a convergence theorem for arrays of l r fuzzy numbers the purpose of this paper is to provide a general law of large numbers for arrays of l r fuzzy numbers

maximum entropy approach to nonlinear modeling abstract we present a technique for building deterministic models of the nonlinear dynamics underlying observed time series it is formulated from maximum entropy principle within the framework of information theory two numerical examples of chaotic time series illustrate the method furthermore the theory on which the method is based provides an entropy like quantity that characterizes the suitability of the model it is defined over the space of parameters of the model we illustrate with two applications how this entropy can be a useful tool for exploring the phase space and establishing a criterion for choosing a convenient parametrized functional form

a nonlinear correlation measure for multivariable data set abstract the paper states that the mutual information carried by the rank sequences that are obtained from the original two sequences is a good measure of nonlinear correlation based on that the nonlinear correlation information entropy ncie is proposed for measuring the general relationship of a multivariable data set ncie uses a number in the closed interval 0 1 to indicate the nonlinear correlation degree of the concerned multivariable data set with 0 and 1 denotes the weakest and the strongest relationship respectively data sets generated from an autoregressive model a nonlinear chaotic lorenz system and a logistic function as known correlation sequences are analyzed using the proposed ncie the results testify the suitableness and correctness of the proposed concepts as nonlinear correlation measure

estimating entropy rates with bayesian confidence intervals the entropy rate quantifies the amount of uncertainty or disorder produced by any dynamical system in a spiking neuron this uncertainty translates into the amount of information potentially encoded and thus the subject of intense theoretical and experimental investigation estimating this quantity in observed experimental data is difficult and requires a judicious selection of probabilistic models balancing between two opposing biases we use a model weighting principle originally developed for lossless data compression following the minimum description length principle this weighting yields a direct estimator of the entropy rate which compared to existing methods exhibits significantly less bias and converges faster in simulation with monte carlo techinques we estimate a bayesian confidence interval for the entropy rate in related work we apply these ideas to estimate the information rates between sensory stimuli and neural responses in experimental data shlens kennel abarbanel chichilnisky in preparation

estimating information rates with confidence intervals in neural spike trains information theory provides a natural set of statistics to quantify the amount of knowledge a neuron conveys about a stimulus a related work kennel shlens abarbanel chichilnisky 2005 demonstrated how to reliably estimate with a bayesian confidence interval the entropy rate from a discrete observed time series we extend this method to measure the rate of novel information that a neural spike train encodes about a stimulus the average and specific mutual information rates our estimator makes few assumptions about the underlying neural dynamics shows excellent performance in experimentally relevant regimes and uniquely provides confidence intervals bounding the range of information rates compatible with the observed spike train we validate this estimator with simulations of spike trains and highlight how stimulus parameters affect its convergence in bias and variance finally we apply these ideas to a recording from a guinea pig retinal ganglion cell and compare results to a simple linear decoder

symbolic diffusion entropy rate of chaotic time series as a surrogate measure for the largest lyapunov exponent existing methods for estimating the largest lyapunov exponent from a time series rely on the rate of separation of initially nearby trajectories reconstructed from the time series in phase space according to ueda chaotic dynamical behavior is viewed as a manifestation of random transitions between unstable periodic orbits in a chaotic attractor which are triggered by perturbations due to experimental observation or the roundoff error characteristic of the computing machine and consequently consists of a sequence of piecewise deterministic processes instead of an entirely deterministic process chaotic trajectories might have no physical reality here we propose a mathematical method for estimating a surrogate measure for the largest lyapunov exponent on the basis of the random diffusion of the symbols generated from a time series in a chaotic attractor without resorting to initially nearby trajectories we apply the proposed method to numerical time series generated by chaotic flow models and verify its validity

entropy and local uncertainty of data from sensory neurons we present an empirical comparison between neural interspike interval sequences obtained from two different kinds of sensory receptors both differ in their internal structure as well as in the strength of correlations and the degree of predictability found in the respective spike trains as a further tool in this context we suggest the local uncertainty assigning a well defined predictability to individual spikes the local uncertainty is demonstrated to reveal significant patterns within the interspike interval sequences even when its overall structure is almost random our approach is based on the concept of symbolic dynamics and information theory

on selecting models for nonlinear time series abstract constructing models from time series with nontrivial dynamics involves the problem of how to choose the best model from within a class of models or to choose between competing classes this paper discusses a method of building nonlinear models of possibly chaotic systems from data while maintaining good robustness against noise the models that are built are close to the simplest possible according to a description length criterion the method will deliver a linear model if that has shorter description length than a nonlinear model we show how our models can be used for prediction smoothing and interpolation in the usual way we also show how to apply the results to identification of chaos by detecting the presence of homoclinic orbits directly from time series

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

a t sum bound of lr fuzzy numbers abstract in this paper we provide an upper bound and a lower bound of t sum of lr fuzzy numbers with different spreads where t is archimedean t norm and also show in three examples how close they are to actual membership functions furthermore we study when the membership function of t sum achieves the upper bound

equivalent conditions for laws of large numbers for t related l r fuzzy numbers in this paper we give some necessary and sufficient conditions for laws of large numbers for sequence of mutually t related l r fuzzy numbers when t is an continuous archimedean t norm and diameter of support of the fuzzy numbers is not uniformly bