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

detrended fluctuation analysis made flexible to detect range of cross correlated fluctuations the detrended cross correlation coefficient dcca has recently been proposed to quantify the strength of cross correlations on different temporal scales in bivariate nonstationary time series it is based on the detrended cross correlation and detrended fluctuation analyses dcca and dfa respectively and can be viewed as an analog of the pearson coefficient in the case of the fluctuation analysis the coefficient dcca works well in many practical situations but by construction its applicability is limited to detection of whether two signals are generally cross correlated without the possibility to obtain information on the amplitude of fluctuations that are responsible for those cross correlations in order to introduce some related flexibility here we propose an extension of dcca that exploits the multifractal versions of dfa and dcca multifractal detrended fluctuation analysis and multifractal detrended cross correlation analysis respectively the resulting new coefficient q not only is able to quantify the strength of correlations but also allows one to identify the range of detrended fluctuation amplitudes that are correlated in two signals under study we show how the coefficient q works in practical situations by applying it to stochastic time series representing processes with long memory autoregressive and multiplicative ones such processes are often used to model signals recorded from complex systems and complex physical phenomena like turbulence so we are convinced that this new measure can successfully be applied in time series analysis in particular we present an example of such application to highly complex empirical data from financial markets the present formulation can straightforwardly be extended to multivariate data in terms of the q dependent counterpart of the correlation matrices and then to the network representation

the time singularity multifractal spectrum distribution abstract although the multifractal singularity spectrum revealed the distribution of singularity exponent it failed to consider the temporal information therefore it is hard to describe the dynamic evolving process of non stationary and nonlinear systems in this paper we aim for a multifractal analysis and propose a time singularity multifractal spectrum distribution ts mfsd which will hopefully reveal the spatial dynamic character of fractal systems similar to the wigner ville time frequency distribution the time delayed conjugation of fractal signals is selected as the windows function furthermore the time varying holder exponent and the time varying wavelet singularity exponent are deduced based on the instantaneous self correlation fractal signal the time singularity exponent distribution i e ts mfsd is proposed which involves time varying hausdorff singularity spectrum distribution time varying large deviation multifractal spectrum and time varying legendre spectrum distribution which exhibit the singularity exponent distribution of fractal signal at arbitrary time finally we studied the algorithm of the ts mfsd based on the wavelet transform module maxima method analyzed and discussed the characteristic of ts mfsd based on devil staircase signal stochastic fractional motion and real sea clutter

analysis of the parareal time parallel time integration method the parareal algorithm is a method to solve time dependent problems parallel in time it approximates parts of the solution later in time simultaneously to parts of the solution earlier in time in this paper the relation of the parareal algorithm to space time multigrid and multiple shooting methods is first briefly discussed the focus of the paper is on new convergence results that show superlinear convergence of the algorithm when used on bounded time intervals and linear convergence for unbounded intervals

fractional brownian motion and fractional gaussian noise in subsurface hydrology a review presentation of fundamental properties and extensions recent studies have shown that fractional brownian motion fbm and fractional gaussian noise fgn are useful in characterizing subsurface heterogeneities in addition to geophysical time series although these studies have led to a fairly good understanding of some aspects of fbm fgn a comprehensive introduction to these stochastic fractal functions is still lacking in the subsurface hydrology literature in this paper efforts have been made to define fbm fgn and present a development of their mathematical properties in a direct yet rigorous manner use of the spectral representation theorem allows one to derive spectral representations for fbm fgn even though these functions do not have classical fourier transforms the discrete and truncated forms of these representations have served as a basis for synthetic generation of fbm fgn the discrete spectral representations are developed and various implications discussed in particular it is shown that a discrete form of the fbm spectral representation is equivalent to the well known weierstrass mandelbrot random fractal function although the full implications are beyond the scope of the present paper it is observed that discrete spectral representations of fbm constitute stationary processes even though fbm is nonstationary a new and general spectral density function is introduced for construction of complicated anisotropic 3 d fractals including those characterized by vertical fgn and horizontal fbm such fractals are useful for modeling anisotropic subsurface heterogeneities but cannot be generated with existing schemes finally some basic properties of fractional levy motion and concepts of universal multifractals which can be considered as generalizations of fbm fgn are reviewed briefly

establishing the relation between detrended fluctuation analysis and power spectral density analysis for stochastic processes stochastic fractal signals can be characterized by the hurst coefficient h which is related to the exponents of various power law statistics characteristic of these processes two techniques widely used to estimate h are spectral analysis and detrended fluctuation analysis dfa this paper examines the analytical link between these two measures and shows that they are related through an integral transform numerical simulations confirm this relationship for ideal synthesized fractal signals their performance as estimators of h is compared based on a mean square error criterion and found to be similar dfa measures are derived for physiological signals of heartbeat r ensuremath r intervals through the integral transform of a spectral density estimate these agree with directly calculated dfa estimates indicating that the relationship holds for signals with nonideal fractal properties it is concluded that dfa and spectral measures provide equivalent characterizations of stochastic signals with long term correlation

theoretical foundation of detrending methods for fluctuation analysis such as detrended fluctuation analysis and detrending moving average we present a bottom up derivation of fluctuation analysis with detrending for the detection of long range correlations in the presence of additive trends or intrinsic nonstationarities while the well known detrended fluctuation analysis dfa and detrending moving average dma were introduced ad hoc we claim basic principles for such methods where dfa and dma are then shown to be specific realizations the mean squared displacement of the summed time series contains the same information about long range correlations as the autocorrelation function but has much better statistical properties for large time lags however the scaling exponent of its estimator on a single time series is affected not only by trends on the data but also by intrinsic nonstationarities we therefore define the fluctuation function as mean squared displacement with weighting kernel we require that its estimator be unbiased and exhibit the correct scaling behavior for the random component of a signal which is only achieved if the weighting kernel implies detrending we show how dfa and dma satisfy these requirements and we extract their kernel weights

multifractal spectra of multi affine functions abstract self affine functions f x with multiscaling height correlations cq x xqhq are described in terms of the standard multifractal formalism with a modified assumption for the partition the corresponding quantities and expressions are shown to exhibit some characteristic differences from the standard ones according to our calculations the f type spectra are not uniquely determined by the hq spectrum but depend on the particular choice which is made for the dependence of n on x where n is the number of points over which the average is taken our results are expected to be relevant in the analysis of signal type data obtained in experiments on systems with an underlying multiplicative process

multi fractional generalized cauchy process and its application to teletraffic abstract the contributions given in this paper are in two aspects the first is to introduce a novel random function which we call the multi fractional generalized cauchy mgc process the second is to dissertate its application to network traffic for studying the multi fractal behavior of traffic on a point by point basis the introduced mgc process is with the time varying fractal dimension d t and the time varying hurst parameter h t the representations of the autocorrelation function acf and the power spectrum density psd of the mgc process are proposed besides the asymptotic expressions of the acf and psd of the mgc process are presented the computation formula of d t is given the mgc model may be a new tool to describe the multi fractal behavior of traffic precisely it may be used to reveal the local irregularity or local self similarity lss which is a small time scale behavior of traffic and global long term persistence or long range dependence lrd which is a large time scale behavior of traffic on a point by point basis the cast study with real traffic traces exhibits that the variance of d t is much greater than that of h t thus the present mgc model may provide a novel way to explain the fact that traffic has highly local irregularity while its lrd is robust

modified dfa and dcca approach for quantifying the multiscale correlation structure of financial markets abstract we use multiscale detrended fluctuation analysis msdfa and multiscale detrended cross correlation analysis msdcca to investigate auto correlation ac and cross correlation cc in the us and chinese stock markets during 1997 2012 the results show that us and chinese stock indices differ in terms of their multiscale ac structures stock indices in the same region also differ with regard to their multiscale ac structures we analyze ac and cc behaviors among indices for the same region to determine similarity among six stock indices and divide them into four groups accordingly we choose s p500 nqci hsi and the shanghai composite index as representative samples for simplicity msdfa and msdcca results and average msdfa spectra for local scaling exponents lses for individual series are presented we find that the msdcca spectrum for lse cc between two time series generally tends to be greater than the average msdfa lse spectrum for individual series we obtain detailed multiscale structures and relations for cc between the four representatives msdfa and msdcca with secant rolling windows of different sizes are then applied to reanalyze the ac and cc vertical and horizontal comparisons of different window sizes are made the msdfa and msdcca results for the original window size are confirmed and some new interesting characteristics and conclusions regarding multiscale correlation structures are obtained

self similarity of higher order moving averages in this work higher order moving average polynomials are defined by straightforward generalization of the standard moving average the self similarity of the polynomials is analyzed for fractional brownian series and quantified in terms of the hurst exponent h by using the detrending moving average method we prove that the exponent h of the fractional brownian series and of the detrending moving average variance asymptotically agree for the first order polynomial such asymptotic values are compared with the results obtained by the simulations the higher order polynomials correspond to trend estimates at shorter time scales as the degree of the polynomial increases importantly the increase of polynomial degree does not require to change the moving average window thus trends at different time scales can be obtained on data sets with the same size these polynomials could be interesting for those applications relying on trend estimates over different time horizons financial markets or on filtering at different frequencies image analysis

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

detrended cross correlation analysis consistently extended to multifractality we propose an algorithm multifractal cross correlation analysis mfcca which constitutes a consistent extension of the detrended cross correlation analysis and is able to properly identify and quantify subtle characteristics of multifractal cross correlations between two time series our motivation for introducing this algorithm is that the already existing methods like multifractal extension have at best serious limitations for most of the signals describing complex natural processes and often indicate multifractal cross correlations when there are none the principal component of the present extension is proper incorporation of the sign of fluctuations to their generalized moments furthermore we present a broad analysis of the model fractal stochastic processes as well as of the real world signals and show that mfcca is a robust and selective tool at the same time and therefore allows for a reliable quantification of the cross correlative structure of analyzed processes in particular it allows one to identify the boundaries of the multifractal scaling and to analyze a relation between the generalized hurst exponent and the multifractal scaling parameter ensuremath lambda q this relation provides information about the character of potential multifractality in cross correlations and thus enables a deeper insight into dynamics of the analyzed processes than allowed by any other related method available so far by using examples of time series from the stock market we show that financial fluctuations typically cross correlate multifractally only for relatively large fluctuations whereas small fluctuations remain mutually independent even at maximum of such cross correlations finally we indicate possible utility of mfcca to study effects of the time lagged cross correlations

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

generalized cauchy model of sea level fluctuations with long range dependence abstract this article suggests the contributions with two highlights one is to propose a novel model of sea level fluctuations sea level for short which is called the generalized cauchy gc process it provides a new outlook for the description of local and global behaviors of sea level from a view of fractal in that the fractal dimension d that measures the local behavior of sea level and the hurst parameter h which characterizes the global behavior of sea level are independent of each other the other is to show that sea level appears multi fractal in both spatial and time such a meaning of multi fractal is new in the sense that a pair of fractal parameters d h of sea level is varying with measurement sites and time this research exhibits that the ranges of d and h of sea level in general are 1 d 2 and 0 5 h 1 respectively but d is independent of h with respect to the global behavior of sea level we shall show that h 0 96 for all data records at all measurement sites implying that strong lrd may be a general phenomenon of sea level on the other side regarding with the local behavior we will reveal that there appears d 1 or d 1 for data records at a few stations and at some time but d 0 96 at most stations and at most time meaning that sea level may appear highly local irregularity more frequently than weak local one

comparison of detrending methods for fluctuation analysis abstract we examine several recently suggested methods for the detection of long range correlations in data series based on similar ideas as the well established detrended fluctuation analysis dfa in particular we present a detailed comparison between the regular dfa and two recently suggested methods the centered moving average cma method and a modified detrended fluctuation analysis mdfa we find that cma performs the same as dfa in long data with weak trends and is slightly superior to dfa in short data with weak trends when comparing standard dfa to mdfa we observe that dfa performs slightly better in almost all examples we studied we also discuss how several types of trends affect different types of dfa for weak trends in the data the new methods are comparable with dfa in these respects however if the functional form of the trend in data is not a priori known dfa remains the method of choice only a comparison of dfa results using different detrending polynomials yields full recognition of the trends a comparison with independent methods is recommended for proving long range correlations

levels of complexity in scale invariant neural signals many physical and physiological signals exhibit complex scale invariant features characterized by 1 f scaling and long range power law correlations indicating a possibly common control mechanism specifically it has been suggested that dynamical processes influenced by inputs and feedback on multiple time scales may be sufficient to give rise to 1 f scaling and scale invariance two examples of physiologic signals that are the output of hierarchical multiscale physiologic systems under neural control are the human heartbeat and human gait here we show that while both cardiac interbeat interval and gait interstride interval time series under healthy conditions have comparable 1 f scaling they still may belong to different complexity classes our analysis of the multifractal scaling exponents of the fluctuations in these two signals demonstrates that in contrast to the multifractal behavior found in healthy heartbeat dynamics gait time series exhibit less complex close to monofractal behavior further we find strong anticorrelations in the sign and close to random behavior for the magnitude of gait fluctuations at short and intermediate time scales in contrast to weak anticorrelations in the sign and strong positive correlation for the magnitude of heartbeat interval fluctuations suggesting that the neural mechanisms of cardiac and gait control exhibit different linear and nonlinear features these findings are of interest because they underscore the limitations of traditional two point correlation methods in fully characterizing physiological and physical dynamics in addition these results suggest that different mechanisms of control may be responsible for varying levels of complexity observed in physiological systems under neural regulation and in physical systems that possess similar 1 f scaling

time and frequency domain characteristics of detrending operation based scaling analysis exact dfa and dma frequency responses we develop a general framework to study the time and frequency domain characteristics of detrending operation based scaling analysis methods such as detrended fluctuation analysis dfa and detrending moving average dma analysis in this framework using either the time or frequency domain approach the frequency responses of detrending operations are calculated analytically although the frequency domain approach based on conventional linear analysis techniques is only applicable to linear detrending operations the time domain approach presented here is applicable to both linear and nonlinear detrending operations furthermore using the relationship between the time and frequency domain representations of the frequency responses the frequency domain characteristics of nonlinear detrending operations can be obtained based on the calculated frequency responses it is possible to establish a direct connection between the root mean square deviation of the detrending operation based scaling analysis and the power spectrum for linear stochastic processes here by applying our methods to dfa and dma including higher order cases exact frequency responses are calculated in addition we analytically investigate the cutoff frequencies of dfa and dma detrending operations and show that these frequencies are not optimally adjusted to coincide with the corresponding time scale

establishing a direct connection between detrended fluctuation analysis and fourier analysis to understand methodological features of the detrended fluctuation analysis dfa using a higher order polynomial fitting we establish the direct connection between dfa and fourier analysis based on an exact calculation of the single frequency response of the dfa the following facts are shown analytically 1 in the analysis of stochastic processes exhibiting a power law scaling of the power spectral density psd s f f a higher order detrending in the dfa has no adverse effect in the estimation of the dfa scaling exponent which satisfies the scaling relation 1 2 2 the upper limit of the scaling exponents detectable by the dfa depends on the order of polynomial fit used in the dfa and is bounded by m 1 where m is the order of the polynomial fit 3 the relation between the time scale in the dfa and the corresponding frequency in the psd are distorted depending on both the order of the dfa and the frequency dependence of the psd we can improve the scale distortion by introducing the corrected time scale in the dfa corresponding to the inverse of the frequency scale in the psd in addition our analytical approach makes it possible to characterize variants of the dfa using different types of detrending as an application properties of the detrending moving average algorithm are discussed

