twenty five years of multifractals in fully developed turbulence a tribute to giovanni paladin the paper on the multifractal nature of fully developed turbulence and chaotic systems by benzi et al 1984 j phys a math gen 17 3521 has been a starting point of many investigations on the different faces of self similarity and intermittency in turbulent phenomena since then the multifractal model has become a useful tool for the study of small scale turbulence in particular for detailed predictions of different eulerian and lagrangian statistical properties on the occasion of the 50th birthday of our unforgettable friend and colleague giovanni paladin 1958 1996 we review here the basic concepts and some applications of the multifractal model for turbulence

on the multifractal nature of fully developed turbulence and chaotic systems it is generally argued that the energy dissipation of three dimensional turbulent flow is concentrated on a set with non integer hausdorff dimension recently in order to explain experimental data it has been proposed that this set does not possess a global dilatation invariance it can be considered to be a multifractal set the authors review the concept of multifractal sets in both turbulent flows and dynamical systems using a generalisation of the beta model

numerical methods for the estimation of multifractal singularity spectra on sampled data physical variables in scale invariant systems often show chaotic turbulent like behavior commonly associated to the existence of an underlying fractal or multifractal structure however the assessment of multifractality over experimental discretized data requires of appropriate methods and to establish criteria to measure the confidence degree on the estimates in this paper we have evaluated the quality of different techniques used for multifractal analysis we have tested four different techniques the moment m method the wavelet transform modulus maxima wtmm method the gradient modulus wavelet projection gmwp method and the gradient histogram gh method which are used to estimate the singularity spectra of multifractal signals the test consists in analyzing synthetic multifractal 1d signals with given multifractal spectrum we have compared the results studying the sensibility of each method to the length of the series size of the ensemble and type of spectrum our results show that gmwp method is the one attaining the best performance providing reliable estimates which can be improved when the statistics is increased all the other methods are affected by problems such as the linearization of the right tail of the spectrum and some of them are very demanding in data

gradual multifractal reconstruction of time series formulation of the method and an application to the coupling between stock market indices and their holder exponents abstract a technique termed gradual multifractal reconstruction gmr is formulated a continuum is defined from a signal that preserves the pointwise holder exponent multifractal structure of a signal but randomises the locations of the original data values with respect to this 0 to the original signal itself 1 we demonstrate that this continuum may be populated with synthetic time series by undertaking selective randomisation of wavelet phases using a dual tree complex wavelet transform that is the 0 end of the continuum is realised using the recently proposed iterated amplitude adjusted wavelet transform algorithm keylock 2017 that fully randomises the wavelet phases this is extended to the gmr formulation by selective phase randomisation depending on whether or not the wavelet coefficient amplitudes exceeds a threshold criterion an econophysics application of the technique is presented the relation between the normalised log returns and their holder exponents for the daily returns of eight financial indices are compared one particularly noticeable result is the change for the two american indices nasdaq 100 and s p 500 from a non significant to a strongly significant as determined using gmr cross correlation between the returns and their holder exponents from before the 2008 crash to afterwards this is also reflected in the skewness of the phase difference distributions which exhibit a geographical structure with asian markets not exhibiting significant skewness in contrast to those from elsewhere globally

multiaffine random field model of rainfall a new method for the statistical description and simulation of two dimensional rainfall radar images is proposed it is based on the theory of multiaffine random fields and the bounded lognormal cascade model the model has three free parameters which are shown to provide a parsimonious and robust statistical description of rainfall images the parameters retrieved from the analysis of the real rainfall data by fitting the one and two point statistics are used for simulation the simulated rainfall fields are in good statistical and visual agreement with their real counterparts

the variable selection problem abstract the problem of variable selection is one of the most pervasive model selection problems in statistical applications often referred to as the problem of subset selection it arises when one wants to model the relationship between a variable of interest and a subset of potential explanatory variables or predictors but there is uncertainty about which subset to use this vignette reviews some of the key developments that have led to the wide variety of approaches for this problem

