horizontal visibility graphs exact results for random time series networks this procedure allows us to apply methods of complex network theory for characterizing time series in this work we present the horizontal visibility algorithm a geometrically simpler and analytically solvable version of our former algorithm focusing on the mapping of random series series of independent identically distributed random variables after presenting some properties of the algorithm we present exact results on the topological properties of graphs associated with random series namely the degree distribution the clustering coefficient and the mean path length we show that the horizontal visibility algorithm stands as a simple method to discriminate randomness in time series since any random series maps to a graph with an exponential degree distribution of the shape pk 1 32 3 k 2 independent of the probability distribution from which the series was generated accordingly visibility graphs with other pk are related to nonrandom series numerical simulations confirm the accuracy of the theorems for finite series in a second part we show that the method is able to distinguish chaotic series from independent and identically distributed i i d theory studying the following situations i noise free low dimensional chaotic series ii low dimensional noisy chaotic series even in the presence of large amounts of noise and iii high dimensional chaotic series coupled map lattice without needs for additional techniques such as surrogate data or noise reduction methods finally heuristic arguments are given to explain the topological properties of chaotic series and several sequences that are conjectured to be random are analyzed

testing time reversibility without moment restrictions in this paper we propose a class of new tests for time reversibility it is shown that this test has an asymptotic normal distribution under the null hypothesis and non trivial power under local alternatives a novel feature of this test is that it does not have any moment restriction in contrast with other time reversibility and linearity tests our simulations also confirm that the proposed test is very robust when data do not possess proper moments an empirical study of stock market indices is also included to illustrate the usefulness of the new test

stock return distributions tests of scaling and universality from three distinct stock markets define the log return of stock price st over a time interval t rrtt log st tlog st1 analyses of returns of individual stocks 2 3 and stock indices 4 have shown that the cumulative distribution of returns is well fit by a power law asymptotic behavior with tail exponent r pr x xr 1

smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics this paper reviews various applications of the theory of smooth dynamical systems to conceptual problems of nonequilibrium statistical mecanics we adopt a new point of view which has emerged progressively in recent years and which takes seriously into account the chaotic character of the microscopic time evolution the emphasis is on nonequilibrium steady states rather than the traditional approach to equilibrium point of view of boltzmann the nonequilibrium steady states in presence of a gaussian thermostat are described by srb measures in terms of these one can prove the gallavotti cohen fluctuation theorem one can also prove a general linear response formula and study its consequences which are not restricted to near equilibrium situations at equilibrium one recovers in particular the onsager reciprocity relations under suitable conditions the nonequilibrium steady states satisfy the pairing theorem of dettmann and morriss the results just mentioned hold so far only for classical systems they do not involve large size i e they hold without a thermodynamic limit

entropy production and kullback leibler divergence between stationary trajectories of discrete systems the irreversibility of a stationary time series can be quantified using the kullback leibler divergence kld between the probability of observing the series and the probability of observing the time reversed series moreover this kld is a tool to estimate entropy production from stationary trajectories since it gives a lower bound to the entropy production of the physical process generating the series in this paper we introduce analytical and numerical techniques to estimate the kld between time series generated by several stochastic dynamics with a finite number of states we examine the accuracy of our estimators for a specific example a discrete flashing ratchet and investigate how close the kld is to the entropy production depending on the number of degrees of freedom of the system that are sampled in the trajectories

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

chaotic principle an experimental test abstract the chaotic hypothesis discussed in gallavotti and cohen 1995 is tested experimentally in a simple conduction model besides a confirmation of the hypothesis predictions the results suggest the validity of the hypothesis in the much wider context in which as the forcing strength grows the attractor ceases to be an anosov system and becomes an axiom a attractor a first text of new predictions is also attempted

steady states invariant measures and response theory using the method of computer simulation we test the predictions of nonlinear response theory for classical systems subject to dissipative external fields we provide convincing numerical evidence that kawasaki methods agree with both the transient time correlation function predictions and with a direct measurement of the nonlinear response furthermore this numerical agreement is observed over a time scale which is sufficiently long for the response to have relaxed to within ensuremath sim 1 of its nonequilibrium steady state value this is in spite of the fact that in the steady state the n particle distribution function ultimately becomes fractal we discuss the normalization of the kawasaki distribution and derive a lagrangian form of the kawasaki response function and show that it is consistent with predictions that are obtained using a natural invariant measure for nonequilibrium steady states c 1995 the american physical society

causality response theory and the second law of thermodynamics we show that there is a close connection between the assumption of causality and the second law of thermodynamics we also show that for a class of classical reversible deterministic systems it is overwhelmingly improbable either to find causal steady states that violate the second law or anticausal states that satisfy the second law these arguments indicate that the existence of and the sign associated with the second law of thermodynamics is ultimately determined by causality our discussion employs a green kubo relation that we derive for an anticausal linear transport coefficient

new irreversibility measure and complexity analysis based on singular value decomposition abstract visibility graph algorithm is a powerful tool for visualization and analysis of complex dynamical system kullback leibler divergence kld based on the horizontal visibility graph algorithm has received largely extensive attention in the field of irreversibility analysis in this paper we will propose a new irreversibility measure built on the multi scale theory i e the kld based on singular value decomposition kld svd furthermore since matrix singular value can reflect the basic characteristics of the complex system we further bring forward the definition of shannon entropy with singular value decomposition se svd after considering that the shannon entropy has been well defined as a complexity measure in order to show the advantages of these two new measures in detecting the complexity of systems several simulation and real data experiments are chosen to examine the performance of them through these results we find that the average and variance of kld svd corresponding to each time series is less than that of the kld moreover kld svd s volatility is significantly weaker than kld s as there is a conceptual link between predictability and irreversibility we can argue that kld svd algorithm implies the higher predictability and lower confusion on the other hand se svd can do well in revealing the internal complexity of the different time series meanwhile it is useful to mention that se svd is robust against noise efficiently

