oscillating singularities on cantor sets a grand canonical multifractal formalism the singular behavior of functions is generally characterized by their holder exponent however we show that this exponent poorly characterizes oscillating singularities we thus introduce a second exponent that accounts for the oscillations of a singular behavior and we give a characterization of this exponent using the wavelet transform we then elaborate on a grand canonical multifractal formalism that describes statistically the fluctuations of both the holder and the oscillation exponents we prove that this formalism allows us to recover the generalized singularity spectrum of a large class of fractal functions involving oscillating singularities

scale invariance analysis of the premature ecg signals abstract the multifractal detrended fluctuation analysis and detrending moving average algorithm were introduced in detail and applied to the study of the multifractal characteristics of the normal signals the atrial premature beat apb signals and the premature ventricular contraction pvc signals by analyzing the generalized hurst exponents renyi exponents and multifractal spectrum and comparing the relation of h h q for original signals and their shuffled time series the result indicated that the three signals have multifractality and present long range correlation in a certain range according to the mean value of we found that the strength of the multifractality is varying the pvc signals is the strongest and the normal signals is the weakest it is useful for clinical practice of medicine to distinguish apb signals with pvc signals

multifractality in the random parameter model for multivariate time series abstract the random parameter model was proposed to explain the structure of the covariance matrix in problems where most but not all of the eigenvalues of the covariance matrix can be explained by random matrix theory in this article we explore the scaling properties of the model as observed in the multifractal structure of the simulated time series we use the wavelet transform modulus maxima technique to obtain the multifractal spectrum dependence with the parameters of the model the model shows a scaling structure compatible with the stylized facts for a reasonable choice of the parameter values

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

p exponent and p leaders part i negative pointwise regularity multifractal analysis aims to characterize signals functions images or fields via the fluctuations of their local regularity along time or space hence capturing crucial features of their temporal spatial dynamics multifractal analysis is becoming a standard tool in signal and image processing and is nowadays widely used in numerous applications of different natures its common formulation relies on the measure of local regularity via the h older exponent by nature restricted to positive values and thus to locally bounded functions or signals it is here proposed to base the quantification of local regularity on p exponents a novel local regularity measure potentially taking negative values first the theoretical properties of p exponents are studied in detail second wavelet based multiscale quantities the p leaders are constructed and shown to permit accurate practical estimation of p exponents exploiting the potential dependence with p it is also shown how the collection of p exponents enriches the classification of locally singular behaviors in functions signals or images the present contribution is complemented by a companion article developing the p leader based multifractal formalism associated to p exponents

continuous cascades in the wavelet space as models for synthetic turbulence we introduce a wide family of stochastic processes that are obtained as sums of self similar localized wave forms with multiplicative intensity in the spirit of the richardson cascade picture of turbulence we establish the convergence and the minimum regularity of our construction we show that its continuous wavelet transform is characterized by stochastic self similarity and multifractal scaling properties this model constitutes a stationary grid free extension of w cascades introduced in the past by arneodo bacry and muzy using a wavelet orthogonal basis moreover our approach generically provides multifractal random functions that are not invariant by time reversal and therefore is able to account for skewed multifractal models and for the so called leverage effect in that respect it can be well suited to providing synthetic turbulence models or to reproducing the main observed features of asset price fluctuations in financial markets

painless nonorthogonal expansions in a hilbert space h discrete families of vectors hj with the property that f j hj f hj for every f in h are considered this expansion formula is obviously true if the family is an orthonormal basis of h but also can hold in situations where the hj are not mutually orthogonal and are overcomplete the two classes of examples studied here are i appropriate sets of weyl heisenberg coherent states based on certain non gaussian fiducial vectors and ii analogous families of affine coherent states it is believed that such quasiorthogonal expansions will be a useful tool in many areas of theoretical physics and applied mathematics

singularity detection and processing with wavelets the mathematical characterization of singularities with lipschitz exponents is reviewed theorems that estimate local lipschitz exponents of functions from the evolution across scales of their wavelet transform are reviewed it is then proven that the local maxima of the wavelet transform modulus detect the locations of irregular structures and provide numerical procedures to compute their lipschitz exponents the wavelet transform of singularities with fast oscillations has a particular behavior that is studied separately the local frequency of such oscillations is measured from the wavelet transform modulus maxima it has been shown numerically that one and two dimensional signals can be reconstructed with a good approximation from the local maxima of their wavelet transform modulus as an application an algorithm is developed that removes white noises from signals by analyzing the evolution of the wavelet transform maxima across scales in two dimensions the wavelet transform maxima indicate the location of edges in images

long range dependencies in heart rate signals revisited abstract the arguments are given that local exponents obtained in multifractal analysis by two methods wavelet transform modulus maxima wtmm and multifractal detrended fluctuation analysis mdfa allow to separate statistically hearts of healthy people and subjects suffering from reduced left ventricle systolic function nyha i iii class proposed indices of fractality suggest that a signal of human heart rate is a mixture of two processes monofractal and multifractal ones

should structure functions be used to estimate power laws in turbulence a comparative study second order structure functions are widely used to characterize turbulence in the inertial range because they are simple to estimate particularly in comparison to spectral density functions and wavelet variances structure function estimators however are highly autocorrelated and as a result no suitable theory has been established to provide confidence intervals for turbulence parameters when determined via regression fits in log log space monte carlo simulations were performed to compare the performance of structure function estimators of turbulence parameters with corresponding multitaper spectral and wavelet variance estimators the simulations indicate that these latter estimators have smaller variances than estimators based upon the structure function in contrast to structure function estimators the statistical properties of the multitaper spectral and wavelet variance estimators allow for the construction of confidence intervals for turbulence parameters the monte carlo simulations also confirm the validity of the statistical theory behind the multitaper spectral and wavelet variance estimators the strengths and weaknesses of the various estimators are further illustrated by analyzing an atmospheric temperature time series

the infinite number of generalized dimensions of fractals and strange attractors abstract we show that fractals in general and strange attractors in particular are characterized by an infinite number of generalized dimensions dq q 0 to this aim we develop a rescaling transformation group which yields analytic expressions for all the quantities dq we prove that lim q 0 dq fractal dimension d limq 1dq information dimension and dq 2 correlation exponent v dq with other integer q s correspond to exponents associated with ternary quaternary and higher correlation functions we prove that generally dq dq for any q q for homogeneous fractals dq dq a particularly interesting dimension is dq for two examples feigenbaum attractor generalized baker s transformation we calculate the generalized dimensions and find that d is a non trivial number all the other generalized dimensions are bounded between the fractal dimension and d

multiscaling in multifractals multiscaling is shown to be a consequence of multifractality when a lower cutoff is introduced in calculations of correlation functions after a suitable rescaling the correlation function data for different values of seem to fall onto a single curve in the multiscaling regime however we show that there is not a unique functional form at varying but a spread very close to a single curve for each this curve can be computed analytically in terms of the f a spectrum which characterizes the multifractal part of this spectrum can thus be obtained by computing only one moment of the weights at 0

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

wavelet transforms of self similar processes this paper deals with the wavelet transforms of self similar random processes of the kind assumed in the kolmogorov 1941 theory of turbulence it is shown that after suitable rescaling the wavelet transform at a given position becomes a stationary random function of the logarithm of the scale argument in the transform the rescaling depends on the scaling exponent the statistical properties of the resulting random fluctuations such as its correlation function depend on the choice of the analyzing wavelet some implications are i the presence of fluctuations in log log plots of the absolute value of wavelet transforms versus the scale ii an estimate of the small scale fluctuations generalizing the iterated logarithm law of brownian motion iii an ultraviolet ergodic formula giving the scaling exponents in terms of zoom averages some observations are made on the issue of fluctuations observed in wavelet transforms of turbulence data

dimensions and entropies of strange attractors from a fluctuating dynamics approach abstract it is shown that the fluctuations in the divergence of near by trajectories on strictly deterministic strange attractors can be modelled by stochastic concepts in particular we propose kramers moyal type equations for correlation functions between points on the attractor the drift terms are the lyapunov exponents the diffusion terms depend on the above fluctuations from this we obtain bounds on generalized dimensions and entropies numerical results show that in nearly all studied cases henon map zaslavskii map mackey glass eq the attractors are fractal measures in the sense of farmer information dimension hausdorff dimension metric entropy topological entropy

