measuring the strangeness of strange attractors we study the correlation exponent v introduced recently as a characteristic measure of strange attractors which allows one to distinguish between deterministic chaos and random noise the exponent v is closely related to the fractal dimension and the information dimension but its computation is considerably easier its usefulness in characterizing experimental data which stem from very high dimensional systems is stressed algorithms for extracting v from the time series of a single variable are proposed the relations between the various measures of strange attractors and between them and the lyapunov exponents are discussed it is shown that the conjecture of kaplan and yorke for the dimension gives an upper bound for v various examples of finite and infinite dimensional systems are treated both numerically and analytically

a new method to detect nonlinearity in a time series synthesizing surrogate data using a kolmogorov smirnoff tested hidden markov model abstract a way of statistically testing for nonlinearity in a time series is to employ the method of surrogate data this method often makes use of the fourier transform ft in order to generate the surrogate as various authors have shown this can lead to artefacts in the surrogates and spurious detection of nonlinearity can result this paper documents a new method to synthesize surrogate data using a 1st order hidden markov model hmm combined with a kolmogorov smirnoff test ks test to determine the required resolution of the hmm significance test results for a sinewave henon map and gaussian noise time series are presented it is demonstrated that ks tested hmm surrogates can be successfully used to distinguish between a deterministic and stochastic time series then by applying a simple test for linearity using linear and nonlinear predictors it is possible to determine the nature of the deterministic class and hence conclude whether the system is linear deterministic or nonlinear deterministic furthermore it is demonstrated that the method works for periodic functions too where ft surrogates break down

self organizing dynamics of human erythrocytes under shear stress abstract the characterization of the erythrocytes viscoelastic properties is studied from the perspective of bounded correlated random walk brownian motion based on the assumption that diffractometric data involves both deterministic and stochastic components the photometric readings are obtained by ektacytometry over several millions of shear elongated cells using a home made device called erythrodeformeter the results suggest that the samples from healthy donors are intrinsically unpredictable ordinary brownian motion while when studying beta thalassemic samples these exhibit not only a great sensitivity to initial conditions fractional brownian motion but also chaotic behavior these results could allow us to claim that we have linked nonlinear tools with clinical aspects of the erythrocytes rheological properties

rate of convergence of empirical measures and costs in controlled markov chains and transient optimality the purpose of this paper is two fold first bounds on the rate of convergence of empirical measures in controlled markov chains are obtained under some recurrence conditions these include bounds obtained through large deviations and central limit theorem arguments these results are then applied to optimal control problems bounds on the rate of convergence of the empirical measures that are uniform over different sets of policies are derived resulting in bounds on the rate of convergence of the costs finally new optimal control problems that involve not only average cost criteria but also measures on the transient behavior of the cost namely the rate of convergence are introduced and applied to a problem in telecommunications the solution to these problems rely on the bounds introduced in previous sections

complexity and synchronization in chaotic fiber optic systems abstract in this work we investigate experimentally the complexity of chaotic attractors generated by a semiconductor laser subjected to optical feedback and associate their dimensionality with the synchronization efficiency of the corresponding chaotic transmitter receiver configuration the complexity is characterized by calculating the correlation dimension d 2 of experimental chaotic time series for different values of the optical feedback we present the effect of d 2 on the synchronization efficiency and determine the optimal operating condition that leads to the most complex chaotic carrier and simultaneously to the most successful synchronization lastly we associate and explain our experimental results with theoretical predictions in the research literature

on the hausdorff dimension of fractal attractors we consider such mappingsx n 1 f xn of an interval into itself for which the attractor is a cantor set for the same class of mappings for which the feigenbaum scaling laws hold we show that the hausdorff dimension of the attractor is universally equal tod 0 538

convergence of the k 2 entropy for random noises with power law spectra abstract we discuss a counter example to the traditional expectation that stochastic processes lead to a non convergence of the k 2 entropy in computed or measured time series we consider a simple class of colored random noises with power law spectra which have already been shown to generate finite and predictable values of the correlation dimension we show that the k 2 entropy of these random noises converges as well these results imply that the observation of a finite or null value of the k 2 entropy in the analysis of data is not enough to infer that the system is dominated by a deterministic process such as low dimensional chaos

noisy dynamic simulations in the presence of symmetry data alignment and model reduction abstract we process snapshots of trajectories of evolution equations with intrinsic symmetries and demonstrate the use of recently developed eigenvector based techniques to successfully quotient out the degrees of freedom associated with the symmetries in the presence of noise our illustrative examples include a one dimensional evolutionary partial differential the kuramoto sivashinsky equation with periodic boundary conditions as well as a stochastic simulation of nematic liquid crystals which can be effectively modeled through a nonlinear smoluchowski equation on the surface of a sphere this is a useful first step towards data mining the symmetry adjusted ensemble of snapshots in search of an accurate low dimensional parametrization and the associated reduction of the original dynamical system we also demonstrate a technique vector diffusion maps that combines in a single formulation the symmetry removal step and the dimensionality reduction step

on the empirical state action frequencies in markov decision processes under general policies we consider the empirical state action frequencies and the empirical reward in weakly communicating finite state markov decision processes under general policies we define a certain polytope and establish that every element of this polytope is the limit of the empirical frequency vector under some policy in a strong sense furthermore we show that the probability of exceeding a given distance between the empirical frequency vector and the polytope decays exponentially with time under every policy we provide similar results for vector valued empirical rewards

a nine dimensional lorenz system to study high dimensional chaos we examine the dynamics of three dimensional cells with square planform in dissipative rayleigh benard convection by applying a triple fourier series ansatz up to second order we obtain a system of nine nonlinear ordinary differential equations from the governing hydrodynamic equations depending on two control parameters namely the rayleigh number and the prandtl number the asymptotic behaviour can be stationary periodic quasiperiodic or chaotic a period doubling cascade is identified as a route to chaos hereafter the asymptotic behaviour progressively evolves towards a hyperchaotic attractor for given values of control parameters beyond the accumulation point we observe a low dimensional chaotic attractor as is currently done for dissipative systems although the correlation dimension strongly suggests that this attractor could be embedded in a three dimensional space a topological characterization reveals that a higher dimensional space must be used thus we reconstruct a four dimensional model which is found to be in agreement with the properties of the original dynamics the nine dimensional lorenz model could therefore play a significant role in developing tools to characterize chaotic attractors embedded in phase space with a dimension greater than 3

a note on sargan densities abstract this note reconsiders the general class of sargan densities studied by goldfeld and quandt 1981 and missiakoulis 1983 in the context of approximating normal densities in regression models some errors in these papers are pointed out and the sargan densities are re computed to the contrary of missiakoulis conclusion the first two order densities are not the best choice higher order densities are better approximations though the best choice appears to be the second order density based on the goldfeld quandt criteria

some relations between dimension and lyapounov exponents we consider differentiable maps and compact invariant sets we introduce dimensional quantities related to the ergodic invariant measures and prove some simple relations

on the detection of determinism in a time series the nonlinearly scaled distributions of the strengths of the orthogonal modes in the data of a time series are compared with that derived from its surrogate counterpart to assess its chaoticity or the stochastic nature chaoticity manifests in the relatively decreasing strengths of the weaker modes with increasing dimension of the orthogonal spaces mapping the process the distributions of the strengths of the modes for the data and the surrogate are compared statistically singular value decomposition is used for the implementation the ability of the proposed method to distinguish between chaotic and stochastic dynamics is demonstrated through a number of simulated series including linear and nonlinear stochastic noise and processes described by lorenz rossler and mackey glass equations henon map and ecological map and some real life examples

the error exponent of variable length codes over markov channels with feedback the error exponent of markov channels with feedback is studied in the variable length block coding setting burnashev s classic result is extended to finite state ergodic markov channels for these channels a single letter characterization of the reliability function is presented under the assumption of full causal output feedback and full causal observation of the channel state both at the transmitter and at the receiver side tools from stochastic control theory are used in order to treat channels with intersymbol interference isi specifically the convex analytic approach to markov decision processes is adopted in order to handle problems with stopping time horizons induced by variable length coding schemes

dependent variables in broad band continuous time series in this paper we continue our development of new methods for the analysis of broad band time series by deriving quantities which are able to indicate deterministic dependence of an element in one time series on elements in other time series these methods are very broadly applicable and are particularly well suited to the study of continuous time series in which the value of the function may depend on derivatives of the function itself or on other quantities we apply our methods to a number of mathematical examples including the lorentz equation the h6non heiles equations the forced brusselator and the mackey glass equation we show that our methods are very successful at indicating deterministic dependencies in these systems even if the time series are highly chaotic statistical aspects of our procedure are discussed as are a number of interesting and surprising epistomological implications

uniform framework for the recurrence network analysis of chaotic time series we propose a general method for the construction and analysis of unweighted e recurrence networks from chaotic time series the selection of the critical threshold e c in our scheme is done empirically and we show that its value is closely linked to the embedding dimension m in fact we are able to identify a small critical range e numerically that is approximately the same for the random and several standard chaotic time series for a fixed m this provides us a uniform framework for the nonsubjective comparison of the statistical measures of the recurrence networks constructed from various chaotic attractors we explicitly show that the degree distribution of the recurrence network constructed by our scheme is characteristic to the structure of the attractor and display statistical scale invariance with respect to increase in the number of nodes n we also present two practical applications of the scheme detection of transition between two dynamical regimes in a time delayed system and identification of the dimensionality of the underlying system from real world data with a limited number of points through recurrence network measures the merits limitations and the potential applications of the proposed method are also highlighted

danco an intrinsic dimensionality estimator exploiting angle and norm concentration abstract in the past decade the development of automatic intrinsic dimensionality estimators has gained considerable attention due to its relevance in several application fields however most of the proposed solutions prove to be not robust on noisy datasets and provide unreliable results when the intrinsic dimensionality of the input dataset is high and the manifold where the points are assumed to lie is nonlinearly embedded in a higher dimensional space in this paper we propose a novel intrinsic dimensionality estimator danco and its faster variant fastdanco which exploit the information conveyed both by the normalized nearest neighbor distances and by the angles computed on couples of neighboring points the effectiveness and robustness of the proposed algorithms are assessed by experiments on synthetic and real datasets by the comparative evaluation with state of the art methodologies and by significance tests

intrinsic dimension estimation of manifolds by incising balls dimensionality reduction is a very important tool in data mining intrinsic dimension of data sets is a key parameter for dimensionality reduction however finding the correct intrinsic dimension is a challenging task in this paper a new intrinsic dimension estimation method is presented the estimator is derived by finding the exponential relationship between the radius of an incising ball and the number of samples included in the ball the method is compared with the previous dimension estimation methods experiments have been conducted on synthetic and high dimensional image data sets and on data sets of the santa fe time series competition and the results show that the new method is accurate and robust

a note on sargan densities abstract this note reexamines the general class of sargan densities analyzed by goldfeld and quandt 1981 kafei and schmidt 1985 and others in the context of approximating normal densities in certain econometric models it suggests a third order sargan density which complies with the goldfeld and quandt criteria is unimodal and is the best approximation to the normal for orders less than or equal to 3 some errors in goldfeld and quandt paper are corrected

a search for chaotic behavior in large and mesoscale motions in the pacific ocean the lagrangian trajectories of three satellite tracked bouys deployed in the pacific ocean for a one year period are studied from the standpoint of dynamical systems theory time series of the bouy positions are analyzed to estimate the correlation dimension and kolmogorov entropy we find a small finite value of correlation dimension near 1 4 and a finite kolmogorov entropy of about 0 2 day 1 these results at first suggest that the bouy motions may be dominated by a low dimensional chaotic attractor using standard ideas from the mathematics of fractals and a new method which we call multivariate scaling analysis we are able to demonstrate however that the bouy trajectories are instead fractal curves in the physical plane x y these results may be viewed from the standpoint of multivariate nongaussian convergence of the scales which occur when the fourier spectrum is described by a power law such systems can have a small finite value for the correlation dimension and are therefore fractal even when no underlying attractor is present systems of this type are more easily related to turbulent behavior stochastic many degrees of freedom rather than to strange attractor deterministic few degrees of freedom

econometric modelling with non normal disturbances abstract this paper analyzes a general class of non normal density functions dubbed sargan densities in the context of the ordinary regression model and the simple one market disequilibrium model use of the normal density in disequilibrium models is unwieldy especially for multimarket models since the application of maximum likelihood methods requires numerical evaluation of multiple integrals these difficulties are avoided with the sargan densities and based on both asymptotic results and limited sampling experiments these densities appear to offer a promising alternative to the normal in disequilibrium models

chaotic attractors of an infinite dimensional dynamical system abstract we study the chaotic attractors of a delay differential equation the dimension of several attractors computed directly from the definition agrees to experimental resolution with the dimension computed from the spectrum of lyapunov exponents according to a conjecture of kaplan and yorke assuming this conjecture to be valid as the delay parameter is varied from computations of the spectrum of lyapunov exponents we observe a roughly linear increase from two to twenty in the dimension while the metric entropy remains roughly constant these results are compared to a linear analysis and the asymptotic behavior of the lyapunov exponents is derived

a nonsmooth coulomb friction oscillator abstract a forced coulomb friction oscillator whose frictional force is allowed to vary with displacement is analyzed geometrically the equation of motion for the oscillator is piecewise linear we geometrically observe the nature of the flow in each region of solvability and then see how these solutions interact at the boundary of the regions the dynamics of the flow is viewed in terms of a map on the boundary between the regions for chaotic motion we geometrically construct the strange attractor and show that its exact behavior is that of a one dimensional map the following dynamical properties arise from the nonsmooth nature of the coulomb friction law the flow may not be invertible the flow may reach its attractor in finite time the dimension of the attractor may be less than or equal to two embeddings of an observable may not be diffeomorphic to the full phase flow

sargan densities which one abstract goldfeld and quandt 1981 suggest the use of sargan distribution as a viable alternative to normal distribution in econometric models this paper examines the resemblance of a sargan type family of densities to the normal density and concludes that the first two order cases are the best choice

intrinsic dimension estimation via nearest constrained subspace classifier we consider the problems of classification and intrinsic dimension estimation on image data a new subspace based classifier is proposed for supervised classification or intrinsic dimension estimation the distribution of the data in each class is modeled by a union of a finite number of affine subspaces of the feature space the affine subspaces have a common dimension which is assumed to be much less than the dimension of the feature space the subspaces are found using regression based on the 0 norm the proposed method is a generalisation of classical nn nearest neighbor nfl nearest feature line classifiers and has a close relationship to ns nearest subspace classifier the proposed classifier with an accurately estimated dimension parameter generally outperforms its competitors in terms of classification accuracy we also propose a fast version of the classifier using a neighborhood representation to reduce its computational complexity experiments on publicly available datasets corroborate these claims highlightsa new classifier called ncsc and its fast version called ncsc ii are proposed classifiers of nn nfl and ns are the special cases of ncsc ncsc ncsc ii can be used as intrinsic dimension estimator using well tuned intrinsic parameter ncsc ncsc ii outperforms nn nfl and ns

uniform partitions and a dimensions spectrum for lacunar measures a multifractal analysis has been proposed to investigate complex systems which exhibit a nonuniform local structure multifractal measures are normalized mass distributions inhomogeneous at every scale to every point p is associated the exponent of the power law r describing the asymptotic behaviour as r 0 of the mass of a sphere of radius r and centre p the measure properties are specified by the dimension spectrum f of the set of points p with the same scaling exponent this spectrum is nontrivial even when the mass distribution has no holes measure with euclidean support and the box counting dimension is an integer in this case it can be evaluated by considering a sequence of uniform grids whose spacing decreases as 2 n known as uniform partitions we consider here a family of iterated function system fractal measures with support on the unit square with random defects specified by three parameters a scale a probability p and an attenuation factor given a partition with a diameter less than any cell is chosen with probability p and its mass is multiplied by the corresponding pre measure is obtained after normalization in all the examples examined the f spectrum appears to widen monotonically as decreases this result suggests the possibility of using the method to detect the presence of defects in sponge like structures such as cancellous bone sponge like by analysing their radiographs since the grey intensity is related to the mass density of the structure a widening of the f spectrum is indeed observed in osteoporotic bone radiographs compared to normal bone radiographs this may be attributed to the rarefaction of the cancellous bone

absence of long range order in one dimensional spin systems for a one dimensional ising model with interaction energy e left mu right sum limits 1 leqslant i j leqslant n j j i mu iota mu j left j k geqslant 0 mu iota pm 1 right it is proved that there is no long range order at any temperature when s n sum limits k 1 n kj left k right o left log n 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 right the same result is shown to hold for the corresponding plane rotator model when s n o left left log n mathord left vphantom log n log log n right kern nulldelimiterspace log log n right right

markov decision problems and state action frequencies consider a controlled markov chain with countable state and action spaces basic quantities that determine the values of average cost functionals are identified under some regularity conditions these turn out to be a collection of numbers one for each state action pair describing for each state the relative number of uses of each action these conditional frequencies which are defined pathwise are shown to determine the state action frequencies that in the finite case are known to determine the costs this is extended to the countable case allowing for unbounded costs the space of frequencies is shown to be compact and convex and the extreme points are identified with stationary deterministic policies conditions under which the search for optimality in several optimization problems may be restricted to stationary policies are given these problems include the standard markov decision process as well as constrained optimization both in terms of average cost functionals and variability sensitive optimization an application to a queueing problem is given where these results imply the existence and explicit computation of optimal policies in constrained optimization problems the pathwise definition of the conditional frequencies implies that their values can be controlled directly moreover they depend only on the limiting behavior of the control this has immediate application to adaptive control of markov chains including adaptive control under constraints

on chaotic graphs a different approach for characterizing aperiodic dynamics fractal worlds with limited connectivity are the topological result of growing graphs from chaotic series we show how this model presents original characteristics which cannot be detected by means of the standard network descriptors in detail intrinsic inaccessibility to the fully connected configuration is demonstrated to be a universal feature associated with this family of graphs and strictly related to the fractality of a specific chaotic source here we discuss the potential of our model to be a generator of fractal graphs and also a self consistent tool for differentiating chaotic dynamics from stochastic processes

some results on the behavior and estimation of the fractal dimensions of distributions on attractors the strong interest in recent years in analyzing chaotic dynamical systems according to their asymptotic behavior has led to various definitions of fractal dimension and corresponding methods of statistical estimation in this paper we first provide a rigorous mathematical framework for the study of dimension focusing on pointwise dimension x and the generalized renyi dimensionsd q and give a rigorous proof of inequalities first derived by grassberger and procaccia and hentschel and procaccia we then specialize to the problem of statistical estimation of the correlation dimension and information dimension it has been recognized for some time that the error estimates accompanying the usual procedures which generally involve least squares methods and nearest neighbor calculations grossly underestimate the true statistical error involved in least squares analyses of and we identify sources of error not previously discussed in the literature and address the problem of obtaining accurate error estimates we then develop an estimation procedure for which corrects for an important bias term the local measure density and provides confidence intervals for the general applicability of this method is illustrated with various numerical examples

simulation based optimization of markov decision processes an empirical process theory approach we generalize and build on the pac learning framework for markov decision processes developed in jain and varaiya 2006 we consider the reward function to depend on both the state and the action both the state and action spaces can potentially be countably infinite we obtain an estimate for the value function of a markov decision process which assigns to each policy its expected discounted reward this expected reward can be estimated as the empirical average of the reward over many independent simulation runs we derive bounds on the number of runs needed for the convergence of the empirical average to the expected reward uniformly for a class of policies in terms of the v c or pseudo dimension of the policy class we then propose a framework to obtain an e optimal policy from simulation we provide sample complexity of such an approach

detection and characterization of changes of the correlation structure in multivariate time series we propose a method based on the equal time correlation matrix as a sensitive detector for phase shape correlations in multivariate data sets the key point of the method is that changes of the degree of synchronization between time series provoke level repulsions between eigenstates at both edges of the spectrum of the correlation matrix consequently detailed information about the correlation structure of the multivariate data set is imprinted into the dynamics of the eigenvalues and into the structure of the corresponding eigenvectors the performance of the technique is demonstrated by application to n f tori autoregressive models and coupled chaotic systems the high sensitivity the comparatively small computational effort and the excellent time resolution of the method recommend it for application to the analysis of complex spatially extended nonstationary systems

characterization of local complex structures in a recurrence plot to improve nonlinear dynamic discriminant analysis structures in recurrence plots rps preserving the rich information of nonlinear invariants and trajectory characteristics have been increasingly analyzed in dynamic discrimination studies the conventional analysis of rps is mainly focused on quantifying the overall diagonal and vertical line structures through a method called recurrence quantification analysis rqa this study extensively explores the information in rps by quantifying local complex rp structures to do this an approach was developed to analyze the combination of three major rqa variables determinism laminarity and recurrence rate dlr in a metawindow moving over a rp it was then evaluated in two experiments discriminating 1 ideal nonlinear dynamic series emulated from the lorenz system with different control parameters and 2 data sets of human heart rate regulations with normal sinus rhythms n 18 and congestive heart failure n 29 finally the dlr was compared with seven major rqa variables in terms of discriminatory power measured by standardized mean difference d smd in the two experiments dlr resulted in the highest discriminatory power with d smd 2 53 and 0 98 respectively which were 7 41 and 2 09 times the best performance from rqa the study also revealed that the optimal rp structures for the discriminations were neither typical diagonal structures nor vertical structures these findings indicate that local complex rp structures contain some rich information unexploited by rqa therefore future research to extensively analyze complex rp structures would potentially improve the effectiveness of the rp analysis in dynamic discrimination studies

time delay embedding increases estimation precision of models of intraindividual variability this paper investigates the precision of parameters estimated from local samples of time dependent functions we find that time delay embedding i e structuring data prior to analysis by constructing a data matrix of overlapping samples increases the precision of parameter estimates and in turn statistical power compared to standard independent rows of panel data we show that the reason for this effect is that the sign of estimation bias depends on the position of a misplaced data point if there is no a priori knowledge about initial conditions of the time dependent function hence we reason that the advantage of time delayed embedding is likely to hold true for a wide variety of functions we support these conclusions both by mathematical analysis and two simulations

a permutation entropy based test for causality the volume stock price relation abstract the purpose of this paper is to propose a newly developed non parametric test for linear and nonlinear causality based on permutation entropy and to show its usefulness in analyzing the potential causal relationship between trading volume and security prices most of the empirical applications and tests for causality rely on using granger causality based test for linear models although these tests have high power in uncovering linear causal relations their power against nonlinear causal relations can be low our test is designed to deal with the detection of linear and non linear causality we also compare our permutation entropy based test with other granger causality tests monte carlo simulations show excellent performance in terms of size and power of the new test for detecting linear and non linear causality under different scenarios our conclusions point that there is a bidirectional causal relation from volume to price returns not only in the mean but also in the variance

computation of lyapunov spectra effect of interactive noise and application to a chemical oscillator abstract a refined version of the sano sawada algorithm to compute lyapunov spectra has been implemented and tested with several models which include periodic chaotic and hyperchaotic attractors under conditions designed to mimic experimental situations it was found that reliable lyapunov exponents can be obtained for time series of moderate length and point density also in the presence of interactive noise as long as its amplitude does not exceed 1 of the oscillation amplitude the algorithm was then applied to experimental data from the co oxidation on pt 110 clearly showing the occurrence of chaos as well as hyperchaos

hurst exponent a brownian approach to characterize the nonlinear behavior of red blood cells deformability abstract ektacytometry techniques quantifies red blood cells rbcs deformability by measuring the elongation of suspended rbcs subjected to shear stress raw shear stress elongation plots are difficult to understand thus most research papers apply data reduction methods characterizing the relationship between curve fitting our approach works with the naturally generated photometrically recorded time series of the diffraction pattern of several million of rbcs subjected to shear stress and applies nonlinear quantifiers to study the fluctuations of these elongations the development of new quantitative methods is crucial for restricting the subjectivity in the study of the cells behavior mainly if they are capable of analyze at the same time biological and mechanical aspects of the cells in flowing conditions and compare their dynamics a patented optical system called erythrocyte rheometer was used to evaluate viscoelastic properties of erythrocytes by ektacytometry to analyze cell dynamics we used the technique of time delay coordinates false nearest neighbors the forecasting procedure proposed by sugihara and may and hurst exponent the results have expressive meaning on comparing healthy samples with parasite treated samples suggesting that apparent noise associated with deterministic chaos can be used not only to distinguish but also to characterize biological and mechanical aspects of cells at the same time in flowing conditions

simulation based uniform value function estimates of markov decision processes the value function of a markov decision process mdp assigns to each policy its expected discounted reward this expected reward can be estimated as the empirical average of the reward over many independent simulation runs we derive bounds on the number of runs needed for the uniform convergence of the empirical average to the expected reward for a class of policies in terms of the vapnik chervonenkis or p dimension of the policy class further we show through a counterexample that whether we get uniform convergence or not for an mdp depends on the simulation method used uniform convergence results are also obtained for the average reward case for partially observed markov decision processes and can be easily extended to markov games the results can be viewed as a contribution to empirical process theory and as an extension of the probably approximately correct pac learning theory for partially observable mdps and markov games

kolmogorov sinai entropy from the ordinal viewpoint abstract in the case of ergodicity much of the structure of a one dimensional time discrete dynamical system is already determined by its ordinal structure we generally discuss this phenomenon by considering the distribution of ordinal patterns which describe the ups and downs in the orbits of a borel measurable map on a borel subset of the real line on this base we give a natural ordinal description of the kolmogorov sinai entropy of one dimensional dynamical systems and relate the kolmogorov sinai entropy to the permutation entropy recently introduced by bandt and pompe

the necessity for a time local dimension in systems with time varying attractors abstract we show that a simple non linear system for ordinary differential equations may possess a time varying attractor dimension this indicates that it is infeasible to characterize eeg and meg time series with a single time global dimension we suggest another measure for the description of non stationary attractors

characterization of strange attractors a new measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable the relation of this new measure to fractal dimension and information theoretic entropy is discussed

on determining the dimension of chaotic flows abstract we describe a method for determining the approximate fractal dimension of an attractor our technique fits linear subspaces of appropriate dimension to sets of points on the attractor the deviation between points on the attractor and this local linear subspace is analyzed through standard multilinear regression techniques we show how the local dimension of attractors underlying physical phenomena can be measured even when only a single time varying quantity is available for analysis these methods are applied to several dissipative dynamical systems