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

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

kullback leibler divergence measure of intermittency application to turbulence for generic systems exhibiting power law behaviors and hence multiscale dependencies we propose a simple tool to analyze multifractality and intermittency after noticing that these concepts are directly related to the deformation of a probability density function from gaussian at large scales to non gaussian at smaller scales our framework is based on information theory and uses shannon entropy and kullback leibler divergence we provide an extensive application to three dimensional fully developed turbulence seen here as a paradigmatic complex system where intermittency was historically defined and the concepts of scale invariance and multifractality were extensively studied and benchmarked we compute our quantity on experimental eulerian velocity measurements as well as on synthetic processes and phenomenological models of fluid turbulence our approach is very general and does not require any underlying model of the system although it can probe the relevance of such a model

flow of spatiotemporal turbulentlike random fields we study the lagrangian trajectories of statistically isotropic homogeneous and stationary divergence free spatiotemporal random vector fields we design this advecting eulerian velocity field such that it gets asymptotically rough and multifractal both in space and time as it is demanded by the phenomenology of turbulence at infinite reynolds numbers we then solve numerically the flow equations for a differentiable version of this field we observe that trajectories get also rough characterized by nearly the same hurst exponent as the one of our prescribed advecting field moreover even when considering the simplest situation of the advection by a fractional gaussian field we evidence in the lagrangian framework additional intermittent corrections the present approach involves properly defined random fields and asks for a rigorous treatment that would explain our numerical findings and deepen our understanding of this long lasting problem

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

on strong anomalous diffusion superdiffusive behavior i e x2 t t2 with 1 2 is in general not completely characterized by a unique exponent we study some systems exhibiting strong anomalous diffusion i e x t q tq q where 2 1 2 and q q is not a linear function of q this feature is different from the weak superdiffusive regime i e q const 1 2 occurring in random shear flows strong anomalous diffusion can be generated by nontrivial chaotic dynamics e g lagrangian motion in 2d time dependent incompressible velocity fields 2d symplectic maps and 1d intermittent maps typically the function q q is piecewise linear this is due to two mechanisms a weak anomalous diffusion for the typical events and a ballistic transport for the rare excursions in order to have strong anomalous diffusion one needs a violation of the hypothesis of the central limit theorem this happens only in a very narrow region of the control parameters space in the presence of strong anomalous diffusion one does not have a unique exponent and therefore one has the failure of the usual scaling p x t t f x t of the probability density this implies that the effective equation at large scale and long time for p x t obeys neither the usual fick equation nor other linear equations involving temporal and or spatial fractional derivatives

synthetic turbulence fractal interpolation and large eddy simulation fractal interpolation has been proposed in the literature as an efficient way to construct closure models for the numerical solution of coarse grained navier stokes equations it is based on synthetically generating a scale invariant subgrid scale field and analytically evaluating its effects on large resolved scales in this paper we propose an extension of previous work by developing a multiaffine fractal interpolation scheme and demonstrate that it preserves not only the fractal dimension but also the higher order structure functions and the non gaussian probability density function of the velocity increments extensive a priori analyses of atmospheric boundary layer measurements further reveal that this multiaffine closure model has the potential for satisfactory performance in large eddy simulations the pertinence of this newly proposed methodology in the case of passive scalars is also discussed

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

bounded cascade models as nonstationary multifractals we investigate a class of bounded random cascade models which are multiplicative by construction yet additive with respect to some but not all of their properties we assume the multiplicative weights go to unity as the cascade proceeds then the resulting field has upper and lower bounds two largely complementary multifractal statistical methods of analysis are used singular measures and structure functions yielding respectively the exponent hierarchies ital d sub ital q and ital h sub ital q we study in more detail a specific subclass of one dimensional models with weights 1 plus minus 1 minus 2 ital p ital r sub ital n minus 1 sup ital h at relative scale ital r sub ital n 2 sup minus ital n after ital n cascade steps the parameter ital h gt 0 regulates the degree of nonstationarity at ital h 0 stationarity prevails and singular ital p model cascades meneveau and sreenivasan phys rev lett 59 1424 1987 are retrieved our model has at once large scale stationarity and small scale nonstationarity with stationary increments due to the boundedness the ital d sub ital q all converge to unity with increasing ital n the rate of convergence is estimated and the results are discussed in terms of residual multifractality a spurious singularity spectrum due to finite size effects the structure function exponents are moremore interesting ital h sub ital q min l brace ital h 1 ital q r brace in the limit ital n r arrow infinity we further focus on the cases ital q 1 related to the fractal structure of the graph ital q 2 related to the energy spectrum and ital q 1 ital h the critical order beyond which our multiplicative and multiscaling bounded cascade model can be statistically distinguished from fractional brownian motion the corresponding additive and monoscaling model less