a class of nonautonomous coupled kdv systems a class of nonautonomous coupled korteweg de vries kdv systems in 1 1 dimensions are considered for integrability classification integrability of the systems is associated with the existence of a certain recursion operator some new integrable nonautonomous two component kdv systems are found

on the wavelet transformation of fractal objects the wavelet transformation is briefly presented it is shown how the analysis of the local scaling behavior of fractals can be transformed into the investigation of the scaling behavior of analytic functions over the half plane near the boundary of its domain of analyticity as an example a weierstrass like fractal function is considered for which the wavelet transform is related to a jacobi theta function some of the scalings of this theta function are analyzed and give some information about the scaling behavior of this fractal

velocity probability density functions of high reynolds number turbulence abstract this paper deals with the probability density function pdf of velocity differences between two points separated by distance r measurements of pdfs were made for r lying in the inertial range for two different flows in a jet with r 852 and in a wind tunnel with r 2720 these pdfs have a characteristic non gaussian shape with exponential tails following kolmogorov s general ideas of log normality a new model for the pdf is developed which contains two parameters determined by experiments this empirical model agrees with the experimental results that the tails of the pdf deviate from a truly exponential behaviour in particular for small r in addition the model leads to the general scaling law ln e 2 r r 0 differenr from kolmogorov s third hypothesis ln e 2 ln r r 0 restricted to the inertial range only x is x x we develop also a formalism based on an extremum principle which is consistent with both the log normality of e and the above mentioned power law in this formalism can be interpreted as the codimension of dissipative structures and asymptotically varies as 1 ln r

direct determination approach for the multifractal detrending moving average analysis in the canonical framework we propose an alternative approach for the multifractal analysis based on the detrending moving average method mf dma we define a canonical measure such that the multifractal mass exponent q is related to the partition function and the multifractal spectrum f can be directly determined the performances of the direct determination approach and the traditional approach of the mf dma are compared based on three synthetic multifractal and monofractal measures generated from the one dimensional p model the two dimensional p model and the fractional brownian motions we find that both approaches have comparable performances to unveil the fractal and multifractal nature in other words without loss of accuracy the multifractal spectrum f can be directly determined using the new approach with less computation cost we also apply the new mf dma approach to the volatility time series of stock prices and confirm the presence of multifractality

multifractal detrended cross correlation analysis of coding and non coding dna sequences through chaos game representation abstract we propose a new approach combining the chaos game representation and the two dimensional multifractal detrended cross correlation analysis methods to examine multifractal behavior in power law cross correlation between any pair of nucleotide sequences of unequal lengths in this work we analyzed the characteristic behavior of coding and non coding dna sequences of eight prokaryotes the results show the presence of strong multifractal nature between coding and non coding sequences of all data sets we found that this integrative approach helps us to consider complete dna sequences for characterization and further it may be useful for classification clustering identification of class affiliation of nucleotide sequences etc with high precision

pseudononstationarity in the scaling exponents of finite interval time series the accurate estimation of scaling exponents is central in the observational study of scale invariant phenomena natural systems unavoidably provide observations over restricted intervals consequently a stationary stochastic process time series can yield anomalous time variation in the scaling exponents suggestive of nonstationarity the variance in the estimates of scaling exponents computed from an interval of n observations is known for finite variance processes to vary as ensuremath sim 1 n as n ensuremath rightarrow ensuremath infty for certain statistical estimators however the convergence to this behavior will depend on the details of the process and may be slow we study the variation in the scaling of second order moments of the time series increments with n for a variety of synthetic and real world time series and we find that in particular for heavy tailed processes for realizable n one is far from this ensuremath sim 1 n limiting behavior we propose a semiempirical estimate for the minimum n needed to make a meaningful estimate of the scaling exponents for model stochastic processes and compare these with some real world time series