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

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

waveform relaxation for the computational homogenization of multiscale magnetoquasistatic problems this paper proposes the application of the waveform relaxation method to the homogenization of multiscale magnetoquasistatic problems in the monolithic heterogeneous multiscale method the nonlinear macroscale problem is solved using the newtonraphson scheme the resolution of many mesoscale problems per gau point allows to compute the homogenized constitutive law and its derivative by finite differences in the proposed approach the macroscale problem and the mesoscale problems are weakly coupled and solved separately using the finite element method on time intervals for several waveform relaxation iterations the exchange of information between both problems is still carried out using the heterogeneous multiscale method however the partial derivatives can now be evaluated exactly by solving only one mesoscale problem per gau point

scaling range of power laws that originate from fluctuation analysis we extend our previous study of scaling range properties performed for detrended fluctuation analysis dfa physica a 392 2384 2013 to other techniques of fluctuation analysis fa the new technique called modified detrended moving average analysis mdma is introduced and its scaling range properties are examined and compared with those of detrended moving average analysis dma and dfa it is shown that contrary to dfa dma and mdma techniques exhibit power law dependence of the scaling range with respect to the length of the searched signal and with respect to the accuracy r 2 of the fit to the considered scaling law imposed by dma or mdma methods this power law dependence is satisfied for both uncorrelated and autocorrelated data we find also a simple generalization of this power law relation for series with a different level of autocorrelations measured in terms of the hurst exponent basic relations between scaling ranges for different techniques are also discussed our findings should be particularly useful for local fa in e g econophysics finances or physiology where the huge number of short time series has to be examined at once and wherever the preliminary check of the scaling range regime for each of the series separately is neither effective nor possible

quantifying the cross correlations between online searches and bitcoin market abstract in this paper we quantify the cross correlations between google trends and bitcoin market by employing the multifractal detrended cross correlation analysis mf dcca method we find that the change of google trends cgt and bitcoin market i e returns and changes of volume are overall significantly cross correlated based on the cross correlation test in particular the empirical results show that 1 there exist power law cross correlations between cgt and bitcoin returns as well as cgt and changes of volume 2 the cross correlations between cgt and returns have a higher degree of multifractal in the long term and weak multifractal in the short term while the cross correlations between cgt and change of volume show the opposite trend 3 with the rolling window analysis we further find that there is a decrease trend for the cross correlations between cgt and bitcoin returns over time and the cross correlations scaling exponents are less than 0 5 which indicate that they are both anti persistent cross correlated

multifractal fourier detrended cross correlation analysis of traffic signals abstract multifractal detrended cross correlation analysis mf dxa has been developed to detect the long range power law cross correlation of considered signals in the presence of non stationarity however crossovers arising from extrinsic periodic trends make the scaling behavior difficult to analyze we introduce a fourier filtering method to eliminate the trend effects and systematically investigate the multifractal cross correlation of simulated and real traffic signals the crossover locations are found approximately corresponding to the periods of underlying trend traffic velocity on one road and flows on adjacent roads show strong cross correlation they also present weak multifractality after periodic trends are removed the traffic velocity and flow are cross correlated in opposite directions which is accordant to their actual evolution

the application of the multifractal cross correlation analysis methods in radar target detection within sea clutter abstract many complex systems generate multifractal time series which are long range cross correlated this paper introduces three multifractal cross correlation analysis methods such as multifractal cross correlation analysis based on the partition function approach mfxpf multifractal detrended cross correlation analysis mfdcca methods based on detrended fluctuation analysis mfxdfa and detrended moving average analysis mfxdma which only consider one moment order we do comparative analysis of the artificial time series binomial multiplicative cascades and cantor sets with different probabilities by these methods then we do a feasibility test of the fixed threshold target detection within sea clutter by applying the multifractal cross correlation analysis methods to the ipix radar sea clutter data the results show that it is feasible to use the method of the fixed threshold based on the multifractal feature parameter f by the mfxpf and mfxdfa 1 methods at last we give the main conclusions and provide a valuable reference on how to choose the multifractal algorithms the detection parameters and the target detection methods within sea clutter in practice

wavelet leaders a new method to estimate the multifractal singularity spectra abstract wavelet leaders is a novel alternative based on wavelet analysis for estimating the multifractal spectrum it was proposed by jaffard and co workers improving the usual wavelet methods in this work we analyze and compare it with the well known multifractal detrended fluctuation analysis the latter is a comprehensible and well adapted method for natural and weakly stationary signals alternatively wavelet leaders exploits the wavelet self similarity structures combined with the multiresolution analysis scheme we give a brief introduction on the multifractal formalism and the particular implementation of the above methods and we compare their effectiveness we expose several cases cantor measures binomial multiplicative cascades and also natural series from a tonic clonic epileptic seizure we analyze the results and extract the conclusions

long range anticorrelations and non gaussian behavior of the heartbeat we find that the successive increments in the cardiac beat to beat interals of healthy subjects display scale invariant long range anticorrelations up to 10 4 heart beats furthermore we find that the histogram for the heartbeat intervals increments is well described by a l evy stable distribution for a group of subjects with severe heart disease we find that the distribution is unchanged but the long range correlations vanish therefore the different scaling behavior in health and disease must relate to the underlying dynamics of the heartbeat

performance of a high dimensional r s method for hurst exponent estimation abstract an extension of the r s method to estimate the hurst exponent of high dimensional fractals is proposed the method s performance was adequate when tested with synthetic surfaces having different preset hurst exponent values and different array sizes the two dimensional r s analysis is used to analyze three images from nature and experimental data revealing interesting scaling behavior with physical meaning

multifractal detrended cross correlation analysis of genome sequences using chaos game representation abstract we characterized the multifractal nature and power law cross correlation between any pair of genome sequence through an integrative approach combining 2d multifractal detrended cross correlation analysis and chaos game representation in this paper we have analyzed genomes of some prokaryotes and calculated fractal spectra h q and f from our analysis we observed existence of multifractal nature and power law cross correlation behavior between any pair of genome sequences cluster analysis was performed on the calculated scaling exponents to identify the class affiliation and the same is represented as a dendrogram we suggest this approach may find applications in next generation sequence analysis big data analytics etc