the thermodynamics of fractals revisited with wavelets abstract the multifractal formalism originally introduced to describe statistically the scaling properties of singular measures is revisited using the wavelet transform this new approach is based on the definition of partition functions from the wavelet transform modulus maxima we demonstrate that very much like thermodynamic functions the generalized fractal dimensions d q and the f singularity spectrum can be readily determined from the scaling behavior of these partition functions we show that this method provides a natural generalization of the classical box counting techniques to fractal signals the wavelets playing the role of generalized boxes we illustrate our theoretical considerations on pedagogical examples e g devil s staircases and fractional brownian motions we also report the results of some recent application of the wavelet transform modulus maxima method to fully developed turbulence data that we emphasize the wavelet transform as a mathematical microscope that can be further used to extract microscopic informations about the scaling properties of fractal objects in particular we show that a dynamical system which leaves invariant such an object can be uncovered form the space scale arrangement of its wavelet transform modulus maxima we elaborate on a wavelet based tree matching algorithm that provides a very promising tool for solving the inverse fractal problem this step towards a statistical mechanics of fractals is illustrated on discrete period doubling dynamical systems where the wavelet transform is shown to reveal the renormalization operation which is essential to the understanding of the universal properties of this transition to chaos finally we apply our technique to analyze the fractal hierarchy of dla azimuthal cantor sets defined by intersecting the inner frozen region of large mass off lattice diffusion limited aggregates dla wit a circle this study clearly lets out the existence of an underlying multiplicative process that is likely to account for the fibonacci structural ordering recently discovered in the apparently disordered arborescent dla morphology

estimating optimal feature subsets using efficient estimation of high dimensional mutual information a novel feature selection method using the concept of mutual information mi is proposed in this paper in all mi based feature selection methods effective and efficient estimation of high dimensional mi is crucial in this paper a pruned parzen window estimator and the quadratic mutual information qmi are combined to address this problem the results show that the proposed approach can estimate the mi in an effective and efficient way with this contribution a novel feature selection method is developed to identify the salient features one by one also the appropriate feature subsets for classification can be reliably estimated the proposed methodology is thoroughly tested in four different classification applications in which the number of features ranged from less than 10 to over 15000 the presented results are very promising and corroborate the contribution of the proposed feature selection methodology

tree based iterative input variable selection for hydrological modeling 1 input variable selection is an important issue associated with the development of several hydrological applications determining the optimal input vector from a large set of candidates to characterize a preselected output might result in a more accurate parsimonious and possibly physically interpretable model of the natural process in the hydrological context the modeled system often exhibits nonlinear dynamics and multiple interrelated variables moreover the number of candidate inputs can be very large and redundant especially when the model reproduces the spatial variability of the physical process the ideal input selection algorithm should therefore provide modeling flexibility computational efficiency in dealing with high dimension data set scalability with respect to input dimensionality and minimum redundancy in this paper we propose the tree based iterative input variable selection algorithm a novel hybrid model based model free approach specifically designed to fulfill these four requirements the algorithm structure provides robustness against redundancy while the tree based nature of the underlying model ensures the other key properties the approach is first tested on a well known benchmark case study to validate its accuracy and subsequently applied to a real world streamflow prediction problem in the upper ticino river basin switzerland results indicate that the algorithm is capable of selecting the most significant and nonredundant inputs in different testing conditions including the real world large data set characterized by the presence of several redundant variables this permits one to identify a compact representation of the observational data set which is key to improving the model performance and assisting with the interpretation of the underlying physical processes

synthetic turbulence constructed by spatially randomized fractal interpolation a spatially randomized fractal interpolation algorithm to construct synthetic fields with statistical properties close to real turbulence is proposed it improves the previous works on fractal interpolation so that the position mapping between large and small scales is chosen randomly and the stretching factors are drawn from any chosen random multiplier model in particular using different parameters of log poisson model synthetic fields with absolute scaling properties close to sl94 model for fully developed turbulence k41 model and mb2000 model for magnetohydrodynamic turbulence are obtained respectively to model real turbulence fields which do not obey absolute scaling laws but ess scaling laws a refined technique is added it is shown that both the velocity structure functions and the moments of subgrid scale stress can be precisely predicted when the scale invariance is broken

statistics of fourier modes in a turbulent flow fourier series are often used to discuss the properties of a homogeneous turbulent field we investigate the statistics of fourier modes of the turbulent velocity field and of a passive scalar the statistics of individual fourier modes is known to be gaussian when the size of the system l is much larger that the integral correlation size l 0 the case where the integral size is of the order of the system size l approximately l 0 is studied by direct numerical simulations in the range 20 or approximately rlambda or approximately 80 at a given rlambda we find that the probabilities of large fluctuations become larger when the wave number increases in qualitative agreement with the notion of intermittency as the reynolds number increases however the probability density functions become closer to gaussian in sharp contrast with the behavior of velocity increments we also show that in a simple model of cascade the fourier series decomposition is not appropriate to capture intermittency effects last we discuss other issues related to our results

bootstrap rank ordered conditional mutual information brocmi a nonlinear input variable selection method for water resources modeling the input variable selection problem has recently garnered much interest in the time series modeling community especially within water resources applications demonstrating that information theoretic nonlinear based input variable selection algorithms such as partial mutual information pmi selection pmis provide an improved representation of the modeled process when compared to linear alternatives such as partial correlation input selection pcis pmis is a popular algorithm for water resources modeling problems considering nonlinear input variable selection however this method requires the specification of two nonlinear regression models each with parametric settings that greatly influence the selected input variables other attempts to develop input variable selection methods using conditional mutual information cmi an analog to pmi have been formulated under different parametric pretenses such as k nearest neighbor knn statistics or kernel density estimates kde in this paper we introduce a new input variable selection method based on cmi that uses a nonparametric multivariate continuous probability estimator based on edgeworth approximations ea we improve the ea method by considering the uncertainty in the input variable selection procedure by introducing a bootstrap resampling procedure that uses rank statistics to order the selected input sets we name our proposed method bootstrap rank ordered cmi brocmi we demonstrate the superior performance of brocmi when compared to cmi based alternatives ea kde and knn pmis and pcis input variable selection algorithms on a set of seven synthetic test problems and a real world urban water demand uwd forecasting experiment in ottawa canada

scaling exponents and multifractal dimensions for independent random cascades this paper is concerned with mandelbrot s stochastic cascade measures the problems of i scaling exponents of structure functions of the measure q and ii multifractal dimensions are considered for cascades with a generator vector w1 wc of the general type these problems were previously studied for independent strongly bounded variableswi 0 a wi c consequently a broad class of models used in applications including kolmogorov s long normal model in turbulence long stable universal cascades in atmospheric dynamics has not been covered roughly speaking problems i ii are here solved under the condition that the scaling exists the function is calculated for all arguments previously this was done for positiveq and a new effect emerges the function can generally involve discontinuities in the first derivative as well as in the second

linear stochastic dynamics with nonlinear fractal properties abstract stochastic processes with multiplicative noise have been studied independently in several different contexts over the past decades we focus on the regime found for a generic set of control parameters in which stochastic processes with multiplicative noise produce intermittency of a special kind characterized by a power law probability density distribution we present a review of applications highlight the common physical mechanism and summarize the main known results the distribution and statistical properties of the duration of intermittent bursts are also characterized in detail

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

a stochastic model of human gait dynamics abstract we present a stochastic model of gait rhythm dynamics based on transitions between different neural centers that reproduces distinctive statistical properties of normal human walking by tuning one model parameter the transition hopping range the model can describe alterations in gait dynamics from childhood to adulthood including a decrease in the correlation and volatility exponents with maturation the model also generates time series with multifractal spectra whose broadness depends only on this parameter moreover we find that the volatility exponent increases monotonically as a function of the width of the multifractal spectrum suggesting the possibility of a change in multifractality with maturation

besov spaces and the multifractal hypothesis parisi and frisch proposed some time ago an explanation for multiscaling of turbulent velocity structure functions in terms of a multifractal hypothesis i e they conjecture that the velocity field has local holder exponents in a range hmin hmax with exponents h occurring on a sets h with a fractal dimensiond h heuristic reasoning led them to an expression for the scaling exponentz p ofpth order as the legendre transform of the codimensiond d h we show here that a part of the multifractal hypothesis is correct under even weaker assumptions namely if the velocity field hasl p mean holder indexs i e if it lies in the besov spaceb p s then local holder regularity is satisfied ifs d p then the hypothesis is true in a generalized sense of holder space with negative exponents and we discuss the proper definition of local holder classes of negative index finally if a certain box counting dimension exists then the legendre transform of its codimension gives the scaling exponentz p and more generally the maximal besov index of order p ass p z p p our method of proof is derived from a recent paper of s jaffard using compactly supported orthonormal wavelet bases and gives an extension of his results we discuss implications of the theorems for ensemble average scaling and fluid turbulence

structure function and fractal dissipation for an intermittent inviscid dyadic model we study a generalization of the original tree indexed dyadic model by katz and pavlovic for the turbulent energy cascade of the three dimensional euler equation we allow the coefficients to vary with some restrictions thus giving the model a realistic spatial intermittency by introducing a forcing term on the first component the fixed point of the dynamics is well defined and some explicit computations allow us to prove the rich multifractal structure of the solution in particular the exponent of the structure function is concave in accordance with other theoretical and experimental models moreover anomalous energy dissipation happens in a fractal set of dimension strictly less than 3

