detrending fluctuation analysis based on moving average filtering abstract detrended fluctuation analysis dfa is a scaling method commonly used for detecting long range correlations in nonstationary time series applications range from financial time series to physiological data however as the removal of trends in dfa is based on discontinuous polynomial fitting oscillations in the fluctuation function and significant errors in crossover locations can be introduced to reduce the problems induced by discontinuous fitting moving average ma methods have been proposed previously by alesio et al eur j phys b 27 2002 197 in this work a variant of such ma methods is studied specifically the performance and characteristics of a ma method based on central differences is studied some important properties of this ma method are analyzed and illustrated with several artificial and real time series

can one make any crash prediction in finance using the local hurst exponent idea abstract we apply the hurst exponent idea for investigation of djia index time series data the behavior of the local hurst exponent prior to drastic changes in financial series signal is analyzed the optimal length of the time window over which this exponent can be calculated in order to make some meaningful predictions is discussed our prediction hypothesis is verified with examples of 1929 and 1987 crashes as well as with more recent phenomena in stock market from the period 1995 to 2003 some interesting agreements are found

fractal properties of financial markets abstract we present an analysis of the usa stock market using a simple fractal function financial bubbles preceding the 1987 2000 and 2007 crashes are investigated using the besicovitch ursell fractal function fits show a good agreement with the s p 500 data when a complete financial growth is considered starting at the threshold of the abrupt growth and ending at the peak moving the final time of the fitting interval towards earlier dates causes growing discrepancy between two curves on the basis of a detailed analysis of the financial index behavior we propose a method for identifying the stage of the current financial growth and estimating the time in which the index value is going to reach the maximum

tta a new approach to estimate hurst exponent with less estimation error and computational time abstract investigation of long memory processes in signals can give us an important information about how signals have behaved so far and how will it behave in future hurst exponent estimation is a proper tool to show memory in signals rescaled range analysis r s detrended fluctuation analysis dfa and generalized hurst exponent ghe are most known methods for estimation of hurst exponent which introduced in literature in this paper we propose a new algorithm to estimate hurst exponent based on triangles total areas tta that can be made out of three samples of different lag in time series to test our algorithm performance we used two kinds of synthetic waveforms with known hurst exponents results indicates that the proposed method is superior with respect to data length estimation error computational time and noise sensitivity we also apply our proposed method in epilepsy detection and compare our results with previous works to show outperformance of our algorithm with accuracy of 94 5 in classification between interictal and ictal eeg signals

multifractal characterization of protein contact networks abstract the multifractal detrended fluctuation analysis of time series is able to reveal the presence of long range correlations and at the same time to characterize the self similarity of the series the rich information derivable from the characteristic exponents and the multifractal spectrum can be further analyzed to discover important insights into the underlying dynamical process in this paper we employ multifractal analysis techniques in the study of protein contact networks to this end initially a network is mapped to three different time series each of which is generated by a stationary unbiased random walk to capture the peculiarities of the networks at different levels we accordingly consider three observables at each vertex the degree the clustering coefficient and the closeness centrality to compare the results with suitable references we consider also instances of three well known network models and two typical time series with pure monofractal and multifractal properties the first result of notable interest is that time series associated to protein contact networks exhibit long range correlations strong persistence which are consistent with signals in between the typical monofractal and multifractal behavior successively a suitable embedding of the multifractal spectra allows to focus on ensemble properties which in turn gives us the possibility to make further observations regarding the considered networks in particular we highlight the different role that small and large fluctuations of the considered observables play in the characterization of the network topology

research on the relationship between the multifractality and long memory of realized volatility in the sseci abstract we examine the multifractal properties of the realized volatility rv and realized bipower variation rbv series in the shanghai stock exchange composite index sseci by using the multifractal detrended fluctuation analysis mf dfa method we find that there exist distinct multifractal characteristics in the volatility series the contributions of two different types of source of multifractality namely fat tailed probability distributions and nonlinear temporal correlations are studied by using the unit root test we also find the strength of the multifractality of the volatility time series is insensitive to the sampling frequency but that the long memory of these series is sensitive

