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

statistical properties of daily ensemble variables in the chinese stock markets abstract we study dynamical behavior of the chinese stock markets by investigating the statistical properties of daily ensemble return and variety defined respectively as the mean and the standard deviation of the ensemble daily price return of a portfolio of stocks traded in china s stock markets on a given day the distribution of the daily ensemble return has an exponential form in the center and power law tails while the variety distribution is lognormal in the bulk followed by a power law tail for large variety based on detrended fluctuation analysis r s analysis and modified r s analysis we find evidence of long memory in the ensemble return and strong evidence of long memory in the evolution of variety

upper and lower bounds on the renyi dimensions and the uniformity of multifractals abstract we prove that for arbitrary probability measures the generalized renyi dimensions d q satisfy the inequality d q q 1 q d q q 1 q d q q 1 q q q q q 0 1 sign q sign q i n particular 1 2 d 2 d d 2 and 1 2 d d 1 d the upper bound corresponds to uniform measures whereas the lower bound describes non uniform measures such as measures with a power law singularity in order to characterize intermediate cases we introduce a quantity that measures the uniformity of multifractals we discuss the relation to the spectrum and apply the formalism to several examples such as power law measures the feigenbaum attractor and the attractor and mode locking structure of the critical circle map

multifractal analysis of chinese stock volatilities based on the partition function approach abstract we have performed a detailed multifractal analysis on the 1 min volatility of two indexes and 1139 stocks in the chinese stock markets based on the partition function approach the partition function q s scales as a power law with respect to the box size s the scaling exponents q form a nonlinear function of q statistical tests based on bootstrapping show that the extracted multifractal nature is significant at the 1 significance level the individual securities can be well modeled by the p model in turbulence with p 0 40 0 02 based on the idea of ensemble averaging including quenched and annealed average we treat each stock exchange as a whole and confirm the existence of multifractal nature in the chinese stock markets

multifractal detrended cross correlation analysis for two nonstationary signals we propose a method called multifractal detrended cross correlation analysis to investigate the multifractal behaviors in the power law cross correlations between two time series or higher dimensional quantities recorded simultaneously which can be applied to diverse complex systems such as turbulence finance ecology physiology geophysics and so on the method is validated with cross correlated one and two dimensional binomial measures and multifractal random walks as an example we illustrate the method by analyzing two financial time series

fractional calculus and continuous time finance in this paper we present a rather general phenomenological theory of tick by tick dynamics in financial markets many well known aspects such as the levy scaling form follow as particular cases of the theory the theory fully takes into account the non markovian and non local character of financial time series predictions on the long time behaviour of the waiting time probability density are presented finally a general scaling form is given based on the solution of the fractional diffusion equation

is the dimension of chaotic attractors invariant under coordinate changes several different dimensionlike quantities which have been suggested as being relevant to the study of chaotic attractors are examined in particular we discuss whether these quantities are invariant under changes of variables that are differentiable except at a finite number of points it is found that some are and some are not it is suggested that the word dimension be reversed only for those quantities have this invariance property

phase transitions in thermodynamics of a local lyapunov exponent for fully developed chaotic systems fluctuations in the divergence of nearby orbits are studied at a crisis point of chaos a statistical thermodynamic method for the description of the fluctuations is developed by using symbolic dynamics which can explicitly write a relation between a fluctuation and reference orbit the thermodynamics the free energy and entropy is exactly analyzed on a nonhyperbolic attractor of maps conjugate to the map u u a for 0 u a andu 1 u 1 a fora u 1 te free energy has discontinuities in its slope the entropy is directly calculated from the partition function then it becomes clear that the collision of a chaotic attractor with a particular fixed point yields a singular local structure in the distribution of fluctuations the existence of first order phase transitions depends on the asymmetry of a map it is shown that each of the coexisting states at the phase transition points is realized with the same probability in the thermodynamic limit

a scaling law how an attractor s volume depends on noise level abstract we investigate the meaning of the dimension of a strange attractor for systems with noise more specifically we investigate the effect of adding noise of magnitude e to a deterministic system with d degrees of freedom if the attractor has dimension d and d d then its volume is zero the addition of noise may be an important physical probe for experimental situations useful for determining how much of the observed phenomena in a system is due to noise already present when the noise is added the attractor a e has positive volume we conjecture that the generalized volume of a e is proportional to e d d for e near 0 and show this relationship is valid in several cases for chaotic attractors there are a variety of ways of defining d and the generalized volume definition must be chosen accordingly