log similarity for turbulent flows abstract we critically examine a recently proposed representation for turbulent energy spectra called here log similarity which is predicted both by the multifractal model and the variational approach of the small scale intermittency within this representation spectra are homothetic in the log log plot the homothety factor depending on the taylor scale based reynolds number r experimental results on energy spectra and velocity structure functions of order 2 3 and 6 measured in different flows at several r show that this factor is r a measurable exponent which characterises intermittency in the inertial range experimentally displays the existence of a transition in the neighbourhood of r 400 and for r tending to infinity behaves as 1 in r r

a fractal model for large eddy simulation of turbulent flow abstract a new class of subgrid closures for large eddy simulation les of turbulence is developed based on the construction of synthetic fractal subgrid scale fields the relevant mathematical tool fractal interpolation allows to interpolate the resolved velocity with fields that have fluctuations down to much smaller scales and to compute the required stresses explicitly in one dimension the approach is used in the context of the coarse grained burgers equation then fractal interpolation is extended to three dimensions and is used to formulate a subgrid model for the filtered navier stokes equations the model is applied to les of both steady and freely decaying isotropic turbulence we find that the assumption of fractality per se is not enough to yield physically meaningful results and we explore several variants of the model in which the rules to generate the synthetic fields explicitly incorporate the condition that energy dissipation take place in one dimension this is accomplished by means of an additional transport equation that allows to dynamically determine the fractal dimension in three dimensions good results are obtained only once the fractal dimension is allowed to vary in different eigendirections of the resolved strain rate tensor so as to nearly maximize energy dissipation

hierarchical monte carlo methods for fractal random fields two hierarchical monte carlo methods for the generation of self similar fractal random fields are compared and contrasted the first technique successive random addition sra is currently popular in the physics community despite the intuitive appeal of sra rigorous mathematical reasoning reveals that sra cannot be consistent with any stationary power law gaussian random field for any hurst exponent furthermore there is an inherent ratio of largest to smallest putative scaling constant necessarily exceeding a factor of 2 for a wide range of hurst exponentsh with 0 30 h 0 85 thus sra is inconsistent with a stationary power law fractal random field and would not be useful for problems that do not utilize additional spatial averaging of the velocity field the second hierarchical method for fractal random fields has recently been introduced by two of the authors and relies on a suitable explicit multiwavelet expansion mwe with high moment cancellation this method is described briefly including a demonstration that unlike sra mwe is consistent with a stationary power law random field over many decades of scaling and has low variance

random cascades on wavelet dyadic trees we introduce a new class of random fractal functions using the orthogonal wavelet transform these functions are built recursively in the space scale half plane of the orthogonal wavelet transform cascading from an arbitrary given large scale towards small scales to each random fractal function corresponds a random cascading process referred to as a w cascade on the dyadic tree of its orthogonal wavelet coefficients we discuss the convergence of these cascades and the regularity of the so obtained random functions by studying the support of their singularity spectra then we show that very different statistical quantities such as correlation functions on the wavelet coefficients or the wavelet based multifractal formalism partition functions can be used to characterize very precisely the underlying cascading process we illustrate all our results on various numerical examples

microcanonical multifractal formalism a geometrical approach to multifractal systems part i singularity analysis multifractal formalism in the microcanonical framework has proved to be a valuable approach to understand and analyze complex signals typically associated with natural phenomena in scale invariant systems in this paper we discuss the multifractal microcanonical formalism in a comprehensive unified way including new theoretical proofs and validation tests on real signals so completing some known gaps in the foundations of this theory we also review the latest advances and describe the present perspectives in this field some technical details on the implementation of involved algorithms and relevant open issues are also discussed

intermittency in turbulence multiplicative random process in space and time we present a simple stochastic algorithm for generating multiplicative processes with multiscaling both in space and in time with this algorithm we are able to reproduce a synthetic signal with the same space and time correlation as the one coming from shell models for turbulence and the one coming from a turbulent velocity field in a quasi lagrangian reference frame

properties making a chaotic system a good pseudo random number generator we discuss the properties making a deterministic algorithm suitable to generate a pseudo random sequence of numbers high value of kolmogorov sinai entropy high dimensionality of the parent dynamical system and very large period of the generated sequence we propose the multidimensional anosov symplectic cat map as a pseudo random number generator we show what chaotic features of this map are useful for generating pseudo random numbers and investigate numerically which of them survive in the discrete state version of the map testing and comparisons with other generators are performed

application of the microcanonical multifractal formalism to monofractal systems the design of appropriate multifractal analysis algorithms able to correctly characterize the scaling properties of multifractal systems from experimental discretized data is a major challenge in the study of such scale invariant systems in the recent years a growing interest for the application of the microcanonical formalism has taken place as it allows a precise localization of the fractal components as well as a statistical characterization of the system in this paper we deal with the specific problems arising when systems that are strictly monofractal are analyzed using some standard microcanonical multifractal methods we discuss the adaptations of these methods needed to give an appropriate treatment of monofractal systems