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

runge kutta methods for first order periodic boundary value differential equations with piecewise constant arguments in this paper we deal with the numerical solutions of runge kutta methods for first order periodic boundary value differential equations with piecewise constant arguments the numerical solution is given by the numerical green s function it is shown that runge kutta methods preserve their original order for first order periodic boundary value differential equations with piecewise constant arguments we give the conditions under which the numerical solutions preserve some properties of the analytic solutions e g uniqueness and comparison theorems finally some experiments are given to illustrate our results

on hurst exponent estimation under heavy tailed distributions abstract in this paper we show how the sampling properties of the hurst exponent methods of estimation change with the presence of heavy tails we run extensive monte carlo simulations to find out how rescaled range analysis r s multifractal detrended fluctuation analysis m f d f a detrending moving average d m a and generalized hurst exponent approach g h e estimate hurst exponent on independent series with different heavy tails for this purpose we generate independent random series from stable distribution with stability exponent changing from 1 1 heaviest tails to 2 gaussian normal distribution and we estimate the hurst exponent using the different methods r s and g h e prove to be robust to heavy tails in the underlying process g h e provides the lowest variance and bias in comparison to the other methods regardless the presence of heavy tails in data and sample size utilizing this result we apply a novel approach of the intraday time dependent hurst exponent and we estimate the hurst exponent on high frequency data for each trading day separately we obtain hurst exponents for s p500 index for the period beginning with year 1983 and ending by november 2009 and we discuss the surprising result which uncovers how the market s behavior changed over this long period

time dependent hurst exponent in financial time series abstract we calculate the hurst exponent h t of several time series by dynamical implementation of a recently proposed scaling technique the detrending moving average dma in order to assess the accuracy of the technique we calculate the exponent h t for artificial series simulating monofractal brownian paths with assigned hurst exponents h we next calculate the exponent h t for the return of high frequency tick by tick sampled every minute series of the german market we find a much more pronounced time variability in the local scaling exponent of financial series compared to the artificial ones the dma algorithm allows the calculation of the exponent h t without any a priori assumption on the stochastic process and on the probability distribution function of the random variables as happens for example in the case of the kitagawa grid and the extended kalmann filtering methods the present technique examines the local scaling exponent h t around a given instant of time this is a significant advance with respect to the standard wavelet transform or to the higher order power spectrum technique which instead operate on the global properties of the series by legendre or fourier transform of qth order moments

mixed correlated arfima processes for power law cross correlations abstract we introduce a general framework of the mixed correlated arfima mc arfima processes which allows for various specifications of univariate and bivariate long term memory apart from a standard case when h x y 1 2 h x h y mc arfima also allows for processes with h x y 1 2 h x h y but also for long range correlated processes which are either short range cross correlated or simply correlated the major contribution of mc arfima lies in the fact that the processes have well defined asymptotic properties for h x h y and h x y which are derived in the paper so that the processes can be used in simulation studies comparing various estimators of the bivariate hurst exponent h x y moreover the framework allows for modeling of processes which are found to have h x y 1 2 h x h y

multifractal analysis of shanghai and hong kong stock markets before and after the connect program abstract in this paper we study the multifractal scaling behaviour in shanghai and hong kong stock markets by means of multifractal detrended fluctuation analysis mf dfa and multifractal detrended cross correlation analysis mf dcca the results show that the multifractal degrees of each stock market are larger after the shanghai hong kong stock connect program shscp than before scaling analysis demonstrates that multifractality exists in cross correlations and the cross correlation coefficients after the shscp are larger than those before the shscp moreover an analysis of the origin of multifractality indicates that long range correlation and fat tailed distribution play important roles in the contributions of multifractality finally the results via the sliding window procedure indicate that the multifractal degrees after the shscp are not significantly affected by the stock market turbulence in 2015

finite sample properties of power law cross correlations estimators abstract we study finite sample properties of estimators of power law cross correlations detrended cross correlation analysis dcca height cross correlation analysis hxa and detrending moving average cross correlation analysis dmca with a special focus on short term memory bias as well as power law coherency we present a broad monte carlo simulation study that focuses on different time series lengths specific methods parameter setting and memory strength we find that each method is best suited for different time series dynamics so that there is no clear winner between the three the method selection should be then made based on observed dynamic properties of the analyzed series