detrending moving average algorithm for multifractals the detrending moving average dma algorithm is a widely used technique to quantify the long term correlations of nonstationary time series and the long range correlations of fractal surfaces which contains a parameter ensuremath theta determining the position of the detrending window we develop multifractal detrending moving average mfdma algorithms for the analysis of one dimensional multifractal measures and higher dimensional multifractals which is a generalization of the dma method the performance of the one dimensional and two dimensional mfdma methods is investigated using synthetic multifractal measures with analytical solutions for backward ensuremath theta 0 centered ensuremath theta 0 5 and forward ensuremath theta 1 detrending windows we find that the estimated multifractal scaling exponent ensuremath tau q and the singularity spectrum f ensuremath alpha are in good agreement with the theoretical values in addition the backward mfdma method has the best performance which provides the most accurate estimates of the scaling exponents with lowest error bars while the centered mfdma method has the worse performance it is found that the backward mfdma algorithm also outperforms the multifractal detrended fluctuation analysis the one dimensional backward mfdma method is applied to analyzing the time series of shanghai stock exchange composite index and its multifractal nature is confirmed

a multifractal detrended fluctuation description of iranian rial us dollar exchange rate the miltifractal properties and scaling behaviour of the exchange rate variations of the iranian rial against the us dollar from a daily perspective is numerically investigated for this purpose the multifractal detrended fluctuation analysis mf dfa is used through multifractal analysis the scaling exponents generalized hurst exponents generalized fractal dimensions and singularity spectrum are derived moreover contribution of two major sources of multifractality that is fat tailed probability distributions and nonlinear temporal correlations are studied

crossing of two mobile averages a method for measuring the roughness exponent self affine signals characterized by a defined hurst roughness exponent h can be investigated through mobile averages the density ensuremath rho of crossing points between any two moving averages is a measure of long range power law correlations in the signal the method is compared to the detrended fluctuation analysis we take advantage of our findings in order to propose a practically powerful and accurate technique for determining h and apply it right away to cases with persistent or antipersistent correlations

on thermodynamics of rational mapsi negative spectrum we study the pressure spectrum p t of the maximal measure for arbitrary rational maps we also consider its modified version which is defined by means of the variational principle with respect to non atomic invariant measures it is shown that for negative values of t the modified spectrum has all major features of the hyperbolic case analyticity the existence of a spectral gap for the corresponding transfer operator rigidity properties etc the spectrum p t can be computed in terms of their legendre transforms are the hausdorff and the box counting dimension spectra of the maximal measure respectively this work is closely related to a paper 32 by d ruelle

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

fractal dimensions and homeomorphic conjugacies we investigate the behavior of the spectrum of singularities associated with the invariant measure of some dynamical systems under nonsmooth coordinate changes when the homeomorphic conjugacy is not lipschitz continuous we discuss how its singularities can affect the whole set of generalized fractal dimensions we give applications to homeomorphisms that conjugate critical circle maps with irrational golden mean winding numbers we present numerical studies corroborating the theoretical predictions

statistical properties of the yuan exchange rate index abstract we choice the yuan exchange rate index based on a basket of currencies as the effective exchange rate of the yuan and investigate the statistical properties of the yuan exchange rate index after china s exchange rate system reform on the 21st july 2005 after dividing the time series into two parts according to the change in the yuan exchange rate regime in july 2008 we compare the statistical properties of the yuan exchange rate index during these two periods we find that the distribution of the two return series has the exponential form we also perform the detrending moving average analysis dma and the multifractal detrending moving average analysis mfdma the two periods possess different degrees of long range correlations and the multifractal nature is also unveiled in these two time series significant difference is found in the scaling exponents q and singularity spectra f of the two periods obtained from the mfdma analysis besides in order to detect the sources of multifractality shuffling and phase randomization procedures are applied to destroy the long range temporal correlation and fat tailed distribution of the yuan exchange rate index respectively we find that the fat tailedness plays a critical role in the sources of multifractality in the first period while the long memory is the major cause in the second period the results suggest that the change in china s exchange rate regime in july 2008 gives rise to the different multifractal properties of the yuan exchange rate index in these two periods and thus has an effect on the effective exchange rate of the yuan after the exchange rate reform on the 21st july 2005

measure of predictability many techniques have been developed to measure the difficulty of forecasting data from an observed time series this paper introduces a measure which we call the forecast entropy designed to measure the predictability of a time series we use attractors reconstructed from the time series and the distributions in the regular and tangent spaces of the data which comprise the attractor we then consider these distributions on different scales we present a formula for calculating the forecast entropy to provide a standard of predictability we define an idealized random system whose forecast entropy will be maximal we then use this measure to rescale the forecast entropy to lie in the range 0 1 the time series obtained from several chaotic systems as well as from a pseudorandom system are studied using this measure we present evidence that the forecast entropy can be used as a tool for determining optimal delays and embedding dimensions used for reconstructing better attractors we also show that the forecast entropy of a random system has completely different characteristics from that of a deterministic one