record length requirement of long range dependent teletraffic abstract this article contributes the highlights mainly in two folds on the one hand it presents a formula to compute the upper bound of the variance of the correlation periodogram measurement of teletraffic traffic for short with long range dependence lrd for a given record length t and a given value of the hurst parameter h theorems 1 and 2 on the other hand it proposes two formulas for the computation of the variance upper bound of the correlation periodogram measurement of traffic of fractional gaussian noise fgn type and the generalized cauchy gc type respectively corollaries 1 and 2 they may constitute a reference guideline of record length requirement of traffic with lrd in addition record length requirement for the correlation periodogram measurement of traffic with either the schuster type or the bartlett one is studied and the present results about it show that both types of periodograms may be used for the correlation measurement of traffic with a pre desired variance bound of correlation estimation moreover real traffic in the internet archive by the special interest group on data communication under the association for computing machinery of us acm sigcomm is analyzed in the case study in this topic

modified detrended fluctuation analysis based on empirical mode decomposition for the characterization of anti persistent processes abstract detrended fluctuation analysis dfa is a simple but very efficient method for investigating the power law long term correlations of non stationary time series in which a detrending step is necessary to obtain the local fluctuations at different timescales we propose to determine the local trends through empirical mode decomposition emd and perform the detrending operation by removing the emd based local trends which gives an emd based dfa method similarly we also propose a modified multifractal dfa algorithm called an emd based mfdfa the performance of the emd based dfa and mfdfa methods is assessed with extensive numerical experiments based on fractional brownian motion and multiplicative cascading process we find that the emd based dfa method performs better than the classic dfa method in the determination of the hurst index when the time series is strongly anticorrelated and the emd based mfdfa method outperforms the traditional mfdfa method when the moment order q of the detrended fluctuations is positive we apply the emd based mfdfa to the 1 min data of shanghai stock exchange composite index and the presence of multifractality is confirmed we also analyze the daily austrian electricity prices and confirm its anti persistence

singularity power spectrum distribution abstract fractal and multifractal signal processing are hot topics in current studies at present the research mainly concentrates in fractal dimension multifractal spectrum and time singularity spectrum distribution which are based on differentiability of fractal subsets however fail to reflect the energy and power measurement of fractal singular subset therefore singularity power spectrum distribution spsd is proposed in this paper based on the traditional power spectrum density function and multifractal spectrum in theory basic concept of spsd is put forward based on the related singularity measure the theoretical expression of spsd is deduced and the compatibility between spsd and conventional power spectrum analysis is proved in algorithm the discrete approximation algorithm of spsd for discrete fractal time series is introduced simulation based on the multifractal brownian motion mfbm and actual sea clutter indicates that spsd can reveal effectively the singular power distribution of sea clutter and mfbm and identify fractal signals even for those with approximate multifractal spectra or approximate power spectra the spsd will provide application prospect of multifractal signal processing detection and distinguishing of multifractal time series

detrended fluctuation analysis as a regression framework estimating dependence at different scales we propose a framework combining detrended fluctuation analysis with standard regression methodology the method is built on detrended variances and covariances and it is designed to estimate regression parameters at different scales and under potential nonstationarity and power law correlations the former feature allows for distinguishing between effects for a pair of variables from different temporal perspectives the latter ones make the method a significant improvement over the standard least squares estimation theoretical claims are supported by monte carlo simulations the method is then applied on selected examples from physics finance environmental science and epidemiology for most of the studied cases the relationship between variables of interest varies strongly across scales

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

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

on the scaling ranges of detrended fluctuation analysis for long term memory correlated short series of data abstract we examine the scaling regime for the detrended fluctuation analysis dfa the most popular method used to detect the presence of long term memory in data and the fractal structure of time series first the scaling range for dfa is studied for uncorrelated data as a function of time series length l and the correlation coefficient of the linear regression r 2 at various confidence levels next a similar analysis for artificial short series of data with long term memory is performed in both cases the scaling range is found to change linearly both with l and r 2 we show how this dependence can be generalized to a simple unified model describing the relation l r 2 h where h 1 2 h 1 stands for the hurst exponent of the long range autocorrelated signal our findings should be useful in all applications of dfa technique particularly for instantaneous local dfa where a huge number of short time series has to be analyzed at the same time without possibility of checking the scaling range in each of them separately

detrending moving average algorithm frequency response and scaling performances the detrending moving average dma algorithm has been widely used in its several variants for characterizing long range correlations of random signals and sets one dimensional sequences or high dimensional arrays over either time or space in this paper mainly based on analytical arguments the scaling performances of the centered dma including higher order ones are investigated by means of a continuous time approximation and a frequency response approach our results are also confirmed by numerical tests the study is carried out for higher order dma operating with moving average polynomials of different degree in particular detrending power degree frequency response asymptotic scaling upper limit of the detectable scaling exponent and finite scale range behavior will be discussed

