations from multifractality due to a broad probability density function we also compare our results with the wavelet transform modulus maxima method and show that the results are equivalent

nonuniversal atmospheric persistence different scaling of daily minimum and maximum temperatures an extensive investigation of 61 daily temperature records by means of detrended fluctuation analysis has revealed that the value of correlation exponent is not universal contrary to earlier claims furthermore statistically significant differences are found for daily minimum and maximum temperatures measured at the same station suggesting different degrees of long range correlations for the two extremes numerical tests on synthetic time series demonstrate that a correlated signal interrupted by uncorrelated segments exhibits an apparently lower exponent value over the usual time window of empirical data analysis in order to find statistical differences between the two daily extreme temperatures high frequency 10 min records were evaluated for two distant locations the results show that daily maxima characterize better the dynamic equilibrium state of the atmosphere than daily minima for both stations this provides a conceptual explanation why scaling analysis can yield different exponent values for minima and maxima

multiscale multifractal detrended fluctuation analysis of multivariate time series abstract this work extends the multivariate multifractal detrended fluctuation analysis mv mfdfa method to multiscale case named multiscale multivariate multifractal detrended fluctuation analysis mmv mfdfa the benefits of the proposed approach are illustrated by numerical simulations on synthetic multivariate processes furthermore the proposed mmv mfdfa method is applied to the fractal auto correlation analysis of six pollutants pm2 5 pm10 so2 no2 co and o3 hourly data in different seasons the results show that the seasonal periodicity has robust impact on the auto correlation of pollutants in spring and summer besides we also find that the pollutants in the four seasons possess strong multifractal auto correlation nature even after the removal of the seasonal pattern finally the source of multifractality among more than two series is also discussed and some interesting results are obtained pm2 5 not only dominates the underlying evolution process in fall and winter but also is more correlated to the other pollutants than the other ones to each other except in spring the proposed mmv mfdfa methodology can provide reliable ways of measuring the fractal auto correlation properties of multivariate series and it can be applied to any system with multiple data channels

finite difference approximations for fractional advection dispersion flow equations fractional advection dispersion equations are used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in a porous medium in this paper we develop practical numerical methods to solve one dimensional fractional advection dispersion equations with variable coefficients on a finite domain the practical application of these results is illustrated by modeling a radial flow problem use of the fractional derivative allows the model equations to capture the early arrival of tracer observed at a field site

multifractal detrended cross correlation analysis of sunspot numbers and river flow fluctuations abstract we use the detrended cross correlation analysis dcca to investigate the influence of sun activity represented by sunspot numbers on one of the climate indicators specifically rivers represented by river flow fluctuation for daugava holston nolichucky and french broad rivers the multifractal detrended cross correlation analysis mf dxa shows that there exist some crossovers in the cross correlation fluctuation function versus time scale of the river flow and sunspot series one of these crossovers corresponds to the well known cycle of solar activity demonstrating a universal property of the mentioned rivers the scaling exponent given by dcca for original series at intermediate time scale 12 24 s 130 months is 1 17 0 04 which is almost similar for all underlying rivers at 1 confidence interval showing the second universal behavior of river runoffs to remove the sinusoidal trends embedded in data sets we apply the singular value decomposition svd method our results show that there exists a long range cross correlation between the sunspot numbers and the underlying streamflow records the magnitude of the scaling exponent and the corresponding cross correlation exponent are 0 76 0 85 and 0 30 0 48 respectively different values for scaling and cross correlation exponents may be related to local and external factors such as topography drainage network morphology human activity and so on multifractal cross correlation analysis demonstrates that all underlying fluctuations have almost weak multifractal nature which is also a universal property for data series in addition the empirical relation between scaling exponent derived by dcca and detrended fluctuation analysis dfa h sun h river 2 is confirmed