estimating the distribution of volatility of realized stock returns and exchange rate changes abstract realized stock return volatility is modelled with a distribution based on the laplace distribution the moment properties of suggested volatility distribution are derived the properties of distribution correspond to the empirical regularities found in the finance literature ml estimator for is also provided the advantage of laplace approach lies in estimating from returns distribution f x directly instead of volatility distribution based on bias sensitive standard deviation estimates the goodness to fit tests with 5 day standard deviations of daily hex closing price returns in period 3 1 1983 4 3 2003 daily s p500 closing stock index returns in period 1 3 1950 27 3 2003 and daily usd euro exchange rate changes in period 28 12 1978 28 2 2003 support the suggested volatility distribution model

extraction of dynamical equations from chaotic data abstract a method is described for extracting from a chaotic time series a system of equations whose solution reproduces the general features of the original data even when these are contaminated with noise the equations facilitate calculation of fractal dimension lyapunov exponents and short term predictions the method is applied to data derived from numerical solutions of the logistic equation the henon equations with added noise the lorenz equations and the rossler equations

scale invariant distribution and multifractality of volatility multipliers in stock markets abstract the statistical properties of the multipliers of the absolute returns are investigated using 1 min high frequency data of financial time series the multiplier distribution is found to be independent of the box size s when s is larger than some crossover scale providing direct evidence of the existence of scale invariance in financial data the multipliers with base a 2 are well approximated by a normal distribution and the most probable multiplier scales as a power law with respect to the base a we unravel that the volatility multipliers possess multifractal nature which is independent of construction of the multipliers that is the values of s and a

power law for the calm time interval of price changes abstract in this paper we describe a newly discovered statistical property of time series data for daily price changes we investigated quantitatively the calm time intervals of price changes for 800 companies listed in the tokyo stock exchange and for the nikkei 225 index in the 27 year period from january 1975 to december 2001 a calm time interval is defined as the interval between two successive price changes above a fixed threshold we found that the calm time interval distribution of price changes obeys a power law decay furthermore we show that the power law exponent monotonically decreases with respect to the threshold

nonconcave entropies in multifractals and the thermodynamic formalism we discuss a subtlety involved in the calculation of multifractal spectra when these are expressed as legendre fenchel transforms of functions analogous to free energy functions we show that the legendre fenchel transform of a free energy function yields the correct multifractal spectrum only when the latter is wholly concave if the spectrum has no definite concavity then the transform yields the concave envelope of the spectrum rather than the spectrum itself some mathematical and physical examples are given to illustrate this result which lies at the root of the nonequivalence of the microcanonical and canonical ensembles on a more positive note we also show that the impossibility of expressing nonconcave multifractal spectra through legendre fenchel transforms of free energies can be circumvented with the help of a generalized free energy function which relates to a recently introduced generalized canonical ensemble analogies with the calculation of rate functions in large deviation theory are finally discussed

modeling heart rate variability in healthy humans a turbulence analogy many complex systems share similar statistical characteristics in this letter a turbulence analogy is proposed for the long term heart rate variability of healthy humans based on such an analogy the equivalence of an inertial range is found and a cascade model which captures the statistical properties of the heart rate data is given

inhomogeneous scaling behaviors in malaysian foreign currency exchange rates in this paper we investigate the fractal scaling behaviors of foreign currency exchange rates with respect to malaysian currency ringgit malaysia these time series are examined piecewise before and after the currency control imposed in 1st september 1998 using the monofractal model based on fractional brownian motion the global hurst exponents are determined using the r s analysis the detrended fluctuation analysis and the method of second moment using the correlation coefficients the limitation of these monofractal analyses is discussed the usual multifractal analysis reveals that there exists a wide range of hurst exponents in each of the time series a new method of modelling the multifractal time series based on multifractional brownian motion with time varying hurst exponents is studied

size dependent standard deviation for growth rates empirical results and theoretical modeling we study annual logarithmic growth rates r of various economic variables such as exports imports and foreign debt for each of these variables we find that the distributions of r can be approximated by double exponential laplace distributions in the central parts and power law distributions in the tails for each of these variables we further find a power law dependence of the standard deviation sigma r on the average size of the economic variable with a scaling exponent surprisingly close to that found for the gross domestic product gdp phys rev lett 81 3275 1998 by analyzing annual logarithmic growth rates r of wages of 161 different occupations we find a power law dependence of the standard deviation sigma r on the average value of the wages with a scaling exponent beta 0 14 close to those found for the growth of exports imports debt and the growth of the gdp in contrast to these findings we observe for payroll data collected from 50 states of the usa that the standard deviation sigma r of the annual logarithmic growth rate r increases monotonically with the average value of payroll however also in this case we observe a power law dependence of sigma r on the average payroll with a scaling exponent beta 0 08 based on these observations we propose a stochastic process for multiple cross correlated variables where for each variable i the distribution of logarithmic growth rates decays exponentially in the central part ii the distribution of the logarithmic growth rate decays algebraically in the far tails and iii podobnik the standard deviation of the logarithmic growth rate depends algebraically on the average size of the stochastic variable