on favard s theorems abstract in this paper we improve and extend the classical favard s theorems i e favard s theorem of the module containment favard s theorem of linear differential equations we study favard s theory of linear differential equations with piecewise constant argument an example shows that the new module containment is necessary in the study of differential equations with piecewise constant argument the equivalence between almost automorphic functions and n almost periodic ones is studied

a comment on measuring the hurst exponent of financial time series abstract a fundamental hypothesis of quantitative finance is that stock price variations are independent and can be modeled using brownian motion in recent years it was proposed to use rescaled range analysis and its characteristic value the hurst exponent to test for independence in financial time series theoretically independent time series should be characterized by a hurst exponent of 1 2 however finite brownian motion data sets will always give a value of the hurst exponent larger than 1 2 and without an appropriate statistical test such a value can mistakenly be interpreted as evidence of long term memory we obtain a more precise statistical significance test for the hurst exponent and apply it to real financial data sets our empirical analysis shows no long term memory in some financial returns suggesting that brownian motion cannot be rejected as a model for price dynamics

stability of θ methods for advanced differential equations with piecewise continuous arguments abstract this paper deals with the stability analysis of numerical methods for the solution of advanced differential equations with piecewise continuous arguments we focus on the behaviour of the one leg method and the linear method in the solution of the equation x t ax t a 0 x t a 1 x t 1 with real a a 0 a 1 and designates the greatest integer function the stability regions of two methods are determined the conditions under which the analytic stability region is contained in the numerical stability region are obtained and some numerical experiments are given

time varying hurst exponent for us stock markets abstract in this work the dynamical behavior of the us stock markets is characterized on the basis of the temporal variations of the hurst exponent estimated with detrended fluctuation analysis dfa over moving windows for the historical dow jones 1928 2007 and the s p 500 1950 2007 daily indices according to the results drawn i the hurst exponent displays an erratic dynamics with some episodes alternating low and high persistent behavior ii the major breakthrough of the long term trend of the scaling behavior occurred in 1972 at the end of the bretton woods system when the hurst exponent shifted form a positive to a negative long term trend other effects such as the 1987 crisis and the emergence of anti correlated behavior in the recent two years are also discussed

hurst exponent footprints from activities on a large structural system abstract this paper presents hurst exponent footprints from pseudo dynamic measurements of significantly varied activities on a damaged bridge structure during rehabilitation through continuous monitoring the system is interesting due to associated uncertainty in large scale structures and significant presence of human intervention arising from fundamentally different processes investigations into the variation of computed hurst exponents on time series of limited lengths are carried out in this regard the hurst exponents are compared with respect to specific events during the rehabilitation as well as with the data collection locations the variations of local hurst exponents about the values computed for each activity are presented the scaling of hurst exponents for different activities is also investigated these are representative of the extent of multifractality for each event the extent of multifractality is assessed along with its source and time dependency

the sampling properties of hurst exponent estimates abstract the classical rescaled adjusted range r s statistic is a popular and robust tool for identifying fractal structures and long term dependence in time series data subsequent to mandelbrot and wallis water resour res 4 1968 909 who proposed the statistic be measured over several subseries contained within the whole series length the use overlapping vs contiguous subseries has been a source of some debate amongst r s theorists this study examines the distributional characteristics of rescaled adjusted range and hurst exponent estimates derived using overlapping vs contiguous subseries henceforth closing debate on the issue of relative bias due to either technique confidence intervals for the statistical significance of the hurst exponent are also determined

stability of differential equations with piecewise constant arguments of generalized type abstract in this paper we continue to consider differential equations with piecewise constant argument of generalized type epcag m u akhmet integral manifolds of differential equations with piecewise constant argument of generalized type nonlinear anal tma 66 2007 367 383 a deviating function of a new form is introduced the linear and quasilinear systems are under discussion the structure of the sets of solutions is specified necessary and sufficient conditions for stability of the zero solution are obtained our approach can be fruitfully applied to the investigation of stability oscillations controllability and many other problems of epcag some of the results were announced at the international conference on hybrid systems and applications university of louisiana lafayette 2006