statistical dynamical theory of nonlinear stochastic processes ii time convolutionless projector method in nonequilibrium open systems abstract a new statistical dynamical theory of nonlinear stochastic processes in nonequilibrium open systems is presented it is shown by means of the time convolutionless projector method that multiplicative type stochastic equations of motion for relevant variables ai t can always be transformed exactly into additive type langevin type stochastic equations of motion for ai t and the corresponding master equation for the probability distribution the langevin type equation consists of a drift term and a fluctuating force the fluctuating force is shown to give a new kind of additive stochastic process which has a quite different feature from the ordinary additive processes the importance of langevin equations of this type in the multiplicative stochastic process is pointed out from the statistical dynamical viewpoint a new cumulant expansion of the master equation in powers of stochastic forces is also found the application of the dynamic renormalization group method to steady states far from thermal equilibrium is discussed from the statistical dynamical standpoint developed in the present paper the present theory is applied to renormalize nonlinear stochastic equations by eliminating the short wavelength modes and thus to derive nonlinear langevin equations with renormalized kinetic coefficients and the corresponding master equation the renormalized coefficients are then used not only to find recursion relations but also to calculate explicitly a cutoff dependence of the kinetic coefficients as an actual example fluctuations in the generalized time dependent ginzburg landau model are investigated near its nonequilibrium tricritical point

a new approach to quantify power law cross correlation and its application to commodity markets abstract we proposed a new method detrended moving average cross correlation analysis dmca to detect the power law cross correlation between two correlated non stationary time series by combining detrended cross correlation analysis dcca and detrended moving average dma in order to compare the performance of dmca and dcca in the detection of cross correlation and to estimate the influence of periodic trend we generate two cross correlated time series x i and y i by a periodic two component fractionally autoregressive integrated moving average arfima process then we apply both methods to quantify the cross correlations of the generated series whose theoretical values are already known to us by comparing the results we obtained we find that the performance of this new approach is comparable to dcca with less calculating amounts our method can also reduce the impact of trends furthermore dmca for background and forward moving average case outperforms dcca in more accurate estimation when the analyzed times series are short in length to provide an example we also apply this new method to the time series of the real world data from brent and wti crude oil spot markets to investigate the complex cross market correlation between these commodity markets in all our method is another practical choice to detect the cross correlation between two short period non stationary time series and has potential application to real world problems

multifractal characterization of stochastic resonance we use a multifractal formalism to study the effect of stochastic resonance in a noisy bistable system driven by various input signals to characterize the response of a stochastic bistable system we introduce a new measure based on the calculation of a singularity spectrum for a return time sequence we use wavelet transform modulus maxima method for the singularity spectrum computations it is shown that the degree of multifractality defined as a width of singularity spectrum can be successfully used as a measure of complexity both in the case of periodic and aperiodic stochastic or chaotic input signals we show that in the case of periodic driving force singularity spectrum can change its structure qualitatively becoming monofractal in the regime of stochastic synchronization this fact allows us to consider the degree of multifractality as a new measure of stochastic synchronization also moreover our calculations have shown that the effect of stochastic resonance can be catched by this measure even from a very short return time sequence we use also the proposed approach to characterize the noise enhanced dynamics of a coupled stochastic neurons model

statistical properties of the volatility of price fluctuations we study the statistical properties of volatility measured by locally averaging over a time window t the absolute value of price changes over a short time interval ensuremath delta t we analyze the s 500 stock index for the 13 year period jan 1984 to dec 1996 we find that the cumulative distribution of the volatility is consistent with a power law asymptotic behavior characterized by an exponent ensuremath mu ensuremath approx 3 similar to what is found for the distribution of price changes the volatility distribution retains the same functional form for a range of values of t further we study the volatility correlations by using the power spectrum analysis both methods support a power law decay of the correlation function and give consistent estimates of the relevant scaling exponents also both methods show the presence of a crossover at approximately 1 5 days in addition we extend these results to the volatility of individual companies by analyzing a data base comprising all trades for the largest 500 u s companies over the two year period jan 1994 to dec 1995

finite element method for the space and time fractional fokker planck equation we develop the finite element method for the numerical resolution of the space and time fractional fokker planck equation which is an effective tool for describing a process with both traps and flights the time fractional derivative of the equation is used to characterize the traps and the flights are depicted by the space fractional derivative the stability and error estimates are rigorously established and we prove that the convergent order is o k 2 alpha h mu where k is the time step size and h the space step size numerical computations are presented which demonstrate the effectiveness of the method and confirm the theoretical claims