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

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

the multifractal structure of contrast changes in natural images from sharp edges to textures we present a formalism that leads naturally to a hierarchical description of the different contrast structures in images providing precise definitions of sharp edges and other texture components within this formalism we achieve a decomposition of pixels of the image in sets the fractal components of the image such that each set contains only points characterized by a fixed strength of the singularity of the contrast gradient in its neighborhood a crucial role in this description of images is played by the behavior of contrast differences under changes in scale contrary to naive scaling ideas where the image is thought to have uniform transformation properties field 1987 each of these fractal components has its own transformation law and scaling exponents a conjecture on their biological relevance is also given

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

modeling and estimation of multiresolution stochastic processes an overview is provided of the several components of a research effort aimed at the development of a theory of multiresolution stochastic modeling and associated techniques for optimal multiscale statistical signal and image processing a natural framework for developing such a theory is the study of stochastic processes indexed by nodes on lattices or trees in which different depths in the tree or lattice correspond to different spatial scales in representing a signal or image in particular it is shown how the wavelet transform directly suggests such a modeling paradigm this perspective then leads directly to the investigation of several classes of dynamic models and related notions of multiscale stationarity in which scale plays the role of a time like variable the investigation of models on homogeneous trees is emphasized the framework examined here allows for consideration in a very natural way of the fusion of data from sensors with differing resolutions also thanks to the fact that wavelet transforms do an excellent job of compressing large classes of covariance kernels it is seen that these modeling paradigms appear to have promise in a far broader context than one might expect

three dimensional synthetic turbulence constructed by spatially randomized fractal interpolation a spatially randomized fractal interpolation algorithm to construct three dimensional synthetic turbulence from original coarse field is reported as in the one dimensional case by ding et al phys rev e 82 036311 2010 during the fractal interpolation positions mapping between large and small scale cubes are chosen randomly and the stretching factors are drawn from a log poisson random multiplicative process a linear combination function defined as the base part in fractal interpolation and a theoretical energy spectrum model for fully developed turbulence are introduced into the procedure statistical analysis shows that the synthetic field displays some properties very close to the direct numeric simulated field such as probability distributions of velocity velocity gradient velocity increment and the anomalous scaling behavior of the longitude velocity structure functions which follows the sl94 model precisely after a short time using direct numeric simulation with the synthetic field as initial data the typical local dynamical structures described by the teardrop shape of the q r plane for empirical turbulence can be reproduced

wavelet analysis of a gaussian kolmogorov signal several attempts have been made recently to use wavelets transforms for extracting histograms of scaling exponents from experimental turbulence data similar techniques are here applied to a gaussian signal having a kolmogorov 5 3 energy spectrum this is an instance of the class of fractional brownian motions having scaling exponent 1 3 almost everywhere for the gaussian signal a spurious non trivial histogram is obtained by applying a mexican hat wavelet transform analysis on the other hand we show that the use of complex wavelets and even more so the application of an optimal wavelet transform method strongly reduce the spurious fluctuations observed in the processing of the gaussian signal

translational invariance in turbulent cascade models due to the underlying hierarchical structure spatial correlation functions calculated from multiplicative cascade models are not translationally invariant a scheme is presented that restores translational invariance by averaging over the experimentally unknown spatial location of cascade realizations with respect to the observation window the impact of this scheme on multiplier distributions for the energy dissipation field in fully developed turbulence is analyzed only the experimental multiplier distribution is found to be invariant under a wide range of scales

a random process for the construction of multiaffine fields abstract we define a random process for the construction of multiaffine fields given the scaling exponents for the structure functions the difference with analogous processes for positive defined multifractal measures is stressed in particular our methods can be used for the study of the scaling laws exhibited by the velocity field in three dimensional fully developed turbulence we also discuss the probability distribution functions for the increments of the signal in the scaling range

