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

what is the fractional laplacian a comparative review with new results abstract the fractional laplacian in r d which we write as 2 with 0 2 has multiple equivalent characterizations moreover in bounded domains boundary conditions must be incorporated in these characterizations in mathematically distinct ways and there is currently no consensus in the literature as to which definition of the fractional laplacian in bounded domains is most appropriate for a given application the riesz or integral definition for example admits a nonlocal boundary condition where the value of a function must be prescribed on the entire exterior of the domain in order to compute its fractional laplacian in contrast the spectral definition requires only the standard local boundary condition these differences among others lead us to ask the question what is the fractional laplacian beginning from first principles we compare several commonly used definitions of the fractional laplacian theoretically through their stochastic interpretations as well as their analytical properties then we present quantitative comparisons using a sample of state of the art methods we discuss recent advances on nonzero boundary conditions and present new methods to discretize such boundary value problems radial basis function collocation for the riesz fractional laplacian and nonharmonic lifting for the spectral fractional laplacian in our numerical studies we aim to compare different definitions on bounded domains using a collection of benchmark problems we consider the fractional poisson equation with both zero and nonzero boundary conditions where the fractional laplacian is defined according to the riesz definition the spectral definition the directional definition and the horizon based nonlocal definition we verify the accuracy of the numerical methods used in the approximations for each operator and we focus on identifying differences in the boundary behaviors of solutions to equations posed with these different definitions through our efforts we aim to further engage the research community in open problems and assist practitioners in identifying the most appropriate definition and computational approach to use for their mathematical models in addressing anomalous transport in diverse applications

fractional laplacian in bounded domains the fractional laplacian operator ensuremath ensuremath ensuremath delta ensuremath alpha 2 appears in a wide class of physical systems including l evy flights and stochastic interfaces in this paper we provide a discretized version of this operator which is well suited to deal with boundary conditions on a finite interval the implementation of boundary conditions is justified by appealing to two physical models namely hopping particles and elastic springs the eigenvalues and eigenfunctions in a bounded domain are then obtained numerically for different boundary conditions some analytical results concerning the structure of the eigenvalue spectrum are also obtained

estimates on translations and taylor expansions in fractional sobolev spaces abstract in this paper we study how the normalised gagliardo semi norms u w s p r n control translations in particular we prove that u y u l p r n c u w s p r n y s for n 1 s 0 1 and p 1 where c depends only on n we then obtain a corresponding higher order version of this result we get fractional rates of the error term in the taylor expansion we also present relevant implications of our two results first we obtain a direct proof of several compact embedding of w s p r n where the frechet kolmogorov theorem is applied with known rates we also derive fractional rates of convergence of the convolution of a function with suitable mollifiers thirdly we obtain fractional rates of convergence of finite difference discretisations for w s p r n

numerical methods for the fractional laplacian a finite difference quadrature approach the fractional laplacian delta alpha 2 is a nonlocal operator which depends on the parameter alpha and recovers the usual laplacian as alpha to 2 a numerical method for the fractional laplacian is proposed based on the singular integral representation for the operator the method combines finite differences with numerical quadrature to obtain a discrete convolution operator with positive weights the accuracy of the method is shown to be o h 3 alpha convergence of the method is proven the treatment of far field boundary conditions using an asymptotic approximation to the integral is used to obtain an accurate method numerical experiments on known exact solutions validate the predicted convergence rates computational examples include exponentially and algebraically decaying solutions with varying regularity the generalization to nonlinear equations involving the operator is discussed the obstacle problem for the fractional laplacian is computed

a fractional laplace equation regularity of solutions and finite element approximations this paper deals with the integral version of the dirichlet homogeneous fractional laplace equation for this problem weighted and fractional sobolev a priori estimates are provided in terms of the holder regularity of the data by relying on these results optimal order of convergence for the standard linear finite element method is proved for quasi uniform as well as graded meshes some numerical examples are given showing results in agreement with the theoretical predictions

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

arbitrage with fractional gaussian processes abstract while the arbitrage opportunity in the black scholes model driven by fractional brownian motion has a long history the arbitrage strategy in the black scholes model driven by general fractional gaussian processes is in its infancy the development of stochastic calculus with respect to fractional gaussian processes allowed us to study such models in this paper following the idea of shiryaev 1998 an arbitrage strategy is constructed for the black scholes model driven by fractional gaussian processes when the stochastic integral is interpreted in the riemann stieltjes sense arbitrage opportunities in some fractional gaussian processes including fractional brownian motion sub fractional brownian motion bi fractional brownian motion weighted fractional brownian motion and tempered fractional brownian motion are also investigated

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

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

volatility distribution in the s p500 stock index abstract we study the volatility of the s p500 stock index from 1984 to 1996 and find that the volatility distribution can be very well described by a log normal function further using detrended fluctuation analysis we show that the volatility is power law correlated with hurst exponent 0 9