existence of almost periodic solutions for forced perturbed systems with piecewise constant argument abstract certain almost periodic forced perturbed systems with piecewise argument are considered in this paper by using the contraction mapping principle and some new analysis technique some sufficient conditions are obtained for the existence and uniqueness of almost periodic solution of these systems furthermore we study the harmonic and subharmonic solutions of these systems the obtained results generalize the previous known results such as a m fink almost periodic differential equation lecture notes in math vol 377 springer verlag berlin 1974 c y he almost periodic differential equations higher education press beijing 1992 in chinese z s lin the existence of almost periodic solution of linear system acta math sinica 22 5 1979 515 528 in chinese c y he existence of almost periodic solutions of perturbation systems ann differential equations 9 2 1992 173 181 y h xia m lin j cao the existence of almost periodic solutions of certain perturbation system j math anal appl 310 1 2005 81 96 finally a tangible example and its numeric simulations show the feasibility of our results the comparison between non perturbed system and perturbed system the relation between systems with and without piecewise argument

introducing hurst exponent in pair trading abstract in this paper we introduce a new methodology for pair trading this new method is based on the calculation of the hurst exponent of a pair our approach is inspired by the classical concepts of co integration and mean reversion but joined under a unique strategy we will show how hurst approach presents better results than classical distance method and correlation strategies in different scenarios results obtained prove that this new methodology is consistent and suitable by reducing the drawdown of trading over the classical ones getting as a result a better performance

differential equations with state dependent piecewise constant argument abstract a new class of differential equations with state dependent piecewise constant argument is introduced it is an extension of systems with piecewise constant argument fundamental theoretical results for the equations the existence and uniqueness of solutions the existence of periodic solutions and the stability of the zero solution are obtained appropriate examples are constructed

power law correlations in finance related google searches and their cross correlations with volatility and traded volume evidence from the dow jones industrial components abstract we study power law correlations properties of the google search queries for dow jones industrial average djia component stocks examining the daily data of the searched terms with a combination of the rescaled range and rescaled variance tests together with the detrended fluctuation analysis we show that the searches are in fact power law correlated with hurst exponents between 0 8 and 1 1 the general interest in the djia stocks is thus strongly persistent we further reinvestigate the cross correlation structure between the searches traded volume and volatility of the component stocks using the detrended cross correlation and detrending moving average cross correlation coefficients contrary to the universal power law correlations structure of the related google searches the results suggest that there is no universal relationship between the online search queries and the analyzed financial measures even though we confirm positive correlation for a majority of pairs there are several pairs with insignificant or even negative correlations in addition the correlations vary quite strongly across scales

generalized hurst exponent estimates differentiate eeg signals of healthy and epileptic patients the aim of our current study is to check whether multifractal patterns of the electroencephalographic eeg signals of normal and epileptic patients are statistically similar or different in this regard the generalized hurst exponent ghe method is used for robust estimation of the multifractals in each type of eeg signals and three powerful statistical tests are performed to check existence of differences between estimated ghes from healthy control subjects and epileptic patients the obtained results show that multifractals exist in both types of eeg signals particularly it was found that the degree of fractal is more pronounced in short variations of normal eeg signals than in short variations of eeg signals with seizure free intervals in contrary it is more pronounced in long variations of eeg signals with seizure free intervals than in normal eeg signals importantly both parametric and nonparametric statistical tests show strong evidence that estimated ghes of normal eeg signals are statistically and significantly different from those with seizure free intervals therefore ghes can be efficiently used to distinguish between healthy and patients suffering from epilepsy

scaling behaviors in differently developed markets scaling properties of four di erent stock market indices are studied in terms of a generalized hurst exponent approach we nd that the deviations from pure brownian motion behavior are associated with the degrees of development of the markets and we observe strong di erentiations in the scaling properties of markets at di erent development stage c

estimating long range dependence finite sample properties and confidence intervals abstract a major issue in financial economics is the behavior of asset returns over long horizons various estimators of long range dependence have been proposed even though some have known asymptotic properties it is important to test their accuracy by using simulated series of different lengths we test r s analysis detrended fluctuation analysis and periodogram regression methods on samples drawn from gaussian white noise the dfa statistics turns out to be the unanimous winner unfortunately no asymptotic distribution theory has been derived for this statistics so far we were able however to construct empirical i e approximate confidence intervals for all three methods the obtained values differ largely from heuristic values proposed by some authors for the r s statistics and are very close to asymptotic values for the periodogram regression method

testing for the source of multifractality in water level records abstract the multifractality in hydrologic data has been studied in many literatures but few literatures focus on its source which is helpful for understanding the underlying mechanisms that generate hydrologic data we propose a hypothesis testing procedure for the source of multifractality in water level records based on the empirical distributions of generalized hurst exponent estimated from a set of shuffled or surrogate series the proposed hypothesis testing procedure can show more details about the source of multifractality than previous methods with some statistics especially about the effects of large and small fluctuations on multifractality the generalized hurst exponents are estimated via multifractal detrended fluctuation analysis the data set contains about two million high frequency water level records of a northern china river at its 10 observation stations the testing results show that the multifractality in water levels is mainly caused by nonlinear correlations in small fluctuations and linear correlations in small and large fluctuations and is also related to the probability distribution of small fluctuations this conclusion is validated via some constructed semi surrogate series and values of statistics for source testing

data driven detrending of nonstationary fractal time series with echo state networks abstract in this paper we propose a novel data driven approach for removing trends detrending from nonstationary fractal and multifractal time series we consider real valued time series relative to measurements of an underlying dynamical system that evolves through time we assume that such a dynamical process is predictable to a certain degree by means of a class of recurrent networks called echo state network esn which are capable to model a generic dynamical process in order to isolate the superimposed multi fractal component of interest we define a data driven filter by leveraging on the esn prediction capability to identify the trend component of a given input time series specifically the estimated trend is removed from the original time series and the residual signal is analyzed with the multifractal detrended fluctuation analysis procedure to verify the correctness of the detrending procedure in order to demonstrate the effectiveness of the proposed technique we consider several synthetic time series consisting of different types of trends and fractal noise components with known characteristics we also process a real world dataset the sunspot time series which is well known for its multifractal features and has recently gained attention in the complex systems field results demonstrate the validity and generality of the proposed detrending method based on esns

detrending moving average cross correlation coefficient measuring cross correlations between non stationary series abstract in the paper we introduce a new measure of correlation between possibly non stationary series as the measure is based on the detrending moving average cross correlation analysis dmca we label it as the dmca coefficient d m c a with a moving average window length we analytically show that the coefficient ranges between 1 and 1 as a standard correlation does in the simulation study we show that the values of d m c a very well correspond to the true correlation between the analyzed series regardless the non stationarity level dependence of the newly proposed measure on other parameters correlation level moving average window length and time series length is discussed as well

differential equations alternately of retarded and advanced type we study differential equations with the argument m t k m where denotes the greatest integer function and k and m are positive integers such that k m the argument deviation t t m t k m is a function of period m and equals t for k t m k it changes sign in each interval mn k t m n 1 k vanishing at t mn

two way finite automata with quantum and classical states we introduce 2 way finite automata with quantum and classical states 2qcfa s this is a variant on the 2 way quantum finite automata 2qfa model which may be simpler to implement than unrestricted 2qfa s the internal state of a 2qcfa may include a quantum part that may be in a mixed quantum state but the tape head position is required to be classical we show two languages for which 2qcfa s are better than classical 2 way automata first 2qcfa s can recognize palindromes a language that cannot be recognized by 2 way deterministic or probabilistic finite automata second in polynomial time 2qcfa s can recognize anbn n n a language that can be recognized classically by a 2 way probabilistic automaton but only in exponential time

integral manifolds of differential equations with piecewise constant argument of generalized type abstract in this paper we introduce a general type of differential equations with piecewise constant argument epcag the existence of global integral manifolds of the quasilinear epcag is established when the associated linear homogeneous system has an exponential dichotomy the smoothness of the manifolds is investigated the existence of bounded and periodic solutions is considered a new technique of investigation of equations with piecewise argument based on an integral representation formula is proposed appropriate illustrating examples are given

