diffusion on self avoiding walk networks two properties of self avoiding walk networks constructed by connecting the nearest neighbour points of self avoiding walks on the square lattice are studied numerically the average end to end linear part of such networks is investigated and is found to be proportional to the length of the walk secondly random walks on this structure are studied and the associated spectral dimension of such networks is estimated to be 1 03 or 0 03 which implies that the effect of such connections affects the spectral dimension only weakly if at all

robust methods for model order estimation model order estimation is a subject in time series analysis that deals with fitting a parametric model to a vector of observations this paper focuses on several model order estimators known in the literature and examines their performance under small deviations of the probability distribution of the noise with respect to a nominal distribution assumed in the model it is demonstrated that the standard estimators suffer from high sensitivity to deviations from the nominal distribution and a drastic performance degradation is experienced to overcome this problem robust estimators that are insensitive to small deviations from the nominal distribution are developed these estimators are based on a composition between model order estimation methods and robust statistical inference techniques known in the literature in addition a new estimator based on a locally best test for weak signals is presented both in nonrobust and robust versions the proposed robust model order estimators are developed on a heuristic basis and there is no claim of optimality but experimental results indicate that they provide significant improvement over the standard estimators

the estimation of the model order in exponential families estimators are sought that achieve high exponential rate of decrease in the underestimation probability while keeping the overestimation probability exponent at a certain prescribed level it is assumed that a given integer is known to upper bound the true order some examples are given of possible applications of the proposed method in specific order estimation and hypothesis testing problems

trapping aspects in enhanced diffusion we study the superlinear diffusion x2 t t 1 in layered media containing random velocity fields the superlinear behavior holds in the case of random velocities along thex direction accompanied by diffusional motion in the space transverse to it the transverse space can be either euclidean fractal or ultrametric for a one dimensional transverse space we derive exact expressions for the higher moments of the displacement furthermore we investigate the propagatorp x t along thex direction and establish its scaling behavior our analysis highlights the resemblance between the stretched gaussian behavior of the propagator and the stretched exponential form of the survivial probability in the trapping problem both show late crossover behavior

remarks on the possible universal mechanism of the non linear long term autocorrelations in financial time series the paper consists of two parts i the empirical one where the non linear long term autocorrelations present in high frequency data extracting from the warsaw stock exchange were analyzed and ii theoretical one where predictions of our model quantitative finance 3 2003 201 physica a 2003 chem phys 284 2002 481 phys comm 147 2002 565 physica a 264 1999 84 physica a 264 1999 107 lecture notes in computer science 2657 2003 407 eur phys j b 33 2003 495 were shown and discussed this model introduces the possibility that the weierstrass hierarchical random walk can be occasionally intermitted by momentary localizations the localizations themselves are again described by the weierstrass process in other words this combined walk is a kind of the non separable generalized continuous time random walk formalism to adapt the model to the description of empirical data recorded at time horizon t 1min we applied a discretization procedure into the continuous time series produced by the model we observed that such a procedure generates the non linear long term autocorrelations even in the gaussian regime as turning points of the random walk trajectory are most often incommensurable with discretization time step these autocorrelations appear to be similar to those observed in the financial time series physica a 287 2000 396 physica a 299 2001 1 physica a 299 2001 16 physica a 299 2001 28 although single steps of the walker within continuous time are by definition uncorrelated our approach suggests a surprising origin of the non linear long term autocorrelations alternative to the one proposed very recently cf phys rev e 67 2003 021112 and refs therein although both approaches involve related variants of the well known ctrw formalism applied in yet many different branches of knowledge phys rep 158 1987 263 phys rep 195 1990 127 in a bunde s havlin eds fractals in science springer berlin 1995 pp 119 161

some properties of a random walk on a comb structure abstract we analyze transport properties of a random walk on a comb structure which serves as a model for a random walk on the backbone of a percolation cluster it is shown that the random walk along the x axis which is the analog of the backbone exhibits anomalous diffusion in that x 2 n n 1 2 and the expected number of x sites visited is proportional to n 1 4 for large n the distribution function is found to be a two dimensional gaussian if a field in the x direction so that diffusion is asymmetric the expected displacement is found to be asymptotically proportional to n 1 2

learning high dimensional markov forest distributions analysis of error rates the problem of learning forest structured discrete graphical models from i i d samples is considered an algorithm based on pruning of the chow liu tree through adaptive thresholding is proposed it is shown that this algorithm is both structurally consistent and risk consistent and the error probability of structure learning decays faster than any polynomial in the number of samples under fixed model size for the high dimensional scenario where the size of the model d and the number of edges k scale with the number of samples n sufficient conditions on n d k are given for the algorithm to satisfy structural and risk consistencies in addition the extremal structures for learning are identified we prove that the independent resp tree model is the hardest resp easiest to learn using the proposed algorithm in terms of error rates for structure learning

detection of signals by information theoretic criteria a new approach is presented to the problem of detecting the number of signals in a multichannel time series based on the application of the information theoretic criteria for model selection introduced by akaike aic and by schwartz and rissanen mdl unlike the conventional hypothesis testing based approach the new approach does not requite any subjective threshold settings the number of signals is obtained merely by minimizing the aic or the mdl criteria simulation results that illustrate the performance of the new method for the detection of the number of signals received by a sensor array are presented