effect of extreme data loss on long range correlated and anticorrelated signals quantified by detrended fluctuation analysis detrended fluctuation analysis dfa is an improved method of classical fluctuation analysis for nonstationary signals where embedded polynomial trends mask the intrinsic correlation properties of the fluctuations to better identify the intrinsic correlation properties of real world signals where a large amount of data is missing or removed due to artifacts we investigate how extreme data loss affects the scaling behavior of long range power law correlated and anticorrelated signals we introduce a segmentation approach to generate surrogate signals by randomly removing data segments from stationary signals with different types of long range correlations the surrogate signals we generate are characterized by four parameters i the dfa scaling exponent alpha of the original correlated signal u i ii the percentage p of the data removed from u i iii the average length mu of the removed or remaining data segments and iv the functional form p l of the distribution of the length l of the removed or remaining data segments we find that the global scaling exponent of positively correlated signals remains practically unchanged even for extreme data loss of up to 90 in contrast the global scaling of anticorrelated signals changes to uncorrelated behavior even when a very small fraction of the data is lost these observations are confirmed on two examples of real world signals human gait and commodity price fluctuations we further systematically study the local scaling behavior of surrogate signals with missing data to reveal subtle deviations across scales we find that for anticorrelated signals even 10 of data loss leads to significant monotonic deviations in the local scaling at large scales from the original anticorrelated to uncorrelated behavior in contrast positively correlated signals show no observable changes in the local scaling for up to 65 of data loss while for larger percentage of data loss the local scaling shows overestimated regions with higher local exponent at small scales followed by underestimated regions with lower local exponent at large scales finally we investigate how the scaling is affected by the average length probability distribution and percentage of the remaining data segments in comparison to the removed segments we find that the average length mu r of the remaining segments is the key parameter which determines the scales at which the local scaling exponent has a maximum deviation from its original value interestingly the scales where the maximum deviation occurs follow a power law relationship with mu r whereas the percentage of data loss determines the extent of the deviation the results presented in this paper are useful to correctly interpret the scaling properties obtained from signals with extreme data loss

fast algorithm for scaling analysis with higher order detrending moving average method among scaling analysis methods based on the root mean square deviation from the estimated trend it has been demonstrated that centered detrending moving average dma analysis with a simple moving average has good performance when characterizing long range correlation or fractal scaling behavior furthermore higher order dma has also been proposed it is shown to have better detrending capabilities removing higher order polynomial trends than original dma however a straightforward implementation of higher order dma requires a very high computational cost which would prevent practical use of this method to solve this issue in this study we introduce a fast algorithm for higher order dma which consists of two techniques 1 parallel translation of moving averaging windows by a fixed interval 2 recurrence formulas for the calculation of summations our algorithm can significantly reduce computational cost monte carlo experiments show that the computational time of our algorithm is approximately proportional to the data length although that of the conventional algorithm is proportional to the square of the data length the efficiency of our algorithm is also shown by a systematic study of the performance of higher order dma such as the range of detectable scaling exponents and detrending capability for removing polynomial trends in addition through the analysis of heart rate variability time series we discuss possible applications of higher order dma

probabilistic properties of detrended fluctuation analysis for gaussian processes detrended fluctuation analysis dfa is one of the most widely used tools for the detection of long range dependence in time series although dfa has found many interesting applications and has been shown to be one of the best performing detrending methods its probabilistic foundations are still unclear in this paper we study probabilistic properties of dfa for gaussian processes our main attention is paid to the distribution of the squared error sum of the detrended process we use a probabilistic approach to derive general formulas for the expected value and the variance of the squared fluctuation function of dfa for gaussian processes we also get analytical results for the expected value of the squared fluctuation function for particular examples of gaussian processes such as gaussian white noise fractional gaussian noise ordinary brownian motion and fractional brownian motion our analytical formulas are supported by numerical simulations the results obtained can serve as a starting point for analyzing the statistical properties of dfa based estimators for the fluctuation function and long memory parameter

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

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

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

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

on the spectrum of fractional brownian motions fractional brownian motions fbms provide useful models for a number of physical phenomena whose empirical spectra obey power laws of fractional order however due to the nonstationary nature of these processes the precise meaning of such spectra remains generally unclear two complementary approaches are proposed which are intended to clarify this point the first one based on a time frequency analysis takes into account the nonstationary nature of fbm and puts emphasis on time averaged measurements the second one based on a time scale analysis is matched to self similarity properties of fbm and reveals an underlying stationary structure relative to each time scaling

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

dynamic iteration for coupled problems of electric circuits and distributed devices coupled systems of differential algebraic equations daes may suffer from instabilities during a dynamic iteration we extend the existing analysis on recursion estimates error propagation and stability to semiexplicit index 1 daes in this context we discuss the influence of certain coupling structures and the computational sequence of the subsystems on the rate of convergence furthermore we investigate in detail convergence and divergence for two coupled problems stemming from refined electric circuit simulation these are the semiconductor circuit and field circuit couplings we quantify the convergence rate and behavior also using lipschitz constants and suggest an enhanced modeling of the coupling interface in order to improve convergence

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

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

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

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

equivariant adaptive source separation source separation consists of recovering a set of independent signals when only mixtures with unknown coefficients are observed this paper introduces a class of adaptive algorithms for source separation that implements an adaptive version of equivariant estimation and is henceforth called equivariant adaptive separation via independence easi the easi algorithms are based on the idea of serial updating this specific form of matrix updates systematically yields algorithms with a simple structure for both real and complex mixtures most importantly the performance of an easi algorithm does not depend on the mixing matrix in particular convergence rates stability conditions and interference rejection levels depend only on the normalized distributions of the source signals closed form expressions of these quantities are given via an asymptotic performance analysis the theme of equivariance is stressed throughout the paper the source separation problem has an underlying multiplicative structure the parameter space forms a matrix multiplicative group we explore the favorable consequences of this fact on implementation performance and optimization of easi algorithms

statistical analysis of fluctuations in the ecg morphology abstract we investigate ecg recordings of healthy subjects h subjects and those with heart failure hf subjects based on a new measure derived from the shape of the ecg signal during each heart beat from this measure we construct sequences called morphograms which are complementary to sequences of successive interbeat time intervals commonly called tachograms the morphograms show long range anticorrelations between the increments of their successive elements analogous to the behavior found previously in the tachograms by studying the average strength of the fluctuations of the increments on certain scales we find that it is possible to completely separate the two groups of h and hf subjects the separation is comparable and in certain cases even more pronounced than that found earlier based on the tachograms