multi affine model for the velocity distribution in fully turbulent flows a simple multi affine model for the velocity distribution in fully developed turbulent flows is introduced to capture the essential features of the underlying geometry of the velocity field the authors show that in this model the various relevant quantities characterizing different aspects of turbulence can be readily calculated a simultaneous good agreement is found with the available experimental data for the velocity structure functions the dq spectra obtained from studies of the velocity derivatives and the exponent describing the scaling of the spectrum of the kinetic energy fluctuations their results are obtained analytically assuming a single free parameter the fractal dimension of the region where the dominating contribution to dissipation comes from is estimated to be d approximately 2 88

efficient computation of many eigenvector derivatives using dynamic flexibility method a new approach for computing eigenvector derivatives with repeated eigenvalues of a complex structural system has been successfully developed this method utilizes a dynamic flexibility technique unlike other published methods this method requires one only to solve the system linear algebraic equations once regardless of the number of eigenvector derivatives needed to be computed tremendous computer time can be saved using this method theoretical derivation of this method is presented numerical examples are also presented to verify the analysis this method can be a powerful tool for the design engineer to compute many eigenvector derivatives of a large complex structural system

estimator of a non gaussian parameter in multiplicative log normal models we study non gaussian probability density functions pdf s of multiplicative log normal models in which the multiplication of gaussian and log normally distributed random variables is considered to describe the pdf of the velocity difference between two points in fully developed turbulent flows the non gaussian pdf model was originally introduced by castaing et al physica d 46 177 1990 in practical applications an experimental pdf is approximated with castaing s model by tuning a single non gaussian parameter which corresponds to the logarithmic variance of the log normally distributed variable in the model in this paper we propose an estimator of the non gaussian parameter based on the qth order absolute moments to test the estimator we introduce two types of stochastic processes within the framework of the multiplicative log normal model one is a sequence of independent and identically distributed random variables the other is a log normal cascade type multiplicative process by analyzing the numerically generated time series we demonstrate that the estimator can reliably determine the theoretical value of the non gaussian parameter scale dependence of the non gaussian parameter in multiplicative log normal models is also studied both analytically and numerically as an application of the estimator we demonstrate that non gaussian pdf s observed in the s p500 index fluctuations are well described by the multiplicative log normal model

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

exit times and entropy for dynamical systems stochastic processes and turbulence abstract we present an investigation of entropy h in dynamical systems stochastic processes and turbulence this tool allows for a suitable characterization of dynamical behaviours arising in systems with many different scales of motion particular emphasis is put on a recently proposed approach to the calculation of the entropy based on the exit time statistics the advantages of this method are demonstrated in examples of deterministic diffusive maps intermittent maps stochastic self and multi affine signals and experimental turbulent data concerning turbulence the multifractal formalism applied to the exit time statistics allows us to predict that h 3 for velocity time measurement this power law is independent of the presence of intermittency and has been confirmed by the experimental data analysis moreover we show that the entropy density of a three dimensional velocity field is affected by the correlations induced by the sweeping of large scales

improved dynamic flexibility method for extracting constrained modes from free test data the dynamic flexibility method presented can be used to extract constrained structural modes from free test data under certain conditions the original dynamic flexibility method is not effective 1 when the constrained structural test frequency moves into higher order free free analytical frequency range and 2 when the test frequency moves closer to any value h s of interesting in the higher order analytical frequencies of the free structure the former is the test frequency range of a constrained structure larger than that of a free structure the largest test frequency max of a constrained structure is higher than the smallest frequency h k 1 in the higher order analytical frequencies of a free structure under these two separate situations a power series used in the original dynamic flexibility method is diverging which leads to an invalid method to solve these problems properly an improvement of the dynamic flexibility method is proposed the kernel technique of this improvement is a hybrid shifting frequency procedure from the numerical results it is found that the improved method is better than the old method for all conditions

dynamic flexibility method for extracting constrained structural modes from free test data a method for extracting constrained modes from free boundary modal test data is developed as an alternate approach when the constrained modal survey proves dife cult the transformation relationship between the modes ofafree freestructureand thoseofa corresponding constrained structureisdescribed and a nonlinearcharacteristic equation is derived then an eigenvalue shifting technique which can also improve computational efe ciency is employed to avoid the singularity of the stiffness matrix of free structures the method utilizes a subset of measurable lower order free modes plus the effects of neglected higher order modes on the dynamic e exibility matrix however theanalytical or measured higher order modesof a freestructure are unknown ordife cult to obtain and so the contribution of the higher order modes to the dynamic e exibility matrix is expanded as a power series the constrained modes are obtained by solving the nonlinear characteristic equation it will be seen that this method is a typical substitute for the procedures based on experimental and theoretical modeling numerical simulation demonstrates that the method can converge quickly and has satisfactory precision