functionals of brownian motion localization and metric graphs we review several results related to the problem of a quantum particle in a random environment in an introductory part we recall how several functionals of brownian motion arise in the study of electronic transport in weakly disordered metals weak localization two aspects of the physics of the one dimensional strong localization are reviewed some properties of the scattering by a random potential time delay distribution and a study of the spectrum of a random potential on a bounded domain the extreme value statistics of the eigenvalues then we mention several results concerning the diffusion on graphs and more generally the spectral properties of the schrodinger operator on graphs the interest of spectral determinants as generating functions characterizing the diffusion on graphs is illustrated finally we consider a two dimensional model of a charged particle coupled to the random magnetic field due to magnetic vortices we recall the connection between spectral properties of this model and winding functionals of planar brownian motion

asymptotically optimal classification for multiple tests with empirically observed statistics the decision problem of testing m hypotheses when the source is kth order markov and there are m or fewer training sequences of length n and a single test sequence of length n is considered k m n n are all given it is shown what the requirements are on m n n to achieve vanishing exponential error probabilities and how to determine or bound the exponent a likelihood ratio test that is allowed to produce a no match decision is shown to provide asymptotically optimal error probabilities and minimum no match decisions as an important serial case the binary hypotheses problem without rejection is discussed it is shown that for this configuration only one training sequence is needed to achieve an asymptotically optimal test

non gaussian random walks the authors present an explicit expression for the probability distribution for the position of a continuous time random walker in an arbitrary number of dimensions when the interjump density has a long time tail in contrast to earlier results which require numerical inversion of a fourier integral they replace this numerical procedure by one that relies on the method of steepest descents their results are applied to diffusion on a comb and on a percolation cluster generated on a cayley tree at criticality and are confirmed numerically

review paper efficient numerical methods for the solution of stiff initial value problems and differential algebraic equations in recent years after a prolonged period of intense activity the study of numerical methods for solving stiff initial value problems for ordinary differential equations and differential algebraic equations has reached a certain maturity there now exist some excellent codes which are both efficient and reliable for solving these particular classes of problems in this paper we sketch some of the main theory which underpins stiff integration methods and we use this to describe and put into context some of the best codes currently available by referencing only codes which have been thoroughly tested and are widely available our aim is to direct users of numerical software to those codes which they should try initially if faced with the problem of solving ordinary differential equations of this type an additional feature is that the codes which we propose serve as benchmarks against which any new codes can be evaluated

possible origin of the non linear long term autocorrelations within the gaussian regime abstract in this work we extend the recently considered toy model of weierstrass or levy walks with varying velocity of the walker quantitative finance 3 2003 201 chem phys 284 2002 481 comp phys comm 147 2002 565 phys a 264 1999 84 phys a 264 1999 107 by introducing a more realistic possibility that the walk can be occasionally intermitted by its momentary localization the localizations themselves are again described by the weierstrass or levy process the direct empirical motivation for developing this combined model is for example the dynamics of financial high frequency time series or hydrological and even meteorological ones where variations of the index are randomly intermitted by flat intervals of different length exhibiting no changes in the activity of the system this combined weierstrass walks was developed in the framework of the non separable generalized continuous time random walk formalism developed very recently lecture notes comput sci 2657 2003 407 eur phys j b 33 2003 495 this non markovian two state walking localization model makes possible to cover by the unified treatment a broad band of known up to now types of non biased diffusion from the dispersive one over the normal enhanced ballistic and hyperdiffusion up to the richardson law of diffusion which defines here a part of the borderline which separates the latter from the levy ocean where the total mean square displacement of the walker diverges we observed that anomalous diffusion is characterized here by three fractional exponents one temporal characterizing the localized state and two temporal and spatial characterizing the walking one by considering successive dynamic even exponents we constructed a series of different diffusion phase diagrams on the plane defined by the temporal and spatial partial fractional dynamic exponents characterizing the walking state to adapt the model to the description of empirical data the discrete time series which are collected with a discrete time step we used in the continuous time series produced by the model a discretization procedure we observed that such a procedure generates in general long range non linear autocorrelations even in the gaussian regime which appear to be similar to those observed e g in the financial time series phys a 287 2000 396 phys a 299 2001 1 phys a 299 2001 16 phys a 299 2001 16 although single steps of the walker within continuous time are by definition uncorrelated this suggests a suprising origin of long range non linear autocorrelations alternative to the one proposed very recently cf mosaliver et al phys rev e 67 2003 021112 and refs therein although both approaches involve related variants of the well known continuous time random walk formalism applied yet in many different branches of knowledge phys rep 158 1987 263 phys rep 195 1990 127 in a bunde s havlin eds fractals in science springer berlin 1995 p 1

scale invariant motion in intermittent chaotic systems we investigate the dynamics generated from iterated maps and analyze the motion in terms of the probabilistic continuous time random walk ctrw approach two different ctrw models are considered i particles jump between sites turning points or ii particles move at a constant velocity between sites and choose a new direction at random for both models we study the mean squared displacement mathit r 2 t and the propagator p r t the probability to be at location r at time t having started at the origin at t 0 iterated maps are used to generate both dispersive and enhanced diffusion and the results are analyzed using the ctrw framework and scaling arguments for the case of dispersive motion we discuss the problem of the stationary state and point out its relevance