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

homoclinic solutions for fractional discrete laplacian equations abstract in this paper we introduce a variational framework to a fractional difference equation on z driven by the fractional discrete laplacian and involving a coercive weight function and a positive parameter by means of the critical point theory we prove the existence of at least two nontrivial and nonnegative homoclinic solutions for big enough

nonlocal discrete diffusion equations and the fractional discrete laplacian regularity and applications abstract the analysis of nonlocal discrete equations driven by fractional powers of the discrete laplacian on a mesh of size h 0 h s u f for u f z h r 0 s 1 is performed the pointwise nonlocal formula for h s u and the nonlocal discrete mean value property for discrete s harmonic functions are obtained we observe that a characterization of h s as the dirichlet to neumann operator for a semidiscrete degenerate elliptic local extension problem is valid regularity properties and schauder estimates in discrete holder spaces as well as existence and uniqueness of solutions to the nonlocal dirichlet problem are shown for the latter the fractional discrete sobolev embedding and the fractional discrete poincare inequality are proved which are of independent interest we introduce the negative power fundamental solution u h s f which can be seen as the neumann to dirichlet map for the semidiscrete extension problem we then prove the discrete hardy littlewood sobolev inequality for h s as applications the convergence of our fractional discrete laplacian to the continuous fractional laplacian as h 0 in holder spaces is analyzed indeed uniform estimates for the error of the approximation in terms of h under minimal regularity assumptions are obtained we finally prove that solutions to the poisson problem for the fractional laplacian s u f in r can be approximated by solutions to the dirichlet problem for our fractional discrete laplacian with explicit uniform error estimates in terms of h

applications of statistical physics to economic and financial topics abstract problems in economy and finance have started to attract the interest of statistical physicists fundamental problems pertain to the existence or not of long medium short range power law correlations in economic systems as well as to the presence of financial cycles methods like the extended detrended fluctuation analysis and the multi affine analysis are recalled emphasizing their value in sorting out correlation ranges and predictability among spectacular results the possibility of crash predictions is indicated the well known financial analyst technique the so called moving average is shown to raise questions about fractional brownian motion properties finally the m k zipf method and the i variability diagram technique are presented for sorting out short range correlations analogies with other fields of modern applied statistical physics are also presented in view of some universal openess

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

estimation of incubation period and generation time based on observed length biased epidemic cohort with censoring for covid 19 outbreak in china the incubation period and generation time are key characteristics in the analysis of infectious diseases the commonly used contact tracing based estimation of incubation distribution is highly influenced by the individuals judgment on the possible date of exposure and might lead to significant errors on the other hand interval censoring based methods are able to utilize a much larger set of traveling data but may encounter biased sampling problems the distribution of generation time is usually approximated by observed serial intervals however it may result in a biased estimation of generation time especially when the disease is infectious during incubation in this paper the theory from renewal process is partially adopted by considering the incubation period as the interarrival time and the duration between departure from wuhan and onset of symptoms as the mixture of forward time and interarrival time with censored intervals in addition a consistent estimator for the distribution of generation time based on incubation period and serial interval is proposed for incubation infectious diseases a real case application to the current outbreak of covid 19 is implemented we find that the incubation period has a median of 8 50 days 95 confidence interval ci 7 22 9 15 the basic reproduction number in the early phase of covid 19 outbreak based on the proposed generation time estimation is estimated to be 2 96 95 ci 2 15 3 86

regularity of the obstacle problem for a fractional power of the laplace operator given a function and s 0 1 we will study the solutions of the following obstacle problem u in r n s u 0i nr n s u x 0 for those x such that u x x lim x u x 0 we show that when is c 1 s or smoother the solution u is in the space c 1 for every s in the case where the contact set u is convex we prove the optimal regularity result u c 1 s when is only c 1 for a s we prove that our solution u is c 1 for every c 2006 wiley periodicals inc

on the continuum limit for discrete nls with long range lattice interactions we consider a general class of discrete nonlinear schrodinger equations dnls on the lattice h mathbb z with mesh size h 0 in the continuum limit when h 0 we prove that the limiting dynamics are given by a nonlinear schrodinger equation nls on mathbb r with the fractional laplacian as dispersive symbol in particular we obtain that fractional powers frac 1 2 alpha 1 arise from long range lattice interactions when passing to the continuum limit whereas the nls with the usual laplacian describes the dispersion in the continuum limit for short range or quick decaying interactions e g nearest neighbor interactions

long range correlation analysis of economic news flow intensity abstract the goal of the paper is to examine the auto correlation properties for time series of the news flow intensity using different methods such as the fluctuation analysis the detrended fluctuation analysis and the detrending moving average analysis empirical findings for news analytics data show the presence of long range correlations for the time series of news intensity data

parameter identification for mixed fractional brownian motions with the drift parameter abstract this paper deals with the problem of estimating the unknown parameters in the drift mixed fractional brownian motion based on discrete sampled observations we construct two different estimators for the drift parameter based on the approximation by random walks of the driving noise and prove the consistency of these two estimators some numerical simulations are also presented to illustrate the performance of these maximum likelihood estimators empirical examples are given to illustrate the potential applications of these proposed estimators

correlations in economic time series a financial index of the new york stock exchange the sp here f is called the crossover frequency iii we confirm these results using the dfa detrended fluctuation analysis method see fig 2b which allows accurate estimates of exponents independent of local trends

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