dichotomies and asymptotic equivalence in alternately advanced and delayed differential systems abstract in this paper ordinary and exponential dichotomies are defined in differential equations with equations with piecewise constant argument of general type we prove the asymptotic equivalence between the bounded solutions of a linear system and a perturbed system with integrable and bounded perturbations

clustering of casablanca stock market based on hurst exponent estimates abstract this paper deals with the problem of casablanca stock exchange cse topology modeling as a complex network during three different market regimes general trend characterized by ups and downs increasing trend and decreasing trend in particular a set of seven different hurst exponent estimates are used to characterize long range dependence in each industrial sector generating process they are employed in conjunction with hierarchical clustering approach to examine the co movements of the casablanca stock exchange industrial sectors the purpose is to investigate whether cluster structures are similar across variable increasing and decreasing regimes it is observed that the general structure of the cse topology has been considerably changed over 2009 variable regime 2010 increasing regime and 2011 decreasing regime time periods the most important findings follow first in general a high value of hurst exponent is associated to a variable regime and a small one to a decreasing regime in addition hurst estimates during increasing regime are higher than those of a decreasing regime second correlations between estimated hurst exponent vectors of industrial sectors increase when casablanca stock exchange follows an upward regime whilst they decrease when the overall market follows a downward regime

existence computability and stability for solutions of the diffusion equation with general piecewise constant argument abstract we study the solution of a class of pde with piecewise constant argument of generalized type separation of variables leads to a solution formed by a series of products in previous works the convergence and bounds of the solution could be obtained from the study of the solution on the first constancy interval only in the general case however each term of the series may be unbounded at every interval implying that the solution is not computable we establish conditions where the convergence of the solution can be verified computing a finite number of terms of the series in each constancy interval without requiring any regularity on the initial condition moreover we combine asymptotic properties for each variable of the equation to obtain an exponential bound for the solution

numerical approximation of the solutions of delay differential equations on an infinite interval using piecewise constant arguments the idea of approximating the solutions of delay differential equations by equations with piecewise constant arguments epca has been suggested by gyijri l who proved convergence of the method for linear and nonlinear delay equations on compact intervals and under certain conditions also on the half line the approximating equations with piecewise constant argument can in turn be solved by use of difference equations the latter then also provide an approximation of the original delay equation the theory of epca was initiated and studied by cooke and wiener in 2 and 3 in this paper we begin to investigate the question of whether in this method of approximation the asymptotic dynamics of the delay differential equations are preserved a number of authors have dealt with the problem of showing that discretization of ordinary differential equations does not significantly alter the basic qualitative features readers may refer to the papers of kloeden and lorenz 4 and beyn 5 for some of this work for delay differential equations questions of this sort have been studied by cryer 6 barwell 7 zennaro 8 and later authors the paper of strehmel et al 9 contains an up to date list of references in our method of approximation the delay equation is first replaced by an epca and then by a difference equation and our objective is to relate the qualitative dynamics of these three equations we obtain results for autonomous and nonautonomous equations with one or many delays for autonomous equations the resulting difference algorithm is just a simple euler scheme but for nonautonomous equations it includes other possibilities in order to make the method and the nature of results as clear as possible we begin with a particular test equation k t pz t 7 p 0 7 0

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

dynamical generalized hurst exponent as a tool to monitor unstable periods in financial time series we investigate the use of the hurst exponent dynamically computed over a weighted moving time window to evaluate the level of stability instability of financial firms financial firms bailed out as a consequence of the 2007 2008 credit crisis show a neat increase with time of the generalized hurst exponent in the period preceding the unfolding of the crisis conversely firms belonging to other market sectors which suffered the least throughout the crisis show opposite behaviors we find that the multifractality of the bailed out firms increase at the crisis suggesting that the multi fractal properties of the time series are changing these findings suggest the possibility of using the scaling behavior as a tool to track the level of stability of a firm in this paper we introduce a method to compute the generalized hurst exponent which assigns larger weights to more recent events with respect to older ones in this way large fluctuations in the remote past are less likely to influence the recent past we also investigate the scaling associated with the tails of the log returns distributions and compare this scaling with the scaling associated with the hurst exponent observing that the processes underlying the price dynamics of these firms are truly multi scaling