hierarchical spatio temporal coupling in fractional wanderings i continuous time weierstrass flights abstract the one dimensional continuous time weierstrass flights ctwf model is considered in the framework of the nonseparable continuous time random walks formalism ctrw a novel spatio temporal coupling is introduced by assuming that in each scale the probability density for the flight and for waiting are joined hence we treat the spatio temporal relations in terms of the self similar structure of the weierstrass process this stochastic structure is characterized by the spatial fractional dimension 1 representing the flights and the temporal one 1 representing the waiting time was assumed here as the only independent truncation range in this work we study the asymptotic properties of the ctwf model for example by applying the method of steepest descents we obtained the particle propagator in the approximate scaling form p x t t pt 2 f where the scalingfunction f pt exp const pt while the scaling variable x t pt 2 is large the principal result of our analysis is that the exponents and depend on more fundamental ones and what leads to a novel scaling as a result of competition between exponents and an enhanced dispersive or normal diffusion was recognized in distinction from the prediction of the separable ctrw model where the enhanced diffusion is lost and the dispersive one is strongly limited it should be noted that we compare here partially thermalized versions of both approaches where some initial fluctuations were also included in agreement with the spirit of the theory of the renewal processes having the propogator we calculated for example the particle mean square displacement and found its novel asymptotic scaling with time for enhanced diffusion given by t1 2 1 in distinction from its diverging for

numerical methods for stiff two point boundary value problems we consider the two point boundary value problem for stiff systems of ordinary differential equations for systems that can be transformed to essentially diagonally dominant form with appropriate smoothness conditions a priori estimates are obtained problems with turning points can be treated with this theory and we discuss this in detail we give robust difference approximations and present error estimates for these schemes in particular we give a detailed description of how to transform a general system to essentially diagonally dominant form and then stretch the independent variable so that the system will satisfy the correct smoothness conditions numerical examples are presented for both linear and nonlinear problems

an efficient approach for solving stiff nonlinear boundary value problems abstract a new method for solving stiff two point boundary value problems is described and compared to other known approaches using the troesch s problem as a test example the method is based on the general idea of alternate approximation of either the unknown function or its inverse and has a genuine immunity towards numerical difficulties invoked by the rapid variation stiffness of the unknown solution a c implementation of the proposed method is available at https github com imathsoft mathsoftdevelopment

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

on the estimation of the order of a markov chain and universal data compression the authors estimate the order of a finite markov source based on empirically observed statistics the performance criterion adopted is to minimize the probability of underestimating the model order while keeping the overestimation probability exponent at a prescribed level a universal asymptotically optimal test in the sense just defined is proposed for the case where a given integer is known to be the upper bound of the true order for the case where such a bound is unavailable an alternative rule based on the lempel ziv data compression algorithm is shown to be asymptotically optimal also and computationally more efficient

continuous time random walks and the fractional diffusion equation the average probability density p r t of random walks on fractals is revisited within the continuous time random walks formalism corrections to the accepted asymptotic stretched gaussian decay of p r t of the form ralpha are discussed it is shown that p r t obeys a diffusion equation with a fractional time derivative asymptotically and predictions about the value of alpha are presented

a nonlinear shooting method for two point boundary value problems abstract we study a new nonlinear shooting method for solving two point boundary value problems and show numerical experiments with various initial velocity conditions we discuss and analyze the numerical solutions which are obtained by the shooting method

the determination of optimum structures for the state space representation of multivariate stochastic processes when identifying a model for a multivariate stationary stochastic process an important problem is that of determining the structure of the state variable model several overlapping parameterizations can usually be fitted to a given process and the question arises as to which structure leads to the most accurate parameter estimates the accuracy of parameter estimates is often measured by the determinant of the fisher information matrix we show that all admissible structures will give asymptotically the same value to this criterion for finite data some structures may still be better than others and two heuristic structure estimation methods are analyzed some simulation results are also presented

generalized phase space projection for nonlinear noise reduction improved phase space projection methods adapted from related work in the linear signal processing field based on subspace decomposition are presented for application to the problem of additive noise reduction in the context of phase space analysis these methods improve upon existing methods such as broomhead king singular spectrum analysis projection by minimizing overall signal distortion subject to constraints on the residual error rather than using a direct least squares fit this results in a range of weighted projections which estimate and compensate for the portion of the principal component s singular values corresponding to noise rather than signal energy and which include least squares ls and least minimum mean square error lmmse as subcases the nature of phase space covariance the key element in construction of the projection matrix is examined across global phase spaces as well as within local neighborhood regions the resulting algorithm illustrated on a noisy henon map as well as on the task of speech enhancement is applicable to a wide variety of nonlinear noise reduction tasks

stochastic multiplicative processes for financial markets abstract we study a stochastic multiplicative system composed of finite asynchronous elements to describe the wealth evolution in financial markets we find that the wealth fluctuations or returns of this system can be described by a walk with correlated step sizes obeying truncated levy like distribution and the cross correlation between relative updated wealths is the origin of the nontrivial properties of returns including the power law distribution with exponent outside the stable levy regime and the long range persistence of volatility correlations

unequal twins probability distributions do not determine everything it is the common lore to assume that knowing the equation for the probability distribution function pdf of a stochastic model as a function of time tells the whole picture defining all other characteristics of the model we show that this is not the case by comparing two exactly solvable models of anomalous diffusion due to geometric constraints the comb model and the random walk on a random walk we show that though the two models have exactly the same pdfs they differ in other respects like their first passage time distributions their autocorrelation functions and their aging properties