power spectrum and detrended fluctuation analysis application to daily temperatures the variability measures of fluctuation analysis fa and detrended fluctuation analysis dfa are expressed in terms of the power spectral density and of the autocovariance of a given process the diagnostic potential of these methods is tested on several model power spectral densities in particular we find that both fa and dfa reveal an algebraic singularity of the power spectral density at small frequencies corresponding to an algebraic decay of the autocovariance a scaling behavior of the power spectral density in an intermediate frequency regime is better reflected by dfa than by fa we apply fa and dfa to ambient temperature data from the 20th century with the primary goal to resolve the controversy in literature whether the low frequency behavior of the corresponding power spectral densities are better described by a power law or a stretched exponential as a third possible model we suggest a weibull distribution however it turns out that neither fa nor dfa can reliably distinguish between the proposed models

appropriateness of correlated first order auto regressive processes for modeling daily temperature records abstract the present study investigates linear and volatile nonlinear correlations of first order auto regressive process with uncorrelated ar 1 and long range correlated car 1 gaussian innovations as a function of the process parameter in the light of recent findings a kiraly i m janosi phys rev e 65 2002 0511021 we discuss the choice of car 1 in modeling daily temperature records we demonstrate that while car 1 is able to capture linear correlations it is unable to capture nonlinear volatile correlations in daily temperature records

a space time spectral method for the time fractional diffusion equation in this paper we consider the numerical solution of the time fractional diffusion equation essentially the time fractional diffusion equation differs from the standard diffusion equation in the time derivative term in the former case the first order time derivative is replaced by a fractional derivative making the problem global in time we propose a spectral method in both temporal and spatial discretizations for this equation the convergence of the method is proven by providing a priori error estimate numerical tests are carried out to confirm the theoretical results thanks to the spectral accuracy in both space and time of the proposed method the storage requirement due to the global time dependence can be considerably relaxed and therefore calculation of the long time solution becomes possible

how fast do stock prices adjust to market efficiency evidence from a detrended fluctuation analysis abstract in this paper we analyse price fluctuations with the aim of measuring how long the market takes to adjust prices to weak form efficiency i e how long it takes for prices to adjust to a fractional brownian motion with a hurst exponent of 0 5 the hurst exponent is estimated for different time horizons using detrended fluctuation analysis a method suitable for non stationary series with trends in order to identify at which time scale the hurst exponent is consistent with the efficient market hypothesis using high frequency share price exchange rate and stock data we show how price dynamics exhibited important deviations from efficiency for time periods of up to 15 min thereafter price dynamics was consistent with a geometric brownian motion the intraday behaviour of the series also indicated that price dynamics at trade opening and close was hardly consistent with efficiency which would enable investors to exploit price deviations from fundamental values this result is consistent with intraday volume volatility and transaction time duration patterns

interpretation of scaling properties of electroencephalographic fluctuations via spectral analysis and underlying physiology detrended fluctuation analysis has recently demonstrated the existence of two approximate temporal scaling regimes in locally detrended human electroencephalographic eeg fluctuations and has suggested a connection between the location of the breakpoint between regimes and the alpha resonance near 10 hz it is shown here that these scalings can be explained in terms of the filtering of the underlying power spectrum implied by the detrending process using a recent physiologically based model of eeg generation the main features of the scalings and deviations from them are related to the underlying physiology of dendritic propagation and muscle electrical activity and it is concluded that the effects of such physiological features are usually clearer in spectra