anti correlation and multifractal features of spain electricity spot market we use multifractal detrended fluctuation analysis mf dfa to numerically investigate correlation persistence multifractal properties and scaling behavior of the hourly spot prices for the spain electricity exchange compania o peradora del mercado de electricidad omel through multifractal analysis fluctuations behavior the scaling exponents and generalized hurst exponents are studied moreover contribution of fat tailed probability distributions and nonlinear temporal correlations to multifractality is studied

multifractal analysis of nonhyperbolic coupled map lattices application to genomic sequences symbolic sequences generated by coupled map lattices cmls can be used to model the chaotic like structure of genomic sequences in this study it is shown that diffusively coupled chebyshev maps of order 4 corresponding to a shift of four symbols very closely reproduce the multifractal spectrum d q of human genomic sequences for coupling constant 0 35 0 01 if q 0 the presence of rare configurations causes deviations for q 0 which disappear if the rare event statistics of the cml is modified such rare configurations are known to play specific functional roles in genomic sequences serving as promoters or regulatory elements

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

multifractal detrending moving average cross correlation analysis there are a number of situations in which several signals are simultaneously recorded in complex systems which exhibit long term power law cross correlations the multifractal detrended cross correlation analysis mfdcca approaches can be used to quantify such cross correlations such as the mfdcca based on the detrended fluctuation analysis mfxdfa method we develop in this work a class of mfdcca algorithms based on the detrending moving average analysis called mfxdma the performances of the proposed mfxdma algorithms are compared with the mfxdfa method by extensive numerical experiments on pairs of time series generated from bivariate fractional brownian motions two component autoregressive fractionally integrated moving average processes and binomial measures which have theoretical expressions of the multifractal nature in all cases the scaling exponents h xy extracted from the mfxdma and mfxdfa algorithms are very close to the theoretical values for bivariate fractional brownian motions the scaling exponent of the cross correlation is independent of the cross correlation coefficient between two time series and the mfxdfa and centered mfxdma algorithms have comparative performances which outperform the forward and backward mfxdma algorithms for two component autoregressive fractionally integrated moving average processes we also find that the mfxdfa and centered mfxdma algorithms have comparative performances while the forward and backward mfxdma algorithms perform slightly worse for binomial measures the forward mfxdma algorithm exhibits the best performance the centered mfxdma algorithms performs worst and the backward mfxdma algorithm outperforms the mfxdfa algorithm when the moment order q 0 we apply these algorithms to the return time series of two stock market indexes and to their volatilities for the returns the centered mfxdma algorithm gives the best estimates of h xy q since its h xy 2 is closest to 0 5 as expected and the mfxdfa algorithm has the second best performance for the volatilities the forward and backward mfxdma algorithms give similar results while the centered mfxdma and the mfxdfa algorithms fail to extract rational multifractal nature

algorithm to estimate the hurst exponent of high dimensional fractals we propose an algorithm to estimate the hurst exponent of high dimensional fractals based on a generalized high dimensional variance around a moving average low pass filter as working examples we consider rough surfaces generated by the random midpoint displacement and by the cholesky levinson factorization algorithms the surrogate surfaces have hurst exponents ranging from 0 1 to 0 9 with step 0 1 and different sizes the computational efficiency and the accuracy of the algorithm are also discussed

differentiating intraday seasonalities through wavelet multi scaling it is well documented that strong intraday seasonalities may induce distortions in the estimation of volatility models these seasonalities are also the dominant source for the underlying misspecifications of the various volatility models therefore an obvious route is to filter out the underlying intraday seasonalities from the data in this paper we propose a simple method for intraday seasonality extraction that is free of model selection parameters which may affect other intraday seasonality filtering methods our methodology is based on a wavelet multi scaling approach which decomposes the data into its low and high frequency components through the application of a non decimated discrete wavelet transform it is simple to calculate does not depend on a particular model selection criterion or model specific parameter choices the proposed filtering method is translation invariant has the ability to decompose an arbitrary length series without boundary adjustments is associated with a zero phase filter and is circular being circular helps to preserve the entire sample unlike other two sided filters where data loss occurs from the beginning and the end of the studied sample

multifractality in stock indexes fact or fiction abstract multifractal analysis and extensive statistical tests are performed upon intraday minutely data within individual trading days for four stock market indexes including hsi szsc s p 500 and nasdaq to check whether the indexes instead of the returns possess multifractality we find that the mass exponent q is linear and the singularity q is close to 1 for all trading days and all indexes furthermore we find strong evidence showing that the scaling behaviors of the original data sets cannot be distinguished from those of shuffled time series hence the so called multifractality in the intraday stock market indexes is merely an illusion