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

the dimension spectrum of some dynamical systems we analyze the dimension spectrum previously introduced and measured experimentally by jensen kadanoff and libchaber using large deviation theory we prove for some invariant measures of expanding markov maps that the hausdorff dimensionf of the set on which the measure has a singularity is a well defined concave and regular function in particular we show that this is the case for the accumulation of period doubling and critical mappings of the circle with golden rotation number we also show in these particular cases that the functionf is universal

multifractal formalism for fractal signals the structure function approach versus the wavelet transform modulus maxima method several attempts have been made recently to generalize the multifractal formalism originally introduced for singular measures to fractal signals we report on a systematic comparison between the structure function approach pioneered by parisi and frisch in 2 proceedings of the international school on turbulence and predictability in geophysical fluid dynamics and climate dynamics edited by m ghil r benzi and g parisi north holland amsterdam 1985 p 84 to account for the multifractal nature of fully developed turbulent signals and an alternative method we have developed within the framework of the wavelet transform analysis we comment on the intrinsic limitations of the structure function approach this technique has fundamental drawbacks and does not provide a full characterization of the singularities of a signal in many cases we demonstrate that our method based on the wavelet transform modulus maxima representation works in most situations and is likely to be the ground of a unified multifractal description of self affine distributions our theoretical considerations are both illustrated on pedagogical examples and supported by numerical simulations

singularity spectrum of fractal signals from wavelet analysis exact results the multifractal formalism for singular measures is revisited using the wavelet transform for bernoulli invariant measures of some expanding markov maps the generalized fractal dimensions are proved to be transition points for the scaling exponents of some partition functions defined from the wavelet transform modulus maxima the generalization of this formalism to fractal signals is established for the class of distribution functions of these singular invariant measures it is demonstrated that the hausdorff dimensiond h of the set of singularities of holder exponenth can be directly determined from the wavelet transform modulus maxima the singularity spectrum so obtained is shown to be not disturbed by the presence in the signal of a superimposed polynomial behavior of ordern provided one uses an analyzing wavelet that possesses at leastn n vanishing moments however it is shown that ac behavior generally induces a phase transition in thed h singularity spectrum that somewhat masks the weakest singularities this phase transition actually depends on the numbern of vanishing moments of the analyzing wavelet its observation is emphasized as a reliable experimental test for the existence of nonsingular behavior in the considered signal these theoretical results are illustrated with numerical examples they are likely to be valid for a large class of fractal functions as suggested by recent applications to fractional brownian motions and turbulent velocity signals

wavelet transform of multifractals we present the wavelet transform as a mathematical microscope which is well suited for studying the local scaling properties of fractal objects we apply this technique to probability measures on self similar cantor sets and to the golden mean trajectories on two tori at the onset of chaos

low order variability diagrams for short range correlation evidence in financial data bgl usd exchange rate dow jones industrial average gold ounce price a method to sort out short range correlations and decorrelations in financial data is tested on three typical sets the bulgarian lev usa dollar bgl usd exchange rate the dow jones industrial average the gold ounce price the method makes use of the so called variability diagram technique three toys are used as models in order to understand features our findings indicate that some predictability can be found at short range time intervals c 1999 elsevier science b v all rights reserved

intermittent process analysis with scattering moments scattering moments provide nonparametric models of random processes with stationary increments they are expected values of random variables computed with a nonexpansive operator obtained by iteratively applying wavelet transforms and modulus nonlinearities which preserves the variance first and second order scattering moments are shown to characterize intermittency and self similarity properties of multiscale processes scattering moments of poisson processes fractional brownian motions levy processes and multifractal random walks are shown to have characteristic decay the generalized method of simulated moments is applied to scattering moments to estimate data generating models numerical applications are shown on financial time series and on energy dissipation of turbulent flows

scaling features of texts images and time series in the given paper we consider the scaling features of long letter sequences like human writings discretized images and discretized 3nancial data using several approaches we show that the symbolic strings and time series being analyzed have a complex multiscale structure and demonstrate di5erent scalings for large and small 6uctuations we discuss complex phenomena in the scaling behavior of partition functions in the case of high frequency dax future data c 2001 elsevier science b v all rights reserved

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

1 f noise and multifractality from bristlecone pine growth explained by the statistical convergence of random data tree ring growth records from bristlecone pines reveal an irregular pattern of fluctuations that have been linked to climatic change but otherwise have remained poorly understood we find within th

multifractal estimates of monofractality in rr heart series in power spectrum ranges abstract two popular estimators of multifractal properties the wavelet transform modulus maxima method and multifractal detrended fluctuation analysis are applied to investigate signals consisting of normal rr series in 39 healthy subjects and 90 patients suffering from systolic dysfunction of the left ventricle however differently from standards for multifractal analysis the scaling is performed separately in intervals corresponding to standard power spectral bands low lf very low vlf and ultra low frequencies ulf tests on fractional brownian motions fbm are done to quantify properties of the estimators as detectors of monofractality in lf vlf and ulf bands arguments are given that multifractal analysis of rr series performed in these bands has a physiological meaning the increased activation of the sympathetic nervous system caused by heart disease is detected evidently only by analysis in lf the transition in multifractal characteristics between diurnal and nocturnal activity takes place when the analysis moves from lf and vlf to ulf only in ulf the diurnal heart rate variability can be approximated by fbm with a self similarity index of h 0 20

characterizing cerebrovascular dynamics with the wavelet based multifractal formalism abstract using the wavelet transform modulus maxima wtmm approach we study the dynamics of cerebral blood flow cbf in rats aiming to reveal responses of macro and microcerebral circulations to changes in the peripheral blood pressure we show that the wavelet based multifractal formalism allows quantifying essentially different reactions in the cbf dynamics at the level of large and small cerebral vessels we conclude that unlike the macrocirculation that is nearly insensitive to increased peripheral blood pressure the microcirculation is characterized by essential changes of the cbf complexity

factorization of joint multifractality abstract complex systems often exhibit multifractal characteristics in various forms the study of the joint fluctuation of multifractal objects referred to as joint multifractality is presented in this work we use the joint partition function approach c meneveau et al phys rev a 41 1990 894 to show that joint multifractality admits a factorization into a common factor related to the notion of relative multifractality studied by riedi and scheuring r h riedi i scheuring fractals 5 1997 153 and a remainder term related to the individual multifractality we demonstrated our ideas using binomial measures and applied to the fluctuation of financial data

multifractal formalism by enforcing the universal behavior of scaling functions a b s t r a c t despite its solid foundations multifractal analysis is still a challenging task the inversed singularity spectrum is a major pitfall in standard multifractal analyses especially for empirical signals to resolve this issue we identified the fan like convergent geometry of scaling functions yielding a limit value termed focus for all moments at the largest scale building on this behavior of scaling we introduce the novel concept of focus based multifractal formalism it relies on enforcing this universal behavior when the moment wise scaling exponents are assessed for the scaling functions besides developing focus based variants of the established multifractal detrended fluctuation analysis and the wavelet leader method we present a novel analytical tool of multifractal signal summation conversion all methods are extensively tested on exact multifractal signals synthesized by the generalized binomial multifractal model in terms of precision and incidence of inversed singularity spectra our focus based variants never yielded inversed spectra and their precision was found similar to that of standard methods our approach allowed computing a moment wise and a global error parameter describing the impact of finite size effect and degree of multifractality as compared to that of the fitted exact multifractal model we demonstrate that the standard approach to multifractal analyses contains a central element that is essentially monofractal due to its regression scheme assessing the scaling exponents for each and every moment separately hence these methods can yield reliable estimates only for ideally behaving multifractal signals in contrast our focus based variants due to their genuine multifractal model fitting always yield reliable estimates accompanied by goodness of fit statistics the presented novel multifractal tools offer means of dealing with the consequences of endogenous impurities of potentially multifractal empirical signals

generalized dimensions entropies and liapunov exponents from the pressure function for strange sets for conformal mixing repellers such as julia sets and nonlinear one dimensional cantor sets we connect the pressure of a smooth transformation on the repeller with its generalized dimensions entropies and liapunov exponents computed with respect to a set of equilibrium gibbs measures this allows us to compute the pressure by means of simple numerical algorithms our results are then extended to axiom a attractors and to a nonhyperbolic invariant set of the line in this last case we show that a first order phase transition appears in the pressure