model selection and the principle of minimum description length this article reviews the principle of minimum description length mdl for problems of model selection by viewing statistical modeling as a means of generating descriptions of observed data the mdl framework discriminates between competing models based on the complexity of each description this approach began with kolmogorov s theory of algorithmic complexity matured in the literature on information theory and has recently received renewed attention within the statistics community here we review both the practical and the theoretical aspects of mdl as a tool for model selection emphasizing the rich connections between information theory and statistics at the boundary between these two disciplines we find many interesting interpretations of popular frequentist and bayesian procedures as we show mdl provides an objective umbrella under which rather disparate approaches to statistical modeling can coexist and be compared we illustrate the mdl principle by considering problems in regression nonpar

financial multifractality and its subtleties an example of dax abstract detailed study of multifractal characteristics of the financial time series of asset values and of its returns is performed using a collection of the high frequency deutsche aktienindex data the tail index the renyi exponents based on the box counting algorithm for the graph d q and the generalized hurst exponents h q are computed in parallel for short and daily return times the results indicate a more complicated nature of the stock market dynamics than just consistent multifractal

universal classification for hidden markov models binary hypotheses testing using empirically observed statistics is studied in the neyman pearson formulation for the hidden markov model hmm an asymptotically optimal decision rule is proposed and compared to the generalized likelihood ratio test glrt which has been shown in earlier studies to be asymptotically optimal for simpler parametric families the proof of the main theorem is provided the result can be applied to several types of hmms commonly used in speech recognition and communication applications several applications are demonstrated

levy walks for turbulence a numerical study we present a numerical study of enhanced diffusion for which the mean squared displacement follows asymptotically r2 t t 1 we simulate continuous time random walks with waiting time distributions which couple the spatial and temporal parameters this gives rise to levy walks our results confirm the theoretically predicted long time behavior and demonstrate its temporal regime of validity furthermore the simulations document the appearance of parameter dependent transitions between regular and enhanced diffusion regimes

microscopic models for long ranged volatility correlations we propose a general interpretation for long range correlation effects in the activity and volatility of financial markets this interpretation is based on the fact that the choice between active and inactive strategies is subordinated to random walk like processes we numerically demonstrate our scenario in the framework of simplified market models such as the minority game model with an inactive strategy or a more sophisticated version that includes some price dynamics we show that real market data can be surprisingly well accounted for by these simple models

generalized diffusion equation for anisotropic anomalous diffusion motivated by studies of comblike structures we present a generalization of the classical diffusion equation to model anisotropic anomalous diffusion we assume that the diffusive flux is given by a diffusion tensor acting on the gradient of the probability density where each component of the diffusion tensor can have its own scaling law we also assume scaling laws that have an explicit power law dependence on space and time solutions of the proposed generalized diffusion equation are consistent with previously derived asymptotic results for the probability density on comblike structures

hierarchical spatio temporal coupling in fractional wanderings ii diffusion phase diagram for weierstrass walks abstract the one dimensional weierstrass walks ww model is developed in the framework of the extended nonseparable continuous time random walk ctrw formalism 1 2 3 4 5 6 7 8 9 10 11 12 the ww model is a lacunary foundation of levy walks 6 7 8 9 10 11 12 generalized to a nonconstant velocity this nonconstant velocity is introduced by hierarchical coherent spatio temporal coupling adopted from the continuous time weierstrass flights ctwf model developed in the previous paper 13 hence for the probability density to pass by a walker in a single step a random displacement with finite velocity is constructed as a geometric series of the corresponding probability densities defined within a sequence of spatio temporal scales we calculated analytically and by monte carlo simulations the asymptotic in time mean square displacement msd of the walker obtaining very good agreement between both approaches also comparison with corresponding results of the ctwf model is made considering different types of the diffusion exponents we constructed a diffusion phase diagram on the plane defined by the spatial and temporal fractional dimensions which characterize our coupling we obtained a diffusion exponent as a function of these fractional dimensions covering all types of nonbiased diffusion known up to now from the dispersive one over the normal enhanced ballistic and hyperdiffusion up to the richardson law of diffusion which defines here a part of the borderline of the region where the msd diverges we observed that all kinds of anomalous diffusion are characterized by three types of diffusion exponents only for example we found an asymptotic scaling of msd to occur with time for enhanced diffusion which was discovered by us within the ctwf model but is valid for a much more extended range of spatio and temporal fractional dimensions

topological properties of percolation clusters percolation properties including the total cluster mass s the shell mass b and the linear geometrical size r are studied as a function of the topological chemical distance parameter l all critical exponents are shown to be related to an apparently new exponent nu defined by r approximately lv critical exponents are calculated exactly for percolation clusters on the cayley tree a model for 6d percolation for which s approximately l2 b approximately l and r2 approximately l for diffusion on such clusters one finds that l3 approximately t numerical estimates of the exponents are obtained for other dimensions a conjecture which relates nu to beta and nu is discussed

on the localization of random walks and vibrational excitations in linear fractal structures abstract we study analytically and numerically the mean probability density n of random walks and the mean vibrational amplitudes r w n of localized excitations on linear random fractals averaged over n configurations we find that for large distances r both functions are characterized by crossovers rc and r c respectively which increase logarithmically with n for r below the respective crossover points in n apru and n a r do not depend on n and are characterized by the exponents u dw dw 1 and 1 while above the crossover the coefficients ap and a decrease logarithmically with n and the exponents become u dmindw dw dmin and dmin where dw is the fractal dimension of the random walk and dmin is the fractal dimension of the shortest path on the fractal structure we discuss the applicability of these results to general random fractal structures

biased random walk on a biased random walk abstract we consider the random walk of a particle along topologically linear channels under the influence of a uniform drift force the channels are generated by the usual biased random walk procedure the resulting mean and mean square displacements of a particle are discussed