long range correlation properties of coding and noncoding dna sequences genbank analysis an open question in computational molecular biology is whether long range correlations are present in both coding and noncoding dna or only in the latter to answer this question we consider all 33301 coding and all 29453 noncoding eukaryotic sequences each of length larger than 512 base pairs bp in the present release of the genbank to dtermine whether there is any statistically significant distinction in their long range correlation properties standard fast fourier transform fft analysis indicates that coding sequences have practically no correlations in the range from 10 bp to 100 bp spectral exponent beta 0 00 0 04 where the uncertainty is two standard deviations in contrast for noncoding sequences the average value of the spectral exponent beta is positive 0 16 0 05 which unambiguously shows the presence of long range correlations we also separately analyze the 874 coding and the 1157 noncoding sequences that have more than 4096 bp and find a larger region of power law behavior we calculate the probability that these two data sets coding and noncoding were drawn from the same distribution and we find that it is less than 10 10 we obtain independent confirmation of these findings using the method of detrended fluctuation analysis dfa which is designed to treat sequences with statistical heterogeneity such as dna s known mosaic structure patchiness arising from the nonstationarity of nucleotide concentration the near perfect agreement between the two independent analysis methods fft and dfa increases the confidence in the reliability of our conclusion

detrended fluctuation analysis of time series of a firing fusimotor neuron abstract we study the interspike intervals isi time series of the spontaneous fusimotor neuron activity by applying the detrended fluctuation analysis that is a modification of the random walk model analysis thus we have found evidence for the white noise characteristics of the isi time series which means that the fusimotor activity does not possess temporal correlations we conclude that such an activity represents the requisite noisy component for occurrence of the stochastic resonance mechanism in the neural coordination of muscle spindles

fractional order anisotropic diffusion for image denoising this paper introduces a new class of fractional order anisotropic diffusion equations for noise removal these equations are euler lagrange equations of a cost functional which is an increasing function of the absolute value of the fractional derivative of the image intensity function so the proposed equations can be seen as generalizations of second order and fourth order anisotropic diffusion equations we use the discrete fourier transform to implement the numerical algorithm and give an iterative scheme in the frequency domain it is one important aspect of the algorithm that it considers the input image as a periodic image to overcome this problem we use a folded algorithm by extending the image symmetrically about its borders finally we list various numerical results on denoising real images experiments show that the proposed fractional order anisotropic diffusion equations yield good visual effects and better signal to noise ratio

effect of the signal filtering on detrended fluctuation analysis abstract detrended fluctuation analysis dfa is an effective method to accurately quantify long term correlations embedded in a nonstationary time series in this paper we study the effect of signal filtering of a signal on the dfa method the theoretical and simulated results show that the signal filtering will affect the range of scale in dfa moreover this effect is different for fractal gaussian noise series and fractal brown movement series our study is meaningful for improving accuracy and efficiency of dfa method in theory and practice

dynamics of electroencephalogram entropy and pitfalls of scaling detection in recent studies a number of research groups have determined that human electroencephalograms eeg have scaling properties in particular a crossover between two regions with different scaling exponents has been reported herein we study the time evolution of diffusion entropy to elucidate the scaling of eeg time series for a cohort of 20 awake healthy volunteers with closed eyes we find that the diffusion entropy of eeg increments obtained from eeg waveforms by differencing exhibits three features short time growth an alpha wave related oscillation whose amplitude gradually decays in time and asymptotic saturation which is achieved after approximately 1 s this analysis suggests a linear stochastic ornstein uhlenbeck langevin equation with a quasiperiodic forcing whose frequency and or amplitude may vary in time as the model for the underlying dynamics this model captures the salient properties of eeg dynamics in particular both the experimental and simulated eeg time series exhibit short time scaling which is broken by a strong periodic component such as alpha waves the saturation of eeg diffusion entropy precludes the existence of asymptotic scaling we find that the crossover between two scaling regions seen in detrended fluctuation analysis dfa of eeg increments does not originate from the underlying dynamics but is merely an artifact of the algorithm this artifact is rooted in the failure of the trend plus signal paradigm of dfa

nodal discontinuous galerkin methods for fractional diffusion equations on 2d domain with triangular meshes this paper as the sequel to previous work develops numerical schemes for fractional diffusion equations on a two dimensional finite domain with triangular meshes we adopt the nodal discontinuous galerkin methods for the full spatial discretization by the use of high order nodal basis employing multivariate lagrange polynomials defined on the triangles stability analysis and error estimates are provided which shows that if polynomials of degree n are used the methods are n 1 th order accurate for general triangulations finally the performed numerical experiments confirm the optimal order of convergence