reply to comment on tests of scaling and universality of the distributions of trade size and share volume evidence from three distinct markets analyzing trade by trade data for three distinct markets we showed that the cumulative distributions of trade size display power law tails pqxxq with exponents q in the levy stable domain qq 2 moreover we reported that the exponent values are consistent for all stocks irrespective of stock specific variables such as market capitalization industry sector or the specific market where the stock is traded our conclusions were based on using two distinct estimation methods racz et al now propose that one of the estimators we used has slow convergence for a pure power law particularly as tail exponents approach the boundary q 2 we examine the robustness of our results to specific estimation method by additional analysis using five distinct techniques to estimate q we find results that are fully consistent with those we had reported providing compelling evidence that our conclusions hold regardless of estimation procedure

langevin equations from time series we discuss the link between the approach to obtain the drift and diffusion of one dimensional langevin equations from time series and pope and ching s relationship for stationary signals the two approaches are based on different interpretations of conditional averages of the time derivatives of the time series at given levels the analysis provides a useful indication for the correct application of pope and ching s relationship to obtain stochastic differential equations from time series and shows its validity in a generalized sense for nondifferentiable processes originating from langevin equations

time reversibility from visibility graphs of nonstationary processes visibility algorithms are a family of methods to map time series into networks with the aim of describing the structure of time series and their underlying dynamical properties in graph theoretical terms here we explore some properties of both natural and horizontal visibility graphs associated to several nonstationary processes and we pay particular attention to their capacity to assess time irreversibility nonstationary signals are infinitely irreversible by definition independently of whether the process is markovian or producing entropy at a positive rate and thus the link between entropy production and time series irreversibility has only been explored in nonequilibrium stationary states here we show that the visibility formalism naturally induces a new working definition of time irreversibility which allows us to quantify several degrees of irreversibility for stationary and nonstationary series yielding finite values that can be used to efficiently assess the presence of memory and off equilibrium dynamics in nonstationary processes without the need to differentiate or detrend them we provide rigorous results complemented by extensive numerical simulations on several classes of stochastic processes

a characterization of horizontal visibility graphs and combinatorics on words abstract a horizontal visibility graph hvg is defined in association with an ordered set of non negative reals hvgs realize a methodology in the analysis of time series their degree distribution being a good discriminator between randomness and chaos luque et al b luque l lacasa f ballesteros j luque horizontal visibility graphs exact results for random time series phys rev e 80 2009 046103 we prove that a graph is an hvg if and only if it is outerplanar and has a hamilton path therefore an hvg is a noncrossing graph as defined in algebraic combinatorics flajolet and noy p flajolet m noy analytic combinatorics of noncrossing configurations discrete math 204 1999 203 229 our characterization of hvgs implies a linear time recognition algorithm treating ordered sets as words we characterize subfamilies of hvgs highlighting various connections with combinatorial statistics and introducing the notion of a visible pair with this technique we determine asymptotically the average number of edges of hvgs

note on two theorems in nonequilibrium statistical mechanics an attempt is made to clarify the difference between a theorem derived by evans and searles in 1994 on the statistics of trajectories in phase space and a theorem proved by the authors in 1995 on the statistics of fluctuations on phase space trajectory segments in a nonequilibrium stationary state

test your surrogate data before you test for nonlinearity the schemes for the generation of surrogate data in order to test the null hypothesis of linear stochastic process undergoing nonlinear static transform are investigated as to their consistency in representing the null hypothesis in particular we pinpoint some important caveats of the prominent algorithm of amplitude adjusted fourier transform surrogates aaft and compare it to the iterated aaft which is more consistent in representing the null hypothesis it turns out that in many applications with real data the inferences of nonlinearity after marginal rejection of the null hypothesis were premature and have to be reinvestigated taking into account the inaccuracies in the aaft algorithm mainly concerning the mismatching of the linear correlations in order to deal with such inaccuracies we propose the use of linear together with nonlinear polynomials as discriminating statistics the application of this setup to some well known real data sets cautions against the use of the aaft algorithm

on loss functions which minimize to conditional expected values and posterior probabilities a loss function or objective function is a function used to compare parameters when fitting a model to data the loss function gives a distance between the model output and the desired output two common examples are the squared error loss function and the cross entropy loss function minimizing the mean square error loss function is equivalent to minimizing the mean square difference between the model output and the expected value of the output given a particular input this property of minimization to the expected value is formalized as p admissibility the necessary and sufficient conditions for p admissibility leading to a parametric description of all p admissible loss functions are found in particular it is shown that two of the simplest members of this class of functions are the squared error and the cross entropy loss functions one application of this work is in the choice of a loss function for training neural networks to provide probability estimates

a numerical criterion for uniruledness an n dimensional variety x over an algebraically closed field k is said to be uniruled if there exist an n 1 dimensional k variety w and a dominant rational map f p x w x x is called separably uniruled if f is a separable map i e k p x w is a finite separable extension of k x via f if x is uniniled then there exists a rational curve passing through a general k valued point of x and the converse is true by the countability of the components of the hilbert scheme provided that the ground field k is uncountable uniruled varieties are of importance in the classification of varieties it is conjectured that in characteristic 0 uniruledness is equivalent to k xc this article gives a numerical criterion for a projective variety to be uniruled

multiple testing strategy for the detection of temporal irreversibility in stationary time series we propose a strategy for the detection of temporal irreversibility in stationary time series based on multiple bidimensional tests the test is helpful to evaluate the displacement of irreversibility toward high dimensions the test can be used independently of the theoretical functionals actually utilized to check irreversibility the method was applied to simulated nonlinear signals generated by the delayed henon map and a two loop negative feedback model to show how the presence of a delay could produce the displacement of irreversibility toward higher dimensions the method was applied also to series of a biological variable i e heart period that is known to be regulated by multiple feedback loops simulations and real data support the need of exploring progressively increasing embedding dimensions when assessing temporal irreversibility

an improvement of the measurement of time series irreversibility with visibility graph approach abstract we propose a method to improve the measure of real valued time series irreversibility which contains two tools the directed horizontal visibility graph and the kullback leibler divergence the degree of time irreversibility is estimated by the kullback leibler divergence between the i n and o u t degree distributions presented in the associated visibility graph in our work we reframe the i n and o u t degree distributions by encoding them with different embedded dimensions used in calculating permutation entropy pe with this improved method we can not only estimate time series irreversibility efficiently but also detect time series irreversibility from multiple dimensions we verify the validity of our method and then estimate the amount of time irreversibility of series generated by chaotic maps as well as global stock markets over the period 2005 2015 the result shows that the amount of time irreversibility reaches the peak with embedded dimension d 3 under circumstances of experiment and financial markets

time irreversibility of financial time series based on higher moments and multiscale kullback leibler divergence abstract irreversibility is an important property of time series in this paper we propose the higher moments and multiscale kullback leibler divergence to analyze time series irreversibility the higher moments kullback leibler divergence hmkld can amplify irreversibility and make the irreversibility variation more obvious therefore many time series whose irreversibility is hard to be found are also able to show the variations we employ simulated data and financial stock data to test and verify this method and find that hmkld of stock data is growing in the form of fluctuations as for multiscale kullback leibler divergence mkld it is very complex in the dynamic system so that exploring the law of simulation and stock system is difficult in conventional multiscale entropy method the coarse graining process is non overlapping however we apply a different coarse graining process and obtain a surprising discovery the result shows when the scales are 4 and 5 their entropy is nearly similar which demonstrates mkld is efficient to display characteristics of time series irreversibility

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

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

relation between the probability density and other properties of a stationary random process we consider the pope ching differential equation phys fluids a 5 1529 1993 connecting the probability density p x x of a stationary homogeneous stochastic process x t and the conditional moments of its squared velocity and acceleration we show that the solution of the pope ching equation can be expressed as n x where n x is the mean number of crossings of the x level per unit time and is the mean inverse velocity of crossing this result shows that the probability density at x is fully determined by a one point measurement of crossing velocities and does not imply knowledge of the x t behavior outside of the infinitesimally narrow window near x

irreversibility and fluctuation theorem in stationary time series the relative entropy between the joint probability distribution of backward and forward sequences is used to quantify time asymmetry or irreversibility for stationary time series the parallel with the thermodynamic theory of nonequilibrium steady states allows us to link the degree of asymmetry in the time signal with the distance from equilibrium and the lack of detailed balance among its states we study the statistics of time asymmetry in terms of the fluctuation theorem showing that this type of relationship derives from simple general symmetries valid for any stationary time series

multiscale multifractal time irreversibility analysis of stock markets abstract time irreversibility is one of the most important properties of nonstationary time series complex time series often demonstrate even multiscale time irreversibility such that not only the original but also coarse grained time series are asymmetric over a wide range of scales we study the multiscale time irreversibility of time series in this paper we develop a method called multiscale multifractal time irreversibility analysis mmra which allows us to extend the description of time irreversibility to include the dependence on the segment size and statistical moments we test the effectiveness of mmra in detecting multifractality and time irreversibility of time series generated from delayed henon map and binomial multifractal model then we employ our method to the time irreversibility analysis of stock markets in different regions we find that the emerging market has higher multifractality degree and time irreversibility compared with developed markets in this sense the mmra method may provide new angles in assessing the evolution stage of stock markets

entropy production for quantum markov semigroups an invariant state of a quantum markov semigroup is an equilibrium state if it satisfies a quantum detailed balance condition in this paper we introduce a notion of entropy production for faithful normal invariant states of a quantum markov semigroup on mathcal b mathsf h as a numerical index measuring how far they are from equilibrium the entropy production rate is defined as the derivative of the relative entropy of the one step forward and backward evolution in analogy with the classical probabilistic concept we prove an explicit trace formula expressing the entropy production rate in terms of the completely positive part of the generator of a norm continuous quantum markov semigroup showing that it turns out to be zero if and only if a standard quantum detailed balance condition holds

modeling long range cross correlations in two component arfima and fiarch processes we investigate how simultaneously recorded long range power law correlated multivariate signals cross correlate to this end we introduce a two component arfima stochastic process and a two component fiarch process to generate coupled fractal signals with long range power law correlations which are at the same time long range cross correlated we study how the degree of cross correlations between these signals depends on the scaling exponents characterizing the fractal correlations in each signal and on the coupling between the signals our findings have relevance when studying parallel outputs of multiple component of physical physiological and social systems

on the structure of strict sense bayesian cost functions and its applications in the context of classification problems the paper analyzes the general structure of the strict sense bayesian ssb cost functions those having a unique minimum when the soft decisions are equal to the posterior class probabilities we show that any ssb cost is essentially the sum of a generalized measure of entropy which does not depend on the targets and an error component symmetric cost functions are analyzed in detail our results provide a further insight on the behavior of this family of objective functions and are the starting point for the exploration of novel algorithms two applications are proposed first the use of asymmetric ssb cost functions for posterior probability estimation in non maximum a posteriori map decision problems second a novel entropy minimization principle for hybrid learning use labeled data to minimize the cost function and unlabeled data to minimize the corresponding entropy measure

on the asymptotic convergence of the transient and steady state fluctuation theorems nonequilibrium molecular dynamics simulations are used to demonstrate the asymptotic convergence of the transient and steady state forms of the fluctuation theorem in the case of planar poiseuille flow we find that the transient form valid for all times converges to the steady state predictions on microscopic time scales further we find that the time of convergence for the two theorems coincides with the time required for satisfaction of the asymptotic steady state fluctuation theorem

asymptotics of the entropy production rate for d dimensional ornstein uhlenbeck processes in the context of non equilibrium statistical physics the entropy production rate is an important concept to describe how far a specific state of a system is from its equilibrium state in this paper we establish a central limit theorem and a moderate deviation principle for the entropy production rate of d dimensional ornstein uhlenbeck processes by the techniques of functional inequalities such as poincare inequality and log sobolev inequality as an application we obtain a law of iterated logarithm for the entropy production rate

the minimum description length principle for probability density estimation by regular histograms abstract the minimum description length principle is a general methodology for statistical modeling and inference that selects the best explanation for observed data as the one allowing the shortest description of them application of this principle to the important task of probability density estimation by histograms was previously proposed we review this approach and provide additional illustrative examples and an application to real world data with a presentation emphasizing intuition and concrete arguments we also consider alternative ways of measuring the description lengths that can be found to be more suited in this context we explicitly exhibit analyze and compare the complete forms of the description lengths with formulas involving the information entropy and redundancy of the data and not given elsewhere histogram estimation as performed here naturally extends to multidimensional data and offers for them flexible and optimal subquantization schemes the framework can be very useful for modeling and reduction of complexity of observed data based on a general principle from statistical information theory and placed within a unifying informational perspective

nonparametric density estimation the l 1 view differentiation of integrals consistency lower bounds for rates of convergence rates of convergence in l1 the automatic kernel estimate l1 and pointwise convergence estimates related to the kernel estimate and the histogram estimate simulation inequalities and random variate generation the transformed kernel estimate applications in discrimination operations on density estimates estimators based on orthogonal series index

a universal learning rule that minimizes well formed cost functions in this paper we analyze stochastic gradient learning rules for posterior probability estimation using networks with a single layer of weights and a general nonlinear activation function we provide necessary and sufficient conditions on the learning rules and the activation function to obtain probability estimates also we extend the concept of well formed cost function proposed by wittner and denker to multiclass problems and we provide theoretical results showing the advantages of this kind of objective functions

divergence estimation of continuous distributions based on data dependent partitions we present a universal estimator of the divergence d p spl par q for two arbitrary continuous distributions p and q satisfying certain regularity conditions this algorithm which observes independent and identically distributed i i d samples from both p and q is based on the estimation of the radon nikodym derivative dp dq via a data dependent partition of the observation space strong convergence of this estimator is proved with an empirically equivalent segmentation of the space this basic estimator is further improved by adaptive partitioning schemes and by bias correction the application of the algorithms to data with memory is also investigated in the simulations we compare our estimators with the direct plug in estimator and estimators based on other partitioning approaches experimental results show that our methods achieve the best convergence performance in most of the tested cases

horizontal visibility graphs transformed from fractional brownian motions topological properties versus the hurst index abstract nonlinear time series analysis aims at understanding the dynamics of stochastic or chaotic processes in recent years quite a few methods have been proposed to transform a single time series to a complex network so that the dynamics of the process can be understood by investigating the topological properties of the network we study the topological properties of horizontal visibility graphs constructed from fractional brownian motions with different hurst indexes h 0 1 special attention has been paid to the impact of the hurst index on topological properties it is found that the clustering coefficient c decreases when h increases we also found that the mean length l of the shortest paths increases exponentially with h for fixed length n of the original time series in addition l increases linearly with respect to n when h is close to 1 and in a logarithmic form when h is close to 0 although the occurrence of different motifs changes with h the motif rank pattern remains unchanged for different h adopting the node covering box counting method the horizontal visibility graphs are found to be fractals and the fractal dimension d b decreases with h furthermore the pearson coefficients of the networks are positive and the degree degree correlations increase with degree which indicate that the horizontal visibility graphs are assortative with the increase of h the pearson coefficient decreases first and then increases in which the turning point is around h 0 6 the presence of both fractality and assortativity in the horizontal visibility graphs converted from fractional brownian motions is different from many cases where fractal networks are usually disassortative

testing time symmetry in time series using data compression dictionaries time symmetry often called statistical time reversibility in a dynamical process means that any segment of time series output has the same probability of occurrence in the process as its time reversal a technique based on symbolic dynamics is proposed to distinguish such symmetrical processes from asymmetrical ones given a time series observation of the otherwise unknown process because linear stochastic gaussian processes and static nonlinear transformations of them are statistically reversible but nonlinear dynamics such as dissipative chaos are usually statistically irreversible a test will separate large classes of hypotheses for the data a general purpose and robust statistical test procedure requires adapting to arbitrary dynamics which may have significant time correlation of undetermined form given a symbolization of the observed time series the technology behind adaptive dictionary data compression algorithms offers a suitable estimate of reversibility as well as a statistical likelihood test the data compression methods create approximately independent segments permitting a simple and direct null test without resampling or surrogate data we demonstrate the results on various time series reversible and irreversible systems

the large deviation principle and steady state fluctuation theorem for the entropy production rate of a stochastic process in magnetic fields fluctuation theorem is one of the major achievements in the field of nonequilibrium statistical mechanics during the past two decades there exist very few results for steady state fluctuation theorem of sample entropy production rate in terms of large deviation principle for diffusion processes due to the technical difficulties here we give a proof for the steady state fluctuation theorem of a diffusion process in magnetic fields with explicit expressions of the free energy function and rate function the proof is based on the karhunen loeve expansion of complex valued ornstein uhlenbeck process