real time acoustic blind signal separation system based on the spatio temporal gradient analysis this paper presents an autonomous directivity microphone system for the blind source separation based on the newly proposed spatio temporal gradient algorithm the blind source separation technique principally uses no a priori knowledge about parameters of convolution filtering nor mixing in the simplest case of the blind source separation problems the observed mixing signals are the linear combinations of unknown source signals which are mutually and statistically independent and contain zero mean the proposed blind signal separation algorithm utilizes the linearity among the four signals 1 the sound pressure 2 x 3 y and 4 z directional particle velocities all of which are governed by the wave equation and the equation of motion the proposed method therefore has an ability to simplify the convoluted blind source separation problems into the instantaneous blind source separation problems over the spatio temporal gradient space several acoustical experiments have been performed with a

detrended partial cross correlation analysis of two nonstationary time series influenced by common external forces when common factors strongly influence two power law cross correlated time series recorded in complex natural or social systems using detrended cross correlation analysis dcca without considering these common factors will bias the results we use detrended partial cross correlation analysis dpxa to uncover the intrinsic power law cross correlations between two simultaneously recorded time series in the presence of nonstationarity after removing the effects of other time series acting as common forces the dpxa method is a generalization of the detrended cross correlation analysis that takes into account partial correlation analysis we demonstrate the method by using bivariate fractional brownian motions contaminated with a fractional brownian motion we find that the dpxa is able to recover the analytical cross hurst indices and thus the multiscale dpxa coefficients are a viable alternative to the conventional cross correlation coefficient we demonstrate the advantage of the dpxa coefficients over the dcca coefficients by analyzing contaminated bivariate fractional brownian motions we calculate the dpxa coefficients and use them to extract the intrinsic cross correlation between crude oil and gold futures by taking into consideration the impact of the u s dollar index we develop the multifractal dpxa mf dpxa method in order to generalize the dpxa method and investigate multifractal time series we analyze multifractal binomial measures masked with strong white noises and find that the mf dpxa method quantifies the hidden multifractal nature while the multifractal dcca method fails

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

long range correlation and predictability of chinese stock prices abstract it is well known that the price time series of an efficient market display a brownian motion in such a case the best prediction of a future price is the last known price thus no arbitrage is possible a real market is however possibly not perfectly efficient in this paper we apply an error back propagation neural network to tick by tick high frequency time series of chinese a share stock prices and try to predict their future values the performance of the predictions varies largely for different stocks in order to seek for the dependence of the performance we study the long range correlation lrc of the data by the conventional detrended fluctuation analysis dfa and its generalization the multifractal detrended fluctuation analysis mfdfa clear relevance between the performance and the lrc is observed the prediction is particularly good when the time series are strongly anti persistent

wavelet versus detrended fluctuation analysis of multifractal structures we perform a comparative study of applicability of the multifractal detrended fluctuation analysis mfdfa and the wavelet transform modulus maxima wtmm method in proper detecting of monofractal and multifractal character of data we quantify the performance of both methods by using different sorts of artificial signals generated according to a few well known exactly soluble mathematical models monofractal fractional brownian motion bifractal levy flights and different sorts of multifractal binomial cascades our results show that in the majority of situations in which one does not know a priori the fractal properties of a process choosing mfdfa should be recommended in particular wtmm gives biased outcomes for the fractional brownian motion with different values of hurst exponent indicating spurious multifractality in some cases wtmm can also give different results if one applies different wavelets we do not exclude using wtmm in real data analysis but it occurs that while one may apply mfdfa in a more automatic fashion wtmm must be applied with care in the second part of our work we perform an analogous analysis on empirical data coming from the american and from the german stock market for this data both methods detect rich multifractality in terms of broad f alpha but mfdfa suggests that this multifractality is poorer than in the case of wtmm

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

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

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

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

blind separation of sound sources from the principle of least spatial entropy abstract the aim of the paper is to offer a method for separating incoherent and compact sound sources which may overlap in both the space and frequency domains this is found of interest in acoustical applications involving the identification and ranking of sound sources stemming from different physical origins the principle proceeds in two steps the first one being reminiscent to source reconstruction e g as in near field acoustical holography and the second one to blind source separation specifically the source mixture is first expanded into a linear combination of spatial basis functions whose coefficients are set by backpropagating the pressures measured by an array of microphones to the source domain this leads to a formulation similar but no identical to blind source separation in the second step these coefficients are blindly separated into uncorrelated latent variables assigned to incoherent virtual sources these are shown to be defined up to an arbitrary rotation a unique set of sound sources is finally recovered by searching for that rotation by conjugate gradient descent in the stiefel manifold of unitary matrices which maximizes their spatial compactness as measured either by their spatial variance or their spatial entropy this results in the proposal of two separation criteria coined least spatial variance and least spatial entropy respectively the same concept of spatial entropy which is central to the paper is also exploited in defining a new criterion the entropic l curve dedicated to determining the number of active sound sources the idea consists in considering the number of sources that achieves the best compromise between a low spatial entropy as expected from compact sources and a low statistical entropy as expected from a low residual error the proposed methodology is validated on both laboratory experiments and numerical data and illustrated on an industrial example concerned with the ranking of sound sources on a diesel engine at the same time its robustness to the estimated number of active sources is demonstrated