effect of trends on detrended fluctuation analysis scaling behavior we find that crossovers result from the competition between the scaling of the noise and the apparent scaling of the trend we study how the characteristics of these crossovers depend on i the slope of the linear trend ii the amplitude and period of the periodic trend iii the amplitude and power of the power law trend and iv the length as well as the correlation properties of the noise surprisingly we find that the crossovers in the scaling of noisy signals with trends also follow scaling laws i e long range power law dependence of the position of the crossover on the parameters of the trends we show that the dfa result of noise with a trend can be exactly determined by the superposition of the separate results of the dfa on the noise and on the trend assuming that the noise and the trend are not correlated if this superposition rule is not followed this is an indication that the noise and the superposed trend are not independent so that removing the trend could lead to changes in the correlation properties of the noise in addition we show how to use dfa appropriately to minimize the effects of trends how to recognize if a crossover indicates indeed a transition from one type to a different type of underlying correlation or if the crossover is due to a trend without any transition in the dynamical properties of the noise

random cascades on wavelet dyadic trees we introduce a new class of random fractal functions using the orthogonal wavelet transform these functions are built recursively in the space scale half plane of the orthogonal wavelet transform cascading from an arbitrary given large scale towards small scales to each random fractal function corresponds a random cascading process referred to as a w cascade on the dyadic tree of its orthogonal wavelet coefficients we discuss the convergence of these cascades and the regularity of the so obtained random functions by studying the support of their singularity spectra then we show that very different statistical quantities such as correlation functions on the wavelet coefficients or the wavelet based multifractal formalism partition functions can be used to characterize very precisely the underlying cascading process we illustrate all our results on various numerical examples

autocorrelation type timescale and statistical property in financial time series abstract earlier studies have documented that three types of autocorrelations exist in financial time series sign volatility and return autocorrelation in this paper we examine how each type of the above autocorrelations affects the statistical properties of financial time series and its role in maintaining such statistical properties using three different shuffling series that correspondingly destroy each type of autocorrelation upon different timescales we find that 1 the statistical properties of the shuffling series significantly vary from the original ones 2 volatility and return autocorrelations show greater impacts than sign autocorrelation 3 the effects on the statistical properties are intensified as time scale expands 4 the nonlinear component of autocorrelation is the major drive of the effect

comparison study on the different dynamics between the allen cahn and the cahn hilliard equations abstract we perform a comparison study on the different dynamics between the allen cahn ac and the cahn hilliard ch equations the ac equation describes the evolution of a non conserved order field during anti phase domain coarsening the ch equation describes the process of phase separation of a conserved order field the ac and the ch equations are second order and fourth order nonlinear parabolic partial differential equations respectively linear stability analysis shows that growing and decaying modes for both the equations are the same while the growth rates are monotonically decreasing with respect to the modes for the ac equation the growth rates for the ch equation are increasing and then decreasing with respect to the modes we perform various numerical tests using the fourier spectral method to highlight the different evolutionary dynamics between the ac and the ch equations

fractional cahn hilliard allen cahn and porous medium equations abstract we introduce a fractional variant of the cahn hilliard equation settled in a bounded domain r n and complemented with homogeneous dirichlet boundary conditions of solid type i e imposed in the whole of r n after setting a proper functional framework we prove existence and uniqueness of weak solutions to the related initial boundary value problem then we investigate some significant singular limits obtained as the order of either of the fractional laplacians appearing in the equation is let tend to 0 in particular we can rigorously prove that the fractional allen cahn fractional porous medium and fractional fast diffusion equations can be obtained in the limit finally in the last part of the paper we discuss existence and qualitative properties of stationary solutions of our problem and of its singular limits