variational methods for time dependent classical many particle systems abstract a variational method for the classical liouville equation is introduced that facilitates the development of theories for non equilibrium classical systems the method is based on the introduction of a complex valued auxiliary quantity that is related to the classical position momentum probability density via a functional of is developed whose extrema imply that satisfies the liouville equation multiscale methods are used to develop trial functions to be optimized by the variational principle the present variational principle with multiscale trial functions can capture both the microscopic and the coarse grained descriptions thereby yielding theories that account for the two way exchange of information across multiple scales in space and time equations of the smoluchowski form for the coarse grained state probability density are obtained constraints on the initial state of the n particle probability density for which the aforementioned equation is closed and conserves probability are presented the methodology has applicability to a wide range of systems including macromolecular assemblies ionic liquids and nanoparticles

finite bath fluctuation theorem we demonstrate that a finite bath fluctuation theorem of the crooks type holds for systems that have been thermalized via weakly coupling them to a bath with energy independent finite specific heat we show that this theorem reduces to the known canonical and microcanonical fluctuation theorems in the two respective limiting cases of infinite and vanishing specific heat of the bath the result is elucidated by applying it to a two dimensional hard disk colliding elastically with few other hard disks in a rectangular box with perfectly reflecting walls

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

power law and multiscaling properties of the chinese stock market abstract we investigate the cumulative probability density function pdf and the multiscaling properties of the returns in the chinese stock market by using returns data adjusted for thin trading we find that the distribution has power law tails at shorter microscopic timescales or lags however the distribution follows an exponential law for longer timescales furthermore we investigate the long range correlation and multifractality of the returns in the chinese stock market by the dfa and mfdfa methods we find that all the scaling exponents are between 0 5 and 1 by dfa method which exhibits the long range power law correlations in the chinese stock market moreover we find by mfdfa method that the generalized hurst exponents h q are not constants which shows the multifractality in the chinese stock market we also find that the correlation of shenzhen stock market is stronger than that of shanghai stock market

a local fluctuation theorem abstract a mechanism for the validity of a local version of the fluctuation theorem uniform in the system size is discussed for a reversible chain of weakly coupled anosov systems

local kinetic interpretation of entropy production through reversed diffusion the time reversal of stochastic diffusion processes is revisited with emphasis on the physical meaning of the time reversed drift and the noise prescription in the case of multiplicative noise the local kinematics and mechanics of free diffusion are linked to the hydrodynamic description these properties also provide an interpretation of the pope ching formula for the steady state probability density function along with a geometric interpretation of the fluctuation dissipation relation finally the statistics of the local entropy production rate of diffusion are discussed in the light of local diffusion properties and a stochastic differential equation for entropy production is obtained using the girsanov theorem for reversed diffusion the results are illustrated for the ornstein uhlenbeck process

cost functions to estimate a posteriori probabilities in multiclass problems the problem of designing cost functions to estimate a posteriori probabilities in multiclass problems is addressed we establish necessary and sufficient conditions that these costs must satisfy in one class one output networks whose outputs are consistent with probability laws we focus our attention on a particular subset of the corresponding cost functions which verify two common properties symmetry and separability well known cost functions such as the quadratic cost or the cross entropy are particular cases in this subset finally we present a universal stochastic gradient learning rule for single layer networks in the sense of minimizing a general version of these cost functions for a wide family of nonlinear activation functions

state dependent diffusion thermodynamic consistency and its path integral formulation the friction coefficient of a particle can depend on its position as it does when the particle is near a wall we formulate the dynamics of particles with such state dependent friction coefficients in terms of a general langevin equation with multiplicative noise whose evaluation requires the introduction of specific rules two common conventions the ito and the stratonovich provide alternative rules for evaluation of the noise but other conventions are possible we show that the requirement that a particle s distribution function approach the boltzmann distribution at long times dictates that a drift term must be added to the langevin equation this drift term is proportional to the derivative of the diffusion coefficient times a factor that depends on the convention used to define the multiplicative noise we explore the consequences of this result in a number of examples with spatially varying diffusion coefficients we also derive a path integral representation for arbitrary interpretation of the noise and use it in a perturbative study of correlations in a simple system

context tree modeling of observed symbolic dynamics modern techniques invented for data compression provide efficient automated algorithms for the modeling of the observed symbolic dynamics we demonstrate the relationship between coding and modeling motivating the well known minimum description length mdl principle and give concrete demonstrations of the context tree weighting and context tree maximizing algorithms the predictive modeling technique obviates many of the technical difficulties traditionally associated with the correct mdl analyses these symbolic models representing the symbol generating process as a finite state automaton with probabilistic emission probabilities provide excellent and reliable entropy estimations the resimulations of estimated tree models satisfying the mdl model selection criterion are faithful to the original in a number of measures the modeling suggests that the automated context tree model construction could replace fixed order word lengths in many traditional forms of empirical symbolic analysis of the data we provide an explicit pseudocode for implementation of the context tree weighting and maximizing algorithms as well as for the conversion to an equivalent markov chain