cross correlations between volume change and price change in finance one usually deals not with prices but with growth rates r defined as the difference in logarithm between two consecutive prices here we consider not the trading volume but rather the volume growth rate r the difference in logarithm between two consecutive values of trading volume to this end we use several methods to analyze the properties of volume changes r and their relationship to price changes r we analyze 14 981 daily recordings of the standard and poor s s p 500 index over the 59 year period 1950 2009 and find power law cross correlations between r and r by using detrended cross correlation analysis dcca we introduce a joint stochastic process that models these cross correlations motivated by the relationship between r and r we estimate the tail exponent of the probability density function p r r 1 for both the s p 500 index as well as the collection of 1819 constituents of the new york stock exchange composite index on 17 july 2009 as a new method to estimate we calculate the time intervals q between events where r q we demonstrate that q the average of q obeys q q we find 3 furthermore by aggregating all q values of 28 global financial indices we also observe an approximate inverse cubic law

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

detrended fluctuation analysis for fractals and multifractals in higher dimensions one dimensional detrended fluctuation analysis dfa and multifractal detrended fluctuation analysis mfdfa are widely used in the scaling analysis of fractal and multifractal time series because they are accurate and easy to implement in this paper we generalize the one dimensional dfa and mfdfa to higher dimensional versions the generalization works well when tested with synthetic surfaces including fractional brownian surfaces and multifractal surfaces the two dimensional mfdfa is also adopted to analyze two images from nature and experiment and nice scaling laws are unraveled

a characteristic time scale in dollar yen exchange rates abstract here we analyze tick data of yen dollar exchange using random walk methods we find that there exists a characteristic time scale approximately at 10 min according to the results at time scales shorter than 10 min the market exhibits anti persistence meaning that it self organizes so as to restore a given tendency for time scales longer than 10 min the market approaches a behavior appropriate to pure brownian motion

construction of phenomenological models from numerical scalar time series abstract the problem of reconstruction of ordinary differential equations from numerical scalar time series is discussed techniques are exemplified for rossler and lorenz chaotic attractors with emphasis on improved algorithms with respect to those previously published the steps which are still required in order to investigate experimental noisy data are discussed

a possible scheme for classifying phase transitions in the generalized entropy spectrum of a nonhyperbolic system abstract in this contribution the generalized thermodynamic formalism is applied to a nonhyperbolic dynamical system the system is used to discuss the occurence of combination of phase transitions among different entropy like scaling functions for the discussion the associated generalized entropy function is calculated from an approximation of finite level and for the asymptotic case from this unifying description of the system the information contained in the particular spectra as well as the relation among the different spectra can be derived and made visible in an easy way it is shown how the phase transition like behavior leads to the manifestation of a critical line in the surface described by the generalized entropy function which in turn determines the occurence of phase transitions for the particular spectra furthermore it is indicated how apparent phase transitions are encountered in approximations of a finite level which do not persist in the asymptotic limit

fractal dimension of spatially extended systems properties of the invariant measure are numerically investigated in 1d chains of diffusively coupled maps the coarse grained fractal dimension is carefully computed in various embedding spaces observing an extremely slow convergence towards the asymptotic value this is in contrast with previous simulations where the analysis of an insufficient number of points led the authors to underestimate the increase of fractal dimension with increasing the dimension of the embedding space orthogonal decomposition is also performed confirming that the slow convergence is intrinsically related to local nonlinear properties of the invariant measure finally the kaplan yorke conjecture is tested for short chains showing that despite the noninvertibility of the dynamical system a good agreement is found between lyapunov dimension and information dimension

detrended fluctuation analysis on spot and futures markets of west texas intermediate crude oil abstract 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

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

detrending moving average algorithm a closed form approximation of the scaling law abstract the hurst exponent h of long range correlated series can be estimated by means of the detrending moving average dma method the computational tool on which the algorithm is based is the generalized variance dma 2 1 n n i n n y i y n i 2 with y n i 1 n k 0 n y i k being the average over the moving window n and n the dimension of the stochastic series y i the ability to yield h relies on the property of dma 2 to vary as n 2 h over a wide range of scales e alessio a carbone g castelli v frappietro eur j phys b 27 2002 197 here we give a closed form proof that dma 2 is equivalent to c h n 2 h and provide an explicit expression for c h we furthermore compare the values of c h with those obtained by applying the dma algorithm to artificial self similar signals

brownian motion from deterministic dynamics abstract certain deterministic dynamical systems exhibit a transition from non gaussian chaotic to brownian motion behavior when a suitable scaling limit is applied for a simple hyperbolic model system we analyze the structure of the corresponding strange attractor determine the invariant measure and elucidate the transition scenario leading to a gaussian stochastic process