arbitrary order hilbert spectral analysis for time series possessing scaling statistics comparison study with detrended fluctuation analysis and wavelet leaders in this paper we present an extended version of hilbert huang transform namely arbitrary order hilbert spectral analysis to characterize the scale invariant properties of a time series directly in an amplitude frequency space we first show numerically that due to a nonlinear distortion traditional methods require high order harmonic components to represent nonlinear processes except for the hilbert based method this will lead to an artificial energy flux from the low frequency large scale to the high frequency small scale part thus the power law if it exists is contaminated we then compare the hilbert method with structure functions sf detrended fluctuation analysis dfa and wavelet leader wl by analyzing fractional brownian motion and synthesized multifractal time series for the former simulation we find that all methods provide comparable results for the latter simulation we perform simulations with an intermittent parameter 0 15 we find that the sf underestimates scaling exponent when q 3 the hilbert method provides a slight underestimation when q 5 however both dfa and wl overestimate the scaling exponents when q 5 it seems that hilbert and dfa methods provide better singularity spectra than sf and wl we finally apply all methods to a passive scalar temperature data obtained from a jet experiment with a taylor s microscale reynolds number re 250 due to the presence of strong ramp cliff structures the sf fails to detect the power law behavior for the traditional method the ramp cliff structure causes a serious artificial energy flux from the low frequency large scale to the high frequency small scale part thus dfa and wl underestimate the scaling exponents however the hilbert method provides scaling exponents q quite close to the one for longitudinal velocity indicating a less intermittent passive scalar field than what was believed before 2011 american physical society

efficiency and multifractality analysis of csi 300 based on multifractal detrending moving average algorithm abstract we apply the multifractal detrending moving average mfdma to investigate and compare the efficiency and multifractality of 5 min high frequency china securities index 300 csi 300 the results show that the csi 300 market becomes closer to weak form efficiency after the introduction of csi 300 future we find that the csi 300 is featured by multifractality and there are less complexity and risk after the csi 300 index future was introduced with the shuffling surrogating and removing extreme values procedures we unveil that extreme events and fat distribution are the main origin of multifractality besides we discuss the knotting phenomena in multifractality and find that the scaling range and the irregular fluctuations for large scales in the f q s vs s plot can cause a knot

statistical volatility duration and complexity of financial dynamics on sierpinski gasket lattice percolation abstract a novel agent based financial dynamics is established based on the percolation system on sierpinski gasket lattice to reproduce the statistical characteristics of financial markets sierpinski gasket lattice is a fractal like graph which corresponds to the fractal well known sierpinski gasket modeling a financial dynamics on sierpinski gasket lattice is a new approach in financial micro mechanism construction in an attempt to investigate the statistical properties of returns and prove the feasibility of the proposed model two new nonlinear statistics maximum and average monotonous volatility duration are introduced in this work for the first time which describe the trend of volatility of return time series in financial markets furthermore a new method cmcid is proposed to measure the synchronization and similarity behaviors of return volatility duration in multiscale the results show that the pairs of real data and simulated data have synchronization and the synchronization becomes stronger when the scale increases and the power law distribution is employed to investigate the corresponding statistical properties which shows that the proposed model shares a common power law distribution with regard to monotonous volatility duration the empirical results show that the simulated series from the financial model and the historical data share common statistical properties which indicate that the proposed model is reasonable in terms of volatility behavior

detecting long range correlations with detrended fluctuation analysis abstract we examine the detrended fluctuation analysis dfa which is a well established method for the detection of long range correlations in time series we show that deviations from scaling which appear at small time scales become stronger in higher orders of dfa and suggest a modified dfa method to remove them the improvement is necessary especially for short records that are affected by non stationarities furthermore we describe how crossovers in the correlation behavior can be detected reliably and determined quantitatively and show how several types of trends in the data affect the different orders of dfa

superroughening by linear growth equations with spatiotemporally correlated noise we give an extensive study on a class of interfacial superroughening processes with finite lateral system size in 1 1 dimensions described by linear growth equations with spatiotemporally power law decaying correlated noise since some of these processes have an extremely long relaxation time we first develop a very efficient method capable of simulating the interface morphology of these growth processes even in very late time we numerically observe that this class of superrough growth processes indeed gradually develops macroscopic structures with the lateral size comparable to the lateral system size through the rigorous analytical study of the equal time height difference correlation function the different time height difference correlation function and the local width we explicitly evaluate not only the leading anomalous dynamic scaling term but also all the subleading anomalous dynamic scaling terms which dominate over the ordinary dynamic scaling term moreover the relation between the macroscopic structure formation and anomalous interfacial roughening of the superrough growth processes is analytically investigated in detail