universal coding information prediction and estimation a connection between universal codes and the problems of prediction and statistical estimation is established a known lower bound for the mean length of universal codes is sharpened and generalized and optimum universal codes constructed the bound is defined to give the information in strings relative to the considered class of processes the earlier derived minimum description length criterion for estimation of parameters including their number is given a fundamental information theoretic justification by showing that its estimators achieve the information in the strings it is also shown that one cannot do prediction in gaussian autoregressive moving average arma processes below a bound which is determined by the information in the data

structure of clusters generated by random walks the authors study the cluster structure resulting from a nearest neighbour random walk embedded in a d dimensional space each bond visited by the random walks is regarded as belonging to the cluster the diffusion exponent and the fracton dimensional of the fractal cluster in d 3 is found to be dw 3 5 or 0 1 and d 0 57 or 0 02 using a method of exact enumeration of random walks on these fractals

stochastic complexity and modeling on demontre un theoreme fondamental qui donne une borne inferieure pour la longueur de code et donc pour les erreurs de prediction on definit les notions d information a priori et d information utile dans les donnees

probability densities for the displacement of random walks on percolation clusters the probability density of the displacement or end to end distance of a random walk on the incipient infinite percolation cluster in d 2 dimensions is studied by an exact enumeration method the numerical data suggest specific forms for the probability density both in the chemical distance variable l and the geometric distance r

levy flights in random searches we review the general search problem of how to find randomly located objects that can only be detected in the limited vicinity of a forager and discuss its quantitative description using the theory of random walks we illustrate levy flight foraging by comparison to brownian random walks and discuss experimental observations of levy flights in biological foraging we review recent findings suggesting that an inverse square probability density distribution p l l 2 of step lengths l can lead to optimal searches finally we survey the explanations put forth to account for these unexpected findings

a variational iteration method for solving troesch s problem troesch s problem is an inherently unstable two point boundary value problem a new and efficient algorithm based on the variational iteration method and variable transformation is proposed to solve troesch s problem the underlying idea of the method is to convert the hyperbolic type nonlinearity in the problem into polynomial type nonlinearities by variable transformation and the variational iteration method is then directly used to solve this transformed problem only the second order iterative solution is required to provide a highly accurate analytical solution as compared with those obtained by other analytical and numerical methods

structural and dynamical properties of random walk clusters the authors study the structural and dynamical properties of the clusters generated by a nearest neighbour random walk embedded in a d dimensional space they have focused on the non trivial case in which the cluster is generated in d 3 the structure of this cluster is characterised by loops for all length scales on the one hand and by the fact that deadends are negligible upon scaling on the other hand the cluster is very dilute and is characterised by fractal dimension df 2 and chemical dimension d1 1 29 or 0 04 from these results it follows that v identical to dldf approximately 2 3 which is consistent with the formula v 2 d 2 or d or 4 obtained using a flory type argument the dynamical diffusion exponents dw and dlw were calculated using the exact enumeration method and found to be dw 3 45 or 0 10 and dlw 2 28 or 0 05 the results suggest that the effect of loops is small but not negligible they also calculated the fracton dimensionality of the cluster and obtained ds 1 14 or 0 02 a scaling function is presented for the end to end mean square displacement of a random walk performed on a random walk cluster this scaling function is supported by their numerical results

the first 20 min in the hong kong stock market based on the minute by minute data of the hang seng index in hong kong and the analysis of probability distribution and autocorrelations we nd that the index uctuations for the rst few minutes of daily opening show behaviors very dierent from those of the other times in particular the properties of tail distribution which will show the power law scaling with exponent about four or an exponential type decay the volatility and its correlations depend on the opening eect of each trading day c 2000 elsevier science b v all rights reserved

on strong consistency of model selection in classification this paper considers model selection in classification in many applications such as pattern recognition probabilistic inference using a bayesian network prediction of the next in a sequence based on a markov chain the conditional probability p y y x x of class yisiny given attribute value xisinx is utilized by model we mean the equivalence relation in x for x x isinxx x harrp y y x x p y y x x forall yisiny by classification we mean the number of such equivalence classes is finite we estimate the model from n samples zn xi yi i 1 n isin xtimesy n using information criteria in the form empirical entropy h plus penalty term k 2 dn the model such that h k 2 dn is minimized is the estimated model where k is the number of independent parameters in the model and dn n 1 infin is a real nonnegative sequence such that lim supndn n 0 for autoregressive processes although the definitions of h and k are different it is known that the estimated model almost surely coincides with the true model as nrarrinfin if dn n 1 infin 2loglogn n 1 infin and that it does not if dn n 1 infin 2loglogn n 1 infin hannan and quinn the problem whether the same property is true for classification was open this paper solves the problem in the affirmative

continuous time random walk model for financial distributions we apply the formalism of the continuous time random walk to the study of financial data the entire distribution of prices can be obtained once two auxiliary densities are known these are the probability densities for the pausing time between successive jumps and the corresponding probability density for the magnitude of a jump we have applied the formalism to data on the u s dollar deutsche mark future exchange finding good agreement between theory and the observed data

how to quantify deterministic and random influences on the statistics of the foreign exchange market it is shown that price changes of the u s dollar german mark exchange rates upon different delay times can be regarded as a stochastic marcovian process furthermore we show how kramers moyal coefficients can be estimated from the empirical data finally we present an explicit fokker planck equation which models very precisely the empirical probability distributions in particular their non gaussian heavy tails

branching rules for e8 so16 branching rules for e8 down arrow so16 are derived from those of e8 down arrow su2 e7 down arrow su8 and so16 down arrow u1 su8 by noting that both group chains share a common u1 su8 subgroup additional branching rules are deduced from kronecker products of e8 and so16 irreps in each case the rules distinguish unambiguously between conjugate irreps of so16 an alternative labelling scheme for the irreps of so2k based on its maximal u1 suk subgroup is outlined

scaling and correlation in financial time series we discuss the results of three recent phenomenological studies focussed on understanding the distinctive statistical properties of nancial time series i the probability distribution of stock price uctuations stock price uctuations occur in all magnitudes in analogy to earthquakes from tiny uctuations to very drastic events such as the crash of 19 october 1987 sometimes referred to as black monday 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 8 orders of magnitude in addition this distribution preserves its functional form for uctuations on time scales that dier by 3 orders of magnitude from 1 min up to approximately 10 days 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 persisting for several months iii volatility and trading activity we quantify the relation between trading activity measured by the number of transactions nt and the price change gt for a given stock over a time interval t t t we nd that nt displays long range power law correlations in time which leads to the interpretation that the long range correlations previously found for jgtj are connected to those of nt c 2000 elsevier science b v all rights reserved

error exponents for ar order testing this paper is concerned with error exponents in testing problems raised by autoregressive ar modeling the tests to be considered are variants of generalized likelihood ratio testing corresponding to traditional approaches to autoregressive moving average arma modeling estimation in several related problems such as markov order or hidden markov model order estimation optimal error exponents have been determined thanks to large deviations theory ar order testing is specially challenging since the natural tests rely on quadratic forms of gaussian processes in sharp contrast with empirical measures of markov chains the large deviation principles ldps satisfied by gaussian quadratic forms do not always admit an information theoretic representation despite this impediment we prove the existence of nontrivial error exponents for gaussian ar order testing and furthermore we exhibit situations where the exponents are optimal these results are obtained by showing that the log likelihood process indexed by ar models of a given order satisfy an ldp upper bound with a weakened information theoretic representation

selecting the order of autoregressive models from small samples the weak parameter criterion wpc is introduced as a means for model order selection it is based on the same principles as mallows c p and the fpe and aic criteria of akaike according to the wpc parameters are weak and should be removed if the squares of their estimates are less than twice the expectation for a white noise signal roughly speaking the square of an estimate must exceed twice its variance due to the conceptual simplicity this criterion remains useful for small samples where the asymptotical properties are no longer valid if the maximum order considered is sqrt n or less the difference between akaike s fpe and aic criteria on one hand and the wpc on the other hand remains small and the use of aic or fpe may be justified however it is advised to use the wpc for higher maximum orders by using different variances in the wpc for yule walker and for burg estimates it is achieved that the average selected wpc order in small samples depends mainly on the given data and no longer on the method of parameter estimation

random walk on a random walk the authors investigate the random walk of a particle on a one dimensional chain which has been constructed by a random walk procedure exact expressions are given for the mean square displacement and the fourth moment after n steps the probability density after n steps is derived in the saddle point approximation for large n these quantities have also been studied by numerical simulation the extension to continuous time has been made where the particle jumps according to a poisson process the exact solution for the self correlation function has been obtained in the fourier and laplace domain the resulting frequency dependent diffusion coefficient and incoherent dynamical structure factor have been discussed the model of random walk on a random walk is applied to self diffusion in the concentrated one dimensional lattice gas where the correct asymptotic behavior is found

a continuous time generalization of the persistent random walk abstract we develop the formalism for a continuous time generalization of the persistent random walk by allowing the sojourn time to deviate from the exponential form found in standard discussions of this subject this generalization leads to evolution equations in the time domain that differ and are of higher order than the telegrapher s equation which is found in the case of the markovian persistent random walk

a numerical algorithm for some singularly perturbed boundary value problems a numerical algorithm is proposed to solve singularly perturbed linear two point value problems the method starts with a partial decoupling of the system to obtain two independent subsystems fast and slow components each subsystem is then solved separately a second order finite difference scheme is used for this purpose numerical examples will be presented to show the efficiency of the method

similarities and differences between physics and economics in this opening talk we discuss some of the similarities between work being done by economists and by physicists seeking to contribute to economics we also mention some of the dierences in the approaches taken and justify these dierent approaches by developing the argument that by approaching the same problem from dierent points of view new results might emerge in particular we review some recent results for example the 1nding that there are two new universal scaling models in economics i the 3uctuation of price changes of any stock market is characterized by a pdf which is a simple power law with exponent 4 that extends over 10 2 standard deviations a factor of 10 8 on the y axis ii for a wide range of economic organizations the histogram that shows how size of organization is inversely correlated to 3uc tuations in size with an exponent 1 6 neither of these two new laws has a 1rm theoretical foundation we also discuss results that are reminiscent of phase transitions in spin systems where the divergent behavior of the response function at the critical point zero magnetic 1eld leads to large 3uctuations c 2001 published by elsevier science b v

a random matrix theory approach to financial cross correlations it is common knowledge that any two rms in the economy are correlated even rms belonging to dierent sectors of an industry may be correlated because of indirect correlations how can we analyze and understand these correlations this article reviews recent results regarding cross correlations between stocks specically we use methods of random matrix theory rmt which originated from the need to understand the interactions between the constituent elements of complex interacting systems to analyze the cross correlation matrix c of returns we analyze 30 min returns of the largest 1000 us stocks for the 2 year period 1994 1995 we nd that the statistics of approximately 20 of the largest eigenvalues 2 show deviations from the predictions of rmt to test that the rest of the eigenvalues are genuinely random we test for universal properties such as eigenvalue spacings and eigenvalue correlations and demonstrate that c shares universal properties with the gaussian orthogonal ensemble of random matrices the statistics of the eigenvectors of c conrm the deviations of the largest few eigenvalues from the rmt prediction we also nd that these deviating eigenvectors are stable in time in addition we quantify the number of rms that participate signicantly to an eigenvector using the concept of inverse participation ratio borrowed from localization theory c 2000 published by elsevier science b v all rights reserved

numerical solution of troesch s problem by simple shooting method this paper describes a simple and efficient approach to the troesch s problem in this approach the hyperbolic nonlinear term in the equation is first converted into polynomial nonlinear terms by variable transformation and a simple shooting method is then used directly to solve this transformed problem the calculated results are in excellent agreement with those obtained by other analytical and numerical methods

transport properties on a random comb we study the random walk of a particle in a random comb structure both in the presence of a biasing field and an the field free case we show that the mean field treatment of the quenched disorder can be exactly mapped on to a continuous time random walk ctrw on the backbone of the comb with a definite waiting time density we find an exact expression for this central quantity the green function for the ctrw is then obtained its first and second moments determine the drift and diffusion at all times we show that the drift velocity v vanishes asymptotically for power law and stretched exponential distributions of branch lengths on the comb whatever be the biasing field strength for an exponential branch length distribution v is a nonmonotonic function of the bias increasing initially to a maximum and then decreasing to zero at a critical value in the field free case anomalous diffusion occurs for a range of power law distributions of the branch length the corresponding exponent for the mean square displacement is obtained as is the asymptotic form of the positional probability distribution for the random walk we show that normal diffusion occurs whenever the mean branch length is finite and present a simple formula for the effective diffusion constant these results are extended to regular nonrandom combs as well the physical reason for anomalous drift or diffusion is traced to the properties of the distribution of a first passage time on a finite chain that controls the effective waiting time density of the ctrw

fractional diffusion and levy stable processes anomalous diffusion in which the mean square distance between diffusing quantities increases faster than linearly in time has been observed in all manner of physical and biological systems from macroscopic surface growth to dna sequences herein we relate the cause of this nondiffusive behavior to the statistical properties of an underlying process using an exact statistical model this model is a simple two state process with long time correlations and is shown to produce a random walk described by an exact fractional diffusion equation fractional diffusion equations describe anomalous transport and are shown to have exact solutions in terms of fox functions including l evy ensuremath alpha stable processes in the superdiffusive domain 1 21

order estimation for a special class of hidden markov sources and binary renewal processes we consider the estimation of the order i e the number of hidden states of a special class of discrete time finite alphabet hidden markov sources this class can be characterized in terms of equivalent renewal processes no a priori bound is assumed on the maximum permissible order an order estimator based on renewal types is constructed and is shown to be strongly consistent by computing the precise asymptotics of the probability of estimation error the probability of underestimation of the true order decays exponentially in the number of observations while the probability of overestimation goes to zero sufficiently fast it is further shown that this estimator has the best possible error exponent in a large class of estimators our results are also valid for the general class of binary independent renewal processes with finite mean renewal times

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

quantum oscillations in mesoscopic rings and anomalous diffusion we consider the weak localization correction to the conductance of a ring connected to a network we analyse the harmonics content of the al tshuler aronov spivak aas oscillations and we show that the presence of wires connected to the ring is responsible for a behaviour different from the one predicted by aas the physical origin of this behaviour is the anomalous diffusion of brownian trajectories around the ring due to the diffusion in the wires we show that this problem is related to the anomalous diffusion along the skeleton of a comb we study in detail the winding properties of brownian curves around a ring connected to an arbitrary network our analysis is based on the spectral determinant and on the introduction of an effective perimeter probing the different time scales a general expression of this length is derived for arbitrary networks more specifically we consider the case of a ring connected to wires to a square network and to a bethe lattice

cluster validation for unsupervised stochastic model based image segmentation image segmentation is an important and early processing stage in many image analysis problems often this must be done in an unsupervised fashion in that training data is not available and the class conditioned feature vectors must be estimated directly from the data a major problem in such applications is the determination of the number of classes actually present in an image this problem called the cluster validation problem remains essentially unsolved we investigate the cluster validation problem associated with the use of a previously developed unsupervised segmentation algorithm based upon the expectation maximization em algorithm more specifically we consider several well known information theoretic criteria ics as candidate solutions to the validation problem when used in conjunction with this em based segmentation scheme we show that these criteria generally provide inappropriate solutions due to the domination of the penalty term by the associated log likelihood function as an alternative we propose a model fitting technique in which the complete data log likelihood functional is modeled as an exponential function in the number of classes acting the estimated number of classes are then determined in a manner similar to finding the rise time of the exponential function this new validation technique is shown to be robust and outperform the ics in our experiments experimental results for both synthetic and real world imagery are detailed

random walks with infinite spatial and temporal moments the continuous time random walk of montroll and weiss has been modified by scher and lax to include a coupled spatial temporal memory we treat novel cases for the random walk and the corresponding generalized master equation when combinations of both spatial and temporal moments of the memory are infinite the asymptotic properties of the probability distribution for being at any lattice site as a function of time and its variance are calculated the resulting behavior includes localized diffusive wavelike and levy s stable laws for the appropriate scaled variable we show that an infinite mean waiting time can lead to long time diffusive behavior while a finite mean waiting time is not sufficient to ensure the same