some results on the behavior and estimation of the fractal dimensions of distributions on attractors the strong interest in recent years in analyzing chaotic dynamical systems according to their asymptotic behavior has led to various definitions of fractal dimension and corresponding methods of statistical estimation in this paper we first provide a rigorous mathematical framework for the study of dimension focusing on pointwise dimension x and the generalized renyi dimensionsd q and give a rigorous proof of inequalities first derived by grassberger and procaccia and hentschel and procaccia we then specialize to the problem of statistical estimation of the correlation dimension and information dimension it has been recognized for some time that the error estimates accompanying the usual procedures which generally involve least squares methods and nearest neighbor calculations grossly underestimate the true statistical error involved in least squares analyses of and we identify sources of error not previously discussed in the literature and address the problem of obtaining accurate error estimates we then develop an estimation procedure for which corrects for an important bias term the local measure density and provides confidence intervals for the general applicability of this method is illustrated with various numerical examples

statistical thermodynamic approach to a chaotic dynamical system exactly solvable examples the static and dynamic properties of a chaotic attractor of a two dimensional map are studied which belongs to a particular class of piecewise continuous invertible maps coverings of a natural size to cover the attractor are introduced so that the microscopic information of the attractor is written on each box composing the cover the statistical thermodynamics of the scaling indices and the size indices of the boxes is formulated analytic forms of the free energy functions of the scaling indices and the size indices of the boxes are obtained for examples of a hyperbolic and a nonhyperbolic chaotic attractor the statistical thermodynamics of local lyapunov exponents is also studied and a relation between the thermodynamics of scaling indices and of local lyapunov exponents is invetigated for the nonhyperbolic example the free energy and entropy functions of local lyapunov exponents are obtained in analytic forms these results display the existence of phase transitions a phase transition is seen in the thermodynamics of scaling indices also

on the geometric structure of non hyperbolic attractors the authors discuss the f alpha spectrum of non hyperbolic attractors of the henon type they elucidate the origin of the phase transition found in a previous paper and give a lower bound to the spectrum in the non hyperbolic phase where kaplan yorke type formulae no longer hold their results disagree with other recent attempts numerical simulations for the henon map agree with analytical estimates

properties of low variability periods in financial time series abstract properties of low variability periods in the time series are analysed the theoretical approach is used to show the relationship between the multi scaling of low variability periods and multi affinity of the time series it is shown that this technically simple method is capable of revealing more details about time series than the traditional multi affine analysis we have applied this scaling analysis to financial time series a number of daily currency and stock index time series the results show a good scaling behaviour for different model parameters the analysis of high frequency usd eur exchange rate data confirmed the theoretical expectations

quantum field theory on fractal spacetime a new regularisation method spacetime is modelled as a fractal subset of rn analysis on homogeneous sets with non integer hausdorff dimensions is applied to the low order perturbative renormalisation of quantum electrodynamics this new regularisation method implements the dirac matrices and tensors in r4 without difficulties is gauge invariant covariant and differs from dimensional regularisation in some aspects

analysis of clusters formed by the moving average of a long range correlated time series we analyze the stochastic function c n i ensuremath equiv y i ensuremath y n i where y i is a long range correlated time series of length n mathrm max and y n i ensuremath equiv 1 n ensuremath sum k 0 n ensuremath 1 y i ensuremath k is the moving average with window n we argue that c n i generates a stationary sequence of self affine clusters mathcal c with length mathcal l lifetime ensuremath tau and area s the length and the area are related to the lifetime by the relationships mathcal l ensuremath sim ensuremath tau ensuremath psi mathcal l and s ensuremath sim ensuremath tau ensuremath psi s where ensuremath psi mathcal l 1 and ensuremath psi s 1 h we also find that mathcal l ensuremath tau and s are power law distributed with exponents depending on h p mathcal l ensuremath sim mathcal l ensuremath ensuremath alpha p ensuremath tau ensuremath sim ensuremath tau ensuremath ensuremath beta and p s ensuremath sim s ensuremath ensuremath gamma with ensuremath alpha ensuremath beta 2 ensuremath h and ensuremath gamma 2 1 h these predictions are tested by extensive simulations on series generated by the midpoint displacement algorithm of assigned hurst exponent h ranging from 0 05 to 0 95 of length up to n mathrm max 2 21 and n up to 2 13

scaling analysis of multi variate intermittent time series abstract the scaling properties of the time series of asset prices and trading volumes of stock markets are analysed it is shown that similar to the asset prices the trading volume data obey multi scaling length distribution of low variability periods in the case of asset prices such scaling behaviour can be used for risk forecasts the probability of observing next day a large price movement is super universally inversely proportional to the length of the ongoing low variability period finally a method is devised for a multi factor scaling analysis we apply the simplest two factor model to equity index and trading volume time series