asymmetric multifractal detrending moving average analysis in time series of pm2 5 concentration abstract in this paper we propose the asymmetric multifractal detrending moving average analysis a mfdma method to explore the asymmetric correlation in non stationary time series the proposed method is applied to explore the asymmetric correlation of pm2 5 daily average concentration with uptrends or downtrends in china in addition shuffling and phase randomization procedures are applied to detect the sources of multifractality the results show that existences of asymmetric correlations and the asymmetric correlations are multifractal further the multifractal scaling behavior in the chinese pm2 5 is caused not only by long range correlation but also by fat tailed distribution but the major source of multifractality is fat tailed distribution

exponentially accurate spectral and spectral element methods for fractional odes current discretizations of fractional differential equations lead to numerical solutions of low order of accuracy here we present different methods for fractional odes that lead to exponentially fast decay of the error first we develop a petrov galerkin pg spectral method for fractional initial value problems fivps of the form d t 0 u t f t and fractional final value problems ffvps d t t u t g t where 0 1 subject to dirichlet initial final conditions these schemes are developed based on a new spectral theory for fractional sturm liouville problems fslps which has been recently developed in 1 specifically we obtain solutions to fivps and ffvps in terms of the new fractional non polynomial basis functions called jacobi polyfractonomials which are the eigenfunctions of the fslp of first kind fslp i correspondingly we employ another space of test functions as the span of polyfractonomial eigenfunctions of the fslp of second kind fslp ii subsequently we develop a discontinuous spectral method dsm of petrov galerkin sense for the aforementioned fivps and ffvps where the basis functions do not satisfy the initial final conditions finally we extend the dsm scheme to a discontinuous spectral element method dsem for efficient longer time integration and adaptive refinement in these discontinuous schemes we employ the asymptotic eigensolutions to fslp i ii which are of jacobi polynomial forms as basis and test functions our numerical tests confirm the exponential algebraic convergence respectively in p and h refinements for various test cases with integer and fractional order solutions

on the well posedness of a spectral fractional forward backward pseudo parabolic equation abstract we prove existence uniqueness and stability of solution of a forward backward pseudo parabolic equation with spectral fractional laplacian then we define and prove existence of young measure valued solution for fractional forward backward problem by the vanishing viscosity method

are the phases in the anderson model long range correlated we investigate the local cumulative phases at single sites of the lattice for time dependent wave functions in the anderson model in d 2 and 3 in addition to a local linear trend the phases exhibit some fluctuations we study the time correlations of these fluctuations using detrended fluctuation analysis our results suggest that the phase fluctuations are long range correlated decaying as a power law with time it seems that the exponent depends on the degree of disorder in d 3 close to the critical disorder wc 16 5 the correlation exponent exhibits a maximum value of 0 6 which is significantly above random fluctuations 0 5

effect of nonstationarities on detrended fluctuation analysis detrended fluctuation analysis dfa is a scaling analysis method used to quantify long range power law correlations in signals many physical and biological signals are noisy heterogeneous and exhibit different types of nonstationarities which can affect the correlation properties of these signals we systematically study the effects of three types of nonstationarities often encountered in real data specifically we consider nonstationary sequences formed in three ways i stitching together segments of data obtained from discontinuous experimental recordings or removing some noisy and unreliable parts from continuous recordings and stitching together the remaining parts a cutting procedure commonly used in preparing data prior to signal analysis ii adding to a signal with known correlations a tunable concentration of random outliers or spikes with different amplitudes and iii generating a signal comprised of segments with different properties e g different standard deviations or different correlation exponents we compare the difference between the scaling results obtained for stationary correlated signals and correlated signals with these three types of nonstationarities we find that introducing nonstationarities to stationary correlated signals leads to the appearance of crossovers in the scaling behavior and we study how the characteristics of these crossovers depend on a the fraction and size of the parts cut out from the signal b the concentration of spikes and their amplitudes c the proportion between segments with different standard deviations or different correlations and d the correlation properties of the stationary signal we show how to develop strategies for preprocessing raw data prior to analysis which will minimize the effects of nonstationarities on the scaling properties of the data and how to interpret the results of dfa for complex signals with different local characteristics