reliability of the walking speed and gait dynamics variables while walking on a feedback controlled treadmill the purpose of this study is to identify the reliability of walking speed and gait dynamics measured with a feedback controlled treadmill and to assess the applicability of the treadmill to gait dynamics studies the intraclass correlation coefficient icc and the standard error of measurement sem for the walking speed and the mean variability coefficient of variance cv and fractal dynamics the scaling exponent of detrended fluctuation analysis dfa of the stride time and stride length were used to evaluate the within day and between day reliability fifteen subjects walked on a feedback controlled treadmill for three trials that were each more than 10 min in length within day this protocol was repeated on another day to identify the between day reliability the results showed that all variables were consistent for within day and between day reliability icc 0 633 0 982 p 0 05 sem 0 02 0 43 the within and between day reliability of the walking speed and the mean variability and fractal dynamics for the stride time and stride length were identified good iccs and low sems for within day and between day reliability were obtained for all variables therefore it is concluded that it is possible to use a feedback controlled treadmill to the study of gait dynamics

nonlinear filtering properties of detrended fluctuation analysis abstract detrended fluctuation analysis dfa has been widely used for quantifying long range correlation and fractal scaling behavior in dfa to avoid spurious detection of scaling behavior caused by a nonstationary trend embedded in the analyzed time series a detrending procedure using piecewise least squares fitting has been applied however it has been pointed out that the nonlinear filtering properties involved with detrending may induce instabilities in the scaling exponent estimation to understand this issue we investigate the adverse effects of the dfa detrending procedure on the statistical estimation we show that the detrending procedure using piecewise least squares fitting results in the nonuniformly weighted estimation of the root mean square deviation and that this property could induce an increase in the estimation error in addition for comparison purposes we investigate the performance of a centered detrending moving average analysis with a linear detrending filter and sliding window dfa and show that these methods have better performance than the standard dfa

following a trend with an exponential moving average analytical results for a gaussian model we investigate how price variations of a stock are transformed into profits and losses p ls of a trend following strategy in the frame of a gaussian model we derive the probability distribution of p ls and analyze its moments mean variance skewness and kurtosis and asymptotic behavior quantiles we show that the asymmetry of the distribution with often small losses and less frequent but significant profits is reminiscent to trend following strategies and less dependent on peculiarities of price variations at short times trend following strategies admit larger losses than one may anticipate from standard gaussian estimates while smaller losses are ensured at longer times simple explicit formulas characterizing the distribution of p ls illustrate the basic mechanisms of momentum trading while general matrix representations can be applied to arbitrary gaussian models we also compute explicitly annualized risk adjusted p l and strategy turnover to account for transaction costs we deduce the trend following optimal timescale and its dependence on both auto correlation level and transaction costs theoretical results are illustrated on the dow jones index

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

three dimensional wavelet based multifractal method the need for revisiting the multifractal description of turbulence dissipation data we generalize the wavelet transform modulus maxima wtmm method to multifractal analysis of 3d random fields this method is calibrated on synthetic 3d monofractal fractional brownian fields and on 3d multifractal singular cascade measures as well as their random function counterpart obtained by fractional integration then we apply the 3d wtmm method to the dissipation field issued from 3d isotropic turbulence simulations we comment on the need to revisit previous box counting analyses which have failed to estimate correctly the corresponding multifractal spectra because of their intrinsic inability to master nonconservative singular cascade measures

a comparative study of two dimensional multifractal detrended fluctuation analysis and two dimensional multifractal detrended moving average algorithm to estimate the multifractal spectrum multifractal detrended fluctuation analysis mfdfa and multifractal detrended moving average mfdma algorithm have been established as two important methods to estimate the multifractal spectrum of the one dimensional random fractal signal they have been generalized to deal with two dimensional and higher dimensional fractal signals this paper gives a brief introduction of the two dimensional multifractal detrended fluctuation analysis 2d mfdfa and two dimensional multifractal detrended moving average 2d mfdma algorithm and a detailed description of the application of the two dimensional fractal signal processing by using the two methods by applying the 2d mfdfa and 2d mfdma to the series generated from the two dimensional multiplicative cascading process we systematically do the comparative analysis to get the advantages disadvantages and the applicabilities of the two algorithms for the first time from six aspects such as the similarities and differences of the algorithm models the statistical accuracy the sensitivities of the sample size the selection of scaling range the choice of the q orders and the calculation amount the results provide a valuable reference on how to choose the algorithm from 2d mfdfa and 2d mfdma and how to make the schemes of the parameter settings of the two algorithms when dealing with specific signals in practical applications

the fractal energy measurement and the singularity energy spectrum analysis abstract the singularity exponent se is the characteristic parameter of fractal and multifractal signals based on se the fractal dimension reflecting the global self similar character the instantaneous se reflecting the local self similar character the multifractal spectrum mfs reflecting the distribution of se and the time varying mfs reflecting pointwise multifractal spectrum were proposed however all the studies were based on the depiction of spatial or differentiability characters of fractal signals taking the se as the independent dimension this paper investigates the fractal energy measurement fem and the singularity energy spectrum ses theory firstly we study the energy measurement and the energy spectrum of a fractal signal in the singularity domain propose the conception of fem and ses of multifractal signals and investigate the hausdorff measure and the local direction angle of the fractal energy element then we prove the compatibility between fem and traditional energy and point out that ses can be measured in the fractal space finally we study the algorithm of ses under the condition of a continuous signal and a discrete signal and give the approximation algorithm of the latter and the estimations of fem and ses of the gaussian white noise fractal brownian motion and the multifractal brownian motion show the theoretical significance and application value of fem and ses

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