probability of large movements in financial markets abstract based on empirical financial time series we show that the silence breaking probability follows a super universal power law the probability of observing a large movement is inversely proportional to the length of the on going low variability period such a scaling law has been previously predicted theoretically r kitt j kalda physica a 353 2005 480 assuming that the length distribution of the low variability periods follows a multi scaling power law

analysis of high resolution foreign exchange data of usd jpy for 13 years we analyze high resolution foreign exchange data consisting of 20 million data points of usd jpy for 13 years to report firm statistical laws in distributions and correlations of exchange rate fluctuations a conditional probability density analysis clearly shows the existence of trend following movements at time scale of 8 ticks about 1 minute

variety and volatility in financial markets we study the price dynamics of stocks traded in a financial market by considering the statistical properties of both a single time series and an ensemble of stocks traded simultaneously we use the n stocks traded on the new york stock exchange to form a statistical ensemble of daily stock returns for each trading day of our database we study the ensemble return distribution we find that a typical ensemble return distribution exists in most of the trading days with the exception of crash and rally days and of the days following these extreme events we analyze each ensemble return distribution by extracting its first two central moments we observe that these moments fluctuate in time and are stochastic processes themselves we characterize the statistical properties of ensemble return distribution central moments by investigating their probability density functions and temporal correlation properties in general time averaged and portfolio averaged price returns have different statistical properties we infer from these differences information about the relative strength of correlation between stocks and between different trading days last we compare our empirical results with those predicted by the single index model and we conclude that this simple model cannot explain the statistical properties of the second moment of the ensemble return distribution

scaling laws for invariant measures on hyperbolic and nonhyperbolic atractors the analysis of dynamical systems in terms of spectra of singularities is extended to higher dimensions and to nonhyperbolic systems prominent roles in our approach are played by the generalized partial dimensions of the invariant measure and by the distribution of effective liapunov exponents for hyperbolic attractors the latter determines the metric entropies and provides one constraint on the partial dimensions for nonhyperbolic attractors there are important modifications we discuss them for the examples of the logistic and henon map we show in particular that the generalized dimensions have singularities with noncontinuous derivative similar to first order phase transitions in statistical mechanics

long term correlations and multifractal nature in the intertrade durations of a liquid chinese stock and its warrant abstract the intertrade duration of equities is an important financial measure characterizing trading activities it is defined as the waiting time between successive trades of an equity using the ultrahigh frequency data of a liquid chinese stock and its associated warrant we perform a comparative investigation of the statistical properties of their intertrade duration time series the distributions of the two equities can be better described by the shifted power law form than the weibull form and their scaled distributions do not collapse onto a single curve although the intertrade durations of the two equities have very different magnitude their intraday patterns exhibit very similar shapes both detrended fluctuation analysis dfa and detrending moving average analysis dma show that the 1 min intertrade duration time series of the two equities are strongly correlated in addition both multifractal detrended fluctuation analysis mfdfa and multifractal detrending moving average analysis mfdma unveil that the 1 min intertrade durations possess multifractal nature however the difference between the two singularity spectra of the two equities obtained from the mfdma is much smaller than that from the mfdfa

multifractal detrended fluctuation analysis of derivative and spot markets abstract we investigate the multifractal properties of price increments in the cases of derivative and spot markets through the multifractal detrended fluctuation analysis we estimate the generalized hurst and the renyi exponents for price fluctuations by deriving the singularity spectrum from the above exponents we quantify the multifractality of a financial time series and compare the multifractal properties of two different markets the different behavior of each agent group in transactions is also discussed in order to identify the nature of the underlying multifractality we apply the method of surrogate data to both sets of financial data it is shown that multifractality due to a fat tailed distribution is significant

scaling properties of foreign exchange volatility in this paper we investigate the scaling properties of foreign exchange volatility our methodology is based on a wavelet multi scaling approach which decomposes the variance of a time series and the covariance between two time series on a scale by scale basis through the application of a discrete wavelet transformation it is shown that foreign exchange rate volatilities follow different scaling laws at different horizons particularly there is a smaller degree of persistence in intra day volatility as compared to volatility at one day and higher scales therefore a common practice in the risk management industry to convert risk measures calculated at shorter horizons into longer horizons through a global scaling parameter may not be appropriate this paper also demonstrates that correlation between the foreign exchange volatilities is the lowest at the intra day scales but exhibits a gradual increase up to a daily scale the correlation coefficient stabilizes at scales one day and higher therefore the benefit of currency diversification is the greatest at the intra day scales and diminishes gradually at higher scales lower frequencies the wavelet cross correlation analysis also indicates that the association between two volatilities is stronger at lower frequencies