multifractal random walk we introduce a class of multifractal processes referred to as multifractal random walks mrws to our knowledge it is the first multifractal process with continuous dilation invariance properties and stationary increments mrws are very attractive alternative processes to classical cascadelike multifractal models since they do not involve any particular scale ratio the mrws are indexed by four parameters that are shown to control in a very direct way the multifractal spectrum and the correlation structure of the increments we briefly explain how in the same way one can build stationary multifractal processes or positive random measures

superconvergence of a discontinuous galerkin method for fractional diffusion and wave equations we consider an initial boundary value problem for partial tu partial t alpha nabla 2u f t that is for a fractional diffusion 1 alpha 0 or wave 0 alpha 1 equation a numerical solution is found by applying a piecewise linear discontinuous galerkin dg method in time combined with a piecewise linear conforming finite element method in space the time mesh is graded appropriately near t 0 but the spatial mesh is quasi uniform previously we proved that the error measured in the spatial l 2 norm is of order k 2 alpha h 2 ell k uniformly in t where k is the maximum time step h is the maximum diameter of the spatial finite elements alpha min alpha 0 le0 and ell k max 1 log k here we prove convergence of order k 3 2 alpha ell k h 2 at each time level t n for 1 alpha 1 thus if 1 2 alpha 1 then the dg solution is superconvergent which generalizes a known result for the classical heat equation i e the case alpha 0 a

discontinuous galerkin method for fractional convection diffusion equations we propose a discontinuous galerkin method for fractional convection diffusion equations with a superdiffusion operator of order alpha 1 alpha 2 defined through the fractional laplacian the fractional operator of order alpha is expressed as a composite of first order derivatives and a fractional integral of order 2 alpha the fractional convection diffusion problem is expressed as a system of low order differential integral equations and a local discontinuous galerkin method scheme is proposed for the equations we prove stability and optimal order of convergence cal o h k 1 for the fractional diffusion problem and an order of convergence of cal o h k frac 1 2 is established for the general fractional convection diffusion problem the analysis is confirmed by numerical examples

use and misuse of some hurst parameter estimators applied to stationary and non stationary financial time series abstract the detection of long range dependence lrd is an important task in time series analysis lrd is often summarized by the well known hurst parameter or exponent h 0 1 which can be estimated by a number of methods some of these techniques are designed to be applied to signals behaving as a stationary fractional gaussian noise fgn whereas others imply that the analyzed time series behave as a non stationary fractional brownian motion fbm moreover some estimators do not yield the hurst parameter but indexes related to h and ranging outside the unit interval therefore the fgn or fbm nature of the studied time series has to be preliminarily analyzed before applying any estimation method and the relationships between h and the indexes resulting from the analyses have to be taken into account to obtain coherent results since fgn like series represent the increments of fbm like processes and both the signals are characterized by the same h value by definition estimators designed for fgn like series can be applied to fbm like sequences after preventive differentiation and conversely estimators designed for fbm like processes can be applied to fgn like series after preventive integration the signal characterization is particularly important when h is estimated on financial time series because the returns represent the first difference of price time series which are often assumed to behave like self affine sequences the analysis of simulated fgn and fbm time series shows that all the considered methods yield comparable h values when properly applied the reanalysis of several market price time series already studied in the literature points out that a correct application of the estimators supported by a preventive signal classification yields homogeneous h values allowing for a useful cross validation of results reported in different works moreover some conclusions reported in the literature about the anti persistence of some financial series are shown to be incorrect because of the inappropriate application of the estimation methods

the moving averages demystified a common method in technical analysis is the construction of moving averages along time series of stock prices we show that they present a practical interest for physicists and raise new questions on fundamental ground indeed self ane signals characterized by a dened roughness exponent h can be investigated through moving averages the density of crossing points between two moving averages is shown to be a measure of long range power law correlations in a signal finally we present a specic transform with which various structures in a signal e g trends cycles noise etc can be investigated in a systematic way c 1999 elsevier science

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