econophysics financial time series from a statistical physics point of view in recent years physicists have started applying concepts and methods of statistical physics to study economic problems the word econophysics is sometimes used to refer to this work much recent work is focused on understanding the statistical properties of nancial time series one reason for this interest is that nancial markets are examples of complex interacting systems for which a huge amount of data exist and it is possible that nancial time series viewed from a dierent perspective might yield new results this article reviews the results of three recent phenomenological studies i the probability distribution of stock price uctuations stock price uctuations occur in all magnitudes in analogy to earthquakes from tiny uctuations to drastic events such as market crashes the distribution of price uctuations decays with a power law tail well outside the levy stable regime and describes uctuations that dier by as much as eight orders of magnitude in addition this distribution preserves its functional form for uctuations on time scales that dier by three orders of magnitude from 1 min up to approximately 10 d ii correlations in nancial time series while price uctuations themselves have rapidly decaying correlations the magnitude of uctuations measured by either the absolute value or the square of the price uctuations has correlations that decay as a power law and persist for several months iii correlations among dierent companies the third result bears on the application of random matrix theory to understand the correlations among price uctuations of any two dierent stocks from a study of the eigenvalue statistics of the cross correlation matrix constructed from price uctuations of the leading 1000 stocks we nd that the largest 1 of the eigenvalues and the corresponding eigenvectors show systematic deviations from the predictions for a random matrix whereas the rest of the eigenvalues conform to random matrix behavior suggesting that these 1 of the eigenvalues contain system specic information about correlated time evolution of dierent companies c 2000 published by elsevier science b v all rights reserved

strong approximation of quantile processes by iterated kiefer processes the notion of a kth iterated kiefer process k t k for k n and v t r is introduced we show that the uniform quantile process n t may be approximated on 0 1 by n 1 2 k n t k at an optimal uniform almost sure rate of o n 1 2 1 2k 1 0 1 for each k n our arguments are based in part on a new functional limit law of independent interest for the increments of the empirical process applications include an extended version of the uniform bahadur kiefer representation together with strong limit theorems for nonparametric functional estimators

equality governing nonequilibrium fluctuations and its information theory and thermodynamic interpretations the master equation is cast in the form of an equality involving the variation in the course of a transformation of a quantity playing the role of a generalized potential weighted with the probability of allowable transformations emanating from different initial states in the most general case this equality cannot be formulated entirely in terms of thermodynamic variables and state functions some conditions under which such a reduction becomes possible are identified and a comparison with fluctuation and work type relationships previously reported in the literature is carried out

dynamics and kinematics of reciprocal diffusions this paper examines the dynamics and kinematics of reciprocal diffusions reciprocal processes were introduced by bernstein in 1932 and were later studied in detail by jamison the reciprocal diffusions are constructed here by specifying their finite joint densities in terms of the green s function of a general heat operator and an end point density a path integral interpretation of the heat operator green s function is provided which is used to derive a stochastic form of newton s law as well as a conditional distribution for the velocity of a diffusing particle given its position these results are then employed to derive two conservation laws expressing the conservation of mass and momentum the conservation laws do not form a closed system of equations in general except for two subclasses of reciprocal diffusions the markov and quantum diffusions

stochastic analysis of symmetry breaking bifurcations master equation approach the multivariate master equation for a general reaction diffusion system is solved perturbatively in the vicinity of a bifurcation point leading to symmetry breaking transitions the possibility to express the result through a brazovskii type of potential is examined and a comparison with the langevin analysis of walgraefet al adv chem phys 49 311 1982 is performed

entropy production fluctuations of finite markov chains for almost every trajectory segment over a finite time span of a finite markov chain with any given initial distribution the logarithm of the ratio of its probability to that of its time reversal converges exponentially to the entropy production rate of the markov chain the large deviation rate function has a symmetry of gallavotti cohen type which is called the fluctuation theorem moreover similar symmetries also hold for the rate functions of the joint distributions of general observables and the logarithmic probability ratio

topological properties of the limited penetrable horizontal visibility graph family the limited penetrable horizontal visibility graph algorithm was recently introduced to map time series in complex networks in this work we extend this algorithm to create a directed limited penetrable horizontal visibility graph and an image limited penetrable horizontal visibility graph we define two algorithms and provide theoretical results on the topological properties of these graphs associated with different types of real value series we perform several numerical simulations to check the accuracy of our theoretical results finally we present an application of the directed limited penetrable horizontal visibility graph to measure real value time series irreversibility and an application of the image limited penetrable horizontal visibility graph that discriminates noise from chaos we also propose a method to measure the systematic risk using the image limited penetrable horizontal visibility graph and the empirical results show the effectiveness of our proposed algorithms

stationary and transient work fluctuation theorems for a dragged brownian particle recently wang et al carried out a laboratory experiment where a brownian particle was dragged through a fluid by a harmonic force with constant velocity of its center this experiment confirmed a theoretically predicted work related integrated transient fluctuation theorem itft which gives an expression for the ratio for the probability to find positive or negative values for the fluctuations of the total work done on the system in a given time in a transient state the corresponding integrated stationary state fluctuation theorem issft was not observed using an overdamped langevin equation and an arbitrary motion for the center of the harmonic force all quantities of interest for these theorems and the corresponding nonintegrated ones tft and ssft respectively are theoretically explicitly obtained in this paper while the tft and the itft are satisfied for all times the ssft and the issft only hold asymptotically in time suggestions for further experiments with arbitrary velocity of the harmonic force and in which also the issft could be observed are given in addition a nontrivial long time relation between the itft and the issft was discovered which could be observed experimentally especially in the case of a resonant circular motion of the center of the harmonic force

fluctuation theorems for entropy production and heat dissipation in periodically driven markov chains asymptotic fluctuation theorems are statements of a gallavotti cohen symmetry in the rate function of either the time averaged entropy production or heat dissipation of a process such theorems have been proved for various general classes of continuous time deterministic and stochastic processes but always under the assumption that the forces driving the system are time independent and often relying on the existence of a limiting ergodic distribution in this paper we extend the asymptotic fluctuation theorem for the first time to inhomogeneous continuous time processes without a stationary distribution considering specifically a finite state markov chain driven by periodic transition rates we find that for both entropy production and heat dissipation the usual gallavotti cohen symmetry of the rate function is generalized to an analogous relation between the rate functions of the original process and its corresponding backward process in which the trajectory and the driving protocol have been time reversed the effect is that spontaneous positive fluctuations in the long time average of each quantity in the forward process are exponentially more likely than spontaneous negative fluctuations in the backward process and vice versa revealing that the distributions of fluctuations in universes in which time moves forward and backward are related as an additional result the asymptotic time averaged entropy production is obtained as the integral of a periodic entropy production rate that generalizes the constant rate pertaining to homogeneous dynamics

a two sided stochastic integral and its calculus let x be a forward diffusion and y a backward diffusion both defined on 0 1 x t and y t being respectively adapted to the past of a wiener process w and to its future increments we construct a two sided stochastic integral of the form mathop smallint limits 0 t phi u x u y u dw u which generalizes the backward and forward ito integrals simultaneously our construction is quite intuitive and leads to a generalized stochastic calculus it is also shown that for each fixed t our integral coincides with that defined by skorohod in 18

a simple discrete model of brownian motors time periodic markov chains in this paper we consider periodically inhomogeneous markov chains which can be regarded as a simple version of physical model brownian motors we introduce for them the concepts of periodical reversibility detailed balance entropy production rate and circulation distribution we prove the equivalence of the following statements the time periodic markov chain is periodically reversible it is in detailed balance kolmogorov s cycle condition is satisfied its entropy production rate vanishes every circuit and its reversed circuit have the same circulation weight hence in our model of markov chains the directed transport phenomenon of brownian motors i e the existence of net circulation can occur only in nonequilibrium and irreversible systems moreover we verify the large deviation property and the gallavotti cohen fluctuation theorem of sample entropy production rates of the markov chain

time irreversibility and intrinsics revealing of series with complex network approach abstract in this work we analyze time series on the basis of the visibility graph algorithm that maps the original series into a graph by taking into account the all round information carried by the signals the time irreversibility and fractal behavior of series are evaluated from a complex network perspective and considered signals are further classified from different aspects the reliability of the proposed analysis is supported by numerical simulations on synthesized uncorrelated random noise short term correlated chaotic systems and long term correlated fractal processes and by the empirical analysis on daily closing prices of eleven worldwide stock indices obtained results suggest that finite size has a significant effect on the evaluation and that there might be no direct relation between the time irreversibility and long range correlation of series similarity and dissimilarity between stock indices are also indicated from respective regional and global perspectives showing the existence of multiple features of underlying systems

transition records of stationary markov chains in any markov chain with finite state space the distribution of transition records always belongs to the exponential family this observation is used to prove a fluctuation theorem and to show that the dynamical entropy of a stationary markov chain is linear in the number of steps three applications are discussed a known result about entropy production is reproduced a thermodynamic relation is derived for equilibrium systems with metropolis dynamics finally a link is made with recent results concerning a one dimensional polymer model

hurst kolmogorov dynamics as a result of extremal entropy production abstract it is demonstrated that extremization of entropy production of stochastic representations of natural systems performed at asymptotic times zero or infinity results in constant derivative of entropy in logarithmic time and in turn in hurst kolmogorov processes the constraints used include preservation of the mean variance and lag 1 autocovariance at the observation time step and an inequality relationship between conditional and unconditional entropy production which is necessary to enable physical consistency an example with real world data illustrates the plausibility of the findings

fluctuation theorem for stochastic systems the fluctuation theorem describes the probability ratio of observing trajectories that satisfy or violate the second law of thermodynamics it has been proved in a number of different ways for thermostatted deterministic nonequilibrium systems in the present paper we show that the fluctuation theorem is also valid for a class of stochastic nonequilibrium systems the theorem is therefore not reliant on the reversibility or the determinism of the underlying dynamics numerical tests verify the theoretical result

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

on the work free energy relations in a driven isothermal system a historical and stochastic perspective equality relationships connecting the work performed in a non equilibrium driven process of an isothermal system with the difference of equilibrium thermodynamic potentials have been established by various authors since jarzynski s work of 1997 mostly based on a classical phase space description quantum considerations have been limited and stochastic derivations have been mainly ad hoc however such processes are better described by an ensemble in the a space for fluctuating state or mesoscopic variables going back to the probabilistic results of boltzmann and einstein and employing a few basic properties of generally non stationary markov processes these relationships follow almost naturally providing the initial and final states are in contact with the heat bath the essential point is that in the exponentiated work exp w the non equilibrium entropy functions of both initial and final states cancel out the results obtained are valid for any particle number n we note that the connection with the microscopic quantum statistical evolution must be considered in order to proceed in addition some aspects of irreversibility are briefly being discussed

universal and nonuniversal allometric scaling behaviors in the visibility graphs of world stock market indices the investigations of financial markets from a complex network perspective have unveiled many phenomenological properties in which the majority of these studies map the financial markets into one complex network in this work we investigate 30 world stock market indices through their visibility graphs by adopting the visibility algorithm to convert each single stock index into one visibility graph a universal allometric scaling law is uncovered in the minimal spanning trees whose scaling exponent is independent of the stock market and the length of the stock index in contrast the maximal spanning trees and the random spanning trees do not exhibit universal allometric scaling behaviors there are marked discrepancies in the allometric scaling behaviors between the stock indices and the brownian motions using surrogate time series we find that these discrepancies are caused by the fat tailedness of the return distribution and the nonlinear long term correlation

symbolic approach for measuring temporal irreversibility we describe a symbolic approach for measuring temporal irreversibility in time series measurements temporal irreversibility is important because it excludes gaussian linear dynamics and static transformations of such dynamics from the set of possible generating processes a symbolic method for measuring temporal irreversibility is attractive because it is computationally efficient robust to noise and simplifies statistical analysis of confidence limits we propose a specific algorithm called false flipped symbols for establishing the presence of temporal irreversibility without the need for generating surrogate data besides characterizing experimental data our results are relevant to the question of selecting alternative models we illustrate our points with numerical model output and experimental measurements

natural boundaries for the smoluchowski equation and affiliated diffusion processes the schrodinger problem of deducing the microscopic dynamics from the input output statistics data is known to admit a solution in terms of markov diffusion processes the uniqueness of the solution is found to be linked to the natural boundaries respected by the underlying random motion by choosing a reference smoluchowski diffusion process we automatically fix the feynman kac potential and the field of local accelerations it induces we generate the family of affiliated diffusion processes with the same local dynamics but different inaccessible boundaries on finite semi infinite and infinite domains for each diffusion process a unique feynman kac kernel is obtained by the constrained dirichlet boundary data wiener path integration as a by product of the discussion we give an overview of the problem of inaccessible boundaries for the diffusion and bring together sometimes viewed from unexpected angles results which are little known and dispersed in publications from scarcely communicating areas of mathematics and physics

testing for nonlinearity in time series the method of surrogate data we describe a statistical approach for identifying nonlinearity in time series the method first specifies some linear process as a null hypothesis then generates surrogate data sets which are consistent with this null hypothesis and finally computes a discriminating statistic for the original and for each of the surrogate data sets if the value computed for the original data is significantly different than the ensemble of values computed for the surrogate data then the null hypothesis is rejected and nonlinearity is detected we discuss various null hypotheses and discriminating statistics the method is demonstrated for numerical data generated by known chaotic systems and applied to a number of experimental time series which arise in the measurement of superfluids brain waves and sunspots we evaluate the statistical significance of the evidence for nonlinear structure in each case and illustrate aspects of the data which this approach identifies

surrogate time series before we apply nonlinear techniques e g those inspired by chaos theory to dynamical phenomena occurring in nature it is necessary to first ask if the use of such advanced techniques is justified by the data while many processes in nature seem very unlikely a priori to be linear the possible nonlinear nature might not be evident in specific aspects of their dynamics the method of surrogate data has become a very popular tool to address such a question however while it was meant to provide a statistically rigorous foolproof framework some limitations and caveats have shown up in its practical use in this paper recent efforts to understand the caveats avoid the pitfalls and to overcome some of the limitations are reviewed and augmented by new material in particular we will discuss specific as well as more general approaches to constrained randomisation providing a full range of examples new algorithms will be introduced for unevenly sampled and multivariate data and for surrogate spike trains the main limitation which lies in the interpretability of the test results will be illustrated through instructive case studies we will also discuss some implementational aspects of the realisation of these methods in the tisean software package

the transient fluctuation theorem of sample entropy production for general stochastic processes the transient fluctuation theorem for stochastic processes was first put forward by d j searles and d j evans 1999 in the present paper it is rigorously proved that the transient fluctuation theorem tft of sample entropy production which is previously defined by jiang et al 2003 and reid et al 2005 and also called the general action functional up to boundary terms by lebowitz and spohn 1999 holds for general stochastic processes without the assumption of markovian homogeneous or stationary properties then the condition of our theorem is verified for various stochastic processes including homogeneous inhomogeneous markov chains and general diffusion processes among these cases the transient fluctuation theorems for inhomogeneous markov chains and general diffusion processes are rigorously derived for the first time

improved surrogate data for nonlinearity tests current tests for nonlinearity compare a time series to the null hypothesis of a gaussian linear stochastic process for this restricted null assumption random surrogates can be constructed which are constrained by the linear properties of the data we propose a more general null hypothesis allowing for nonlinear rescalings of a gaussian linear process we show that such rescalings cannot be accounted for by a simple amplitude adjustment of the surrogates which leads to spurious detection of nonlinearity an iterative algorithm is proposed to make appropriate surrogates which have the same autocorrelations as the data and the same probability distribution

chaotic hypothesis onsager reciprocity and fluctuation dissipation theorem it is shown that the chaoticity hypothesis recently introduced in statistical mechanics which is analogous to ruelle s principle for turbulence implies the onsager reciprocity and the fluctuation dissipation theorem in various reversible models for coexisting transport phenomena

entropy production and information gain in axiom a systems anosov systems are mathematical models for chaotic systems in statistical mechanics and fluid dynamics most of these systems enjoy the property of positive entropy production we introduce the concept of specific information gain or specific relative entropy h in anosov systems and prove that it is identical to the entropy production rate e p defined by ruelle and gallavotti in anosov systems from this point of view the entropy production rate e p characterizes the degree of macroscopic irreversibility of the system

the uniformization theorem for compact kahler manifolds of nonnegative holomorphic bisectional curvature soit x h une variete de kahler compacte a n dimensions de courbure bisectionnelle holomorphe non negative et soit x h son espace de recouvrement universel alors il existe des entiers non negatifs k n 1 n l et des espaces symetriques hermitiens compacts ineductibles m 1 m k de rang 2 tels que x h est isometriquement bi holomorphe a c k g 0 p nu 1 1 p nu l l m 1 g 1 m p g p ou g 0 est la metrique euclidienne sur c k g 1 g p sont des metriques canoniques sur m 1 m p et i 1 i l est une metrique de kahler sur p nu i portant une courbure bisectionnelle holomorphe non negative

on the definition of entropy production via examples we present a definition of entropy production rate for classes of deterministic and stochastic dynamics the point of departure is a gibbsian representation of the steady state path space measure for which the density is determined with respect to the time reversed process the gibbs formalism is used as a unifying algorithm capable of incorporating basic properties of entropy production in nonequilibrium systems our definition is motivated by recent work on the gallavotti cohen local fluctuation theorem and it is illustrated via a number of examples