tree based reinforcement learning for optimal water reservoir operation 1 although being one of the most popular and extensively studied approaches to design water reservoir operations stochastic dynamic programming is plagued by a dual curse that makes it unsuitable to cope with large water systems the computational requirement grows exponentially with the number of state variables considered curse of dimensionality and an explicit model must be available to describe every system transition and the associated rewards costs curse of modeling a variety of simplifications and approximations have been devised in the past which in many cases make the resulting operating policies inefficient and of scarce relevance in practical contexts in this paper a reinforcement learning approach called fitted q iteration is presented it combines the principle of continuous approximation of the value functions with a process of learning off line from experience to design daily cyclostationary operating policies the continuous approximation performed via tree based regression makes it possible to mitigate the curse of dimensionality by adopting a very coarse discretization grid with respect to the dense grid required to design an equally performing policy via stochastic dynamic programming the learning experience in the form of a data set generated combining historical observations and model simulations allows us to overcome the curse of modeling lake como water system italy is used as study site to infer general guidelines on the appropriate setting for the algorithm parameters and to demonstrate the advantages of the approach in terms of accuracy and computational effectiveness compared to traditional stochastic dynamic programming

generalized scaling in fully developed turbulence in this paper we report numerical and experimental results on the scaling properties of the velocity turbulent fields in several flows the limits of a new form of scaling named extended self similarity ess are discussed we show that when a mean shear is absent the self scaling exponents are universal and they do not depend on the specific flow 3d homogeneous turbulence thermal convection mhd in contrast ess is not observed when a strong shear is present we propose a generalized version of self scaling which extends down to the smallest resolvable scales even in cases where ess is not present this new scaling is checked in several laboratory and numerical experiments a possible theoretical interpretation is also proposed a synthetic turbulent signal having most of the properties of a real one has been generated

multifractality of the feigenbaum attractor and fractional derivatives it is shown that fractional derivatives of the integrated invariant measure of the feigenbaum map at the onset of chaos have power law tails in their cumulative distributions whose exponents can be related to the spectrum of singularities f this is a new way of characterizing multifractality in dynamical systems so far applied only to multifractal random functions frisch and matsumoto j stat phys 108 1181 2002 the relation between the thermodynamic approach vul sinai and khanin russian math surveys 39 1 1984 and that based on singularities of the invariant measures is also examined the theory for fractional derivatives is developed from a heuristic point view and tested by very accurate simulations

the random subspace method for constructing decision forests much of previous attention on decision trees focuses on the splitting criteria and optimization of tree sizes the dilemma between overfitting and achieving maximum accuracy is seldom resolved a method to construct a decision tree based classifier is proposed that maintains highest accuracy on training data and improves on generalization accuracy as it grows in complexity the classifier consists of multiple trees constructed systematically by pseudorandomly selecting subsets of components of the feature vector that is trees constructed in randomly chosen subspaces the subspace method is compared to single tree classifiers and other forest construction methods by experiments on publicly available datasets where the method s superiority is demonstrated we also discuss independence between trees in a forest and relate that to the combined classification accuracy

fast algorithm for a three dimensional synthetic model of intermittent turbulence synthetic turbulence models are useful tools that provide realistic representations of turbulence necessary to test theoretical results to serve as background fields in some numerical simulations and to test analysis tools models of one dimensional 1d and 3d synthetic turbulence previously developed still required large computational resources a wavelet based model of synthetic turbulence able to produce a field with tunable spectral law intermittency and anisotropy is presented here the rapid algorithm introduced based on the classic p model of intermittent turbulence allows us to reach a broad spectral range using a modest computational effort the model has been tested against the standard diagnostics for intermittent turbulence i e the spectral analysis the scale dependent statistics of the field increments and the multifractal analysis all showing an excellent response

multifractal geometry in stock market time series it has been recently noticed that time series of returns in stock markets are of multifractal multiscaling character in that context multifractality has been always evidenced by its statistical signature i e the scaling exponents associated to a related variable however a direct geometrical framework much more revealing about the underlying dynamics is possible in this paper we present the techniques allowing the multifractal decomposition we will show that there exists a particular fractal component the most singular manifold msm which contains the relevant information about the dynamics of the series it is possible to reconstruct the series at a given precision from the msm we analyze the dynamics of the msm which shows revealing features about the evolution of this type of series