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

fractal heterogeneous media a method is presented for generating compact fractal disordered media by generalizing the random midpoint displacement algorithm the obtained structures are invasive stochastic fractals with the hurst exponent varying as a continuous parameter as opposed to lacunar deterministic fractals such as the menger sponge by employing the detrending moving average algorithm a carbone phys rev e 76 056703 2007 the hurst exponent of the generated structure can be subsequently checked the fractality of such a structure is referred to a property defined over a three dimensional topology rather than to the topology itself consequently in this framework the hurst exponent should be intended as an estimator of compactness rather than of roughness applications can be envisaged for simulating and quantifying complex systems characterized by self similar heterogeneity across space for example exploitation areas range from the design and control of multifunctional self assembled artificial nanostructures and microstructures to the analysis and modeling of complex pattern formation in biology environmental sciences geomorphological sciences etc

testing for nonlinearity in time series the method of surrogate data we describe a statistical approach for identifying nonlinearity in time series the method first specifies some linear process as a null hypothesis then generates surrogate data sets which are consistent with this null hypothesis and finally computes a discriminating statistic for the original and for each of the surrogate data sets if the value computed for the original data is significantly different than the ensemble of values computed for the surrogate data then the null hypothesis is rejected and nonlinearity is detected we discuss various null hypotheses and discriminating statistics the method is demonstrated for numerical data generated by known chaotic systems and applied to a number of experimental time series which arise in the measurement of superfluids brain waves and sunspots we evaluate the statistical significance of the evidence for nonlinear structure in each case and illustrate aspects of the data which this approach identifies

multifractal moving average analysis and test of multifractal model with tuned correlations we address two common major problems in the study of time series characterizing fluctuations in complex systems multifractal analysis and multifractal modeling specifically we introduce a multi fractal centered moving average mf cma analysis which is computationally easier but equivalently performing compared with the well established multi fractal detrended fluctuation analysis mf dfa with linear detrending in addition we study in detail a generalized binomial multi fractal model gb mfm to conveniently and reliably generate multifractal surrogate data with arbitrary singularity strengths and arbitrary long term persistence we use the data generated by this model as well as realistic by construction monofractal data series with crossovers and trends to test and compare the multifractal analysis methods and discuss finite size effects as well as limitations due to spurious multifractality

fractional calculus and continuous time finance ii the waiting time distribution we complement the theory of tick by tick dynamics of financial markets based on a continuous time random walk ctrw model recently proposed by scalas et al 4 and we point out its consistency with the behaviour observed in the waiting time distribution for bund future prices traded at liffe london

introducing false eur and false eur exchange rates abstract the euro eur is a new currency introduced by the european community its exchange rate is very puzzling we have invented a false euro feur dating back to 1993 and have derived the exchange rates of the feur with respect to currencies not belonging to the eur i e dkk chf jpy and usd this allows us to search for correlations between the fluctuations preexisting to the introduction of eur and present ones in such financial data the detrended fluctuation analysis dfa statistical method is used this leads to assume a power law behavior i e a scaling hypothesis through an exponent the latter has demonstrated its usefulness for the investigations of long range power law correlations in several types of financial sequences our findings show that the exponent interestingly characterizes fractional brownian motion of the currency exchange rates between eur and dkk over a 25 day interval and usual brownian motion otherwise and for the three other investigated exchange rates we can devise an investment strategy based on the local technique and obtain appreciable gains for the time being

universal features in the growth dynamics of complex organizations we analyze the fluctuations in the gross domestic product gdp of 152 countries for the period 1950 1992 we find that i the distribution of annual growth rates for countries of a given gdp decays with fatter tails than for a gaussian and ii the width of the distribution scales as a power law of gdp with a scaling exponent b 0 15 both findings are in surprising agreement with results on firm growth these results are consistent with the hypothesis that the evolution of organizations with complex structure is governed by similar growth mechanisms s0031 9007 98 07339 6 pacs numbers 02 50 ey 01 75 m 05 40 j 05 45 b in the study of physical systems the analysis of the scaling properties of the fluctuations has been shown to give important information regarding the underlying processes responsible for the observed macroscopic behavior in contrast most studies on the time evolution of economic time series have concentrated on average growth rates 1 18 here we investigate the possibility that the study of fluctuations in economics may also lead to a better understanding of the mechanisms responsible for the observed dynamics 19 23 we therefore analyze the fluctuations in the growth rate of the gross domestic product gdp of 152 countries during the period 1950 1992 24 we will show that i the distribution of annual growth rates for countries of a given gdp is consistent for a certain range with an exponential decay and ii the width of the distribution scales as a power law of gdp with a scaling exponent b 0 15 both findings are in surprising agreement with