optimal watermark embedding and detection strategies under limited detection resources an information theoretic approach is proposed to watermark embedding and detection under limited detector resources first the attack free scenario is considered under which asymptotically optimal decision regions in the neyman pearson sense are proposed along with the optimal embedding rule later the case of zero mean independent and identically distributed i i d gaussian covertext distribution is explored with unknown variance under the attack free scenario for this case a lower bound on the exponential decay rate of the false negative probability is proposed it is proven that the optimal embedding and detecting strategy is superior to the customary linear additive embedding strategy in the exponential sense finally these results are extended to the case of memoryless attacks and general worst case attacks optimal decision regions and embedding rules are offered and the worst attack channel is identified

order estimation and sequential universal data compression of a hidden markov source by the method of mixtures we consider first the estimation of the order i e the number of states of a discrete time finite alphabet stationary ergodic hidden markov source hms our estimator uses a description of the observed data in terms of a uniquely decodable code with respect to a mixture distribution obtained by suitably mixing a parametric family of distributions on the observation space this procedure avoids maximum likelihood calculations the order estimator is shown to be strongly consistent with the probability of underestimation decaying exponentially fast in the number n of observations while the probability of overestimation does not exceed cn sup 3 where c is a constant next we present a sequential algorithm for the uniquely decodable universal data compression of the hms which performs an on line estimation of source order followed by arithmetic coding this code asymptotically attains optimum average redundancy

exact correct decoding exponent of the wiretap channel decoder the performance of the achievability scheme for wyner s wiretap channel model is examined from the perspective of the probability of correct decoding pc at the wiretap channel decoder in particular for finite alphabet memoryless channels the exact random coding exponent of pc is derived as a function of the total coding rate r1 and the rate of each subcode r2 two different representations are given for this function and its basic properties are provided we also characterize the region of pairs of rates r1 r2 of full security in the sense of the random coding exponent of pc in other words the region where the exponent of this achievability scheme is the same as that of blind guessing at the eavesdropper side finally an analogous derivation of the correct decoding exponent is outlined for the case of the gaussian channel

stock market context of the levy walks with varying velocity abstract we developed the most general levy walks with varying velocity shorter called the weierstrass walks ww model by which one can describe both stationary and non stationary stochastic time series we considered a non brownian random walk where the walker moves in general with a velocity that assumes a different constant value between the successive turning points i e the velocity is a piecewise constant function this model is a kind of levy walks where we assume a hierarchical self similar in a stochastic sense spatio temporal representation of the main quantities such as waiting time distribution and sojourn probability density which are principal quantities in the continuous time random walk formalism the ww model makes possible to analyze both the structure of the hurst exponent and the power law behavior of kurtosis this structure results from the hierarchical spatio temporal coupling between the walker displacement and the corresponding time of the walks the analysis uses both the fractional diffusion and the super burnett coefficients we constructed the diffusion phase diagram which distinguishes regions occupied by classes of different universality we study only such classes which are characteristic for stationary situations we thus have a model ready for describing the data presented e g in the form of moving averages the operation is often used for stochastic time series especially financial ones the model was inspired by properties of financial time series and tested for empirical data extracted from the warsaw stock exchange since it offers an opportunity to study in an unbiased way several features of stock exchange in its early stage

properties of levy flights on an interval with absorbing boundaries we consider a l evy yer of orderthat starts from a point x0 on an interval 0 l with absorbing boundaries we 7nd a closed form expression for an arbitrary average quantity char acterizing the trajectory of the yer such as mean 7rst passage time average total path length probability to be absorbed by one of the boundaries using fractional di erential equations with a riesz kernel we 7nd exact analytical expressions for these quantities in the continuous limit we 7nd numerically the eigenfunctions and the eigenvalues of these equations we study how the results of monte carlo simulations of the l evy ights with di erent ight length distribu tions converge to the continuous approximations we show that if x0 is placed in the vicinity of absorbing boundaries the average total path length has a minimum near 1 corresponding to the cauchy distribution we discuss the relevance of these results to the problem of biological foraging and transmission of light through cloudy atmosphere c 2001 elsevier science b v all rights reserved

analytical solutions for diffusion on fractal objects a generalization of the euclidean diffusion equation is proposed for diffusion on fractals on the basis of a scaling argument a renormalization group theory for the green s function and numerical tests conjectures on the applicability to natural fractals such as porous media are presented

fractional calculus and continuous time finance ii the waiting time distribution we complement the theory of tick by tick dynamics of financial markets based on a continuous time random walk ctrw model recently proposed by scalas et al 4 and we point out its consistency with the behaviour observed in the waiting time distribution for bund future prices traded at liffe london

probability densities of random walks in random systems abstract we review recent results for the probability distribution of random walkers in random systems where diffusion is anomalous and the mean square displacement scales with time as r 2 t t 2 w d w 2 the random systems are characterized by structural disorder and by random transition rates in general the mean distribution function p r t of the random walkers is a stretched gaussian and scales as log p r t p r 0 r r t u where u d w d w 1 on random fractals the fluctuations of the density distribution p r t for fixed distance r and time t have a broad logarithmic distribution the average moments p q scale in a multifractal way as p q where q q l space the fluctuations of p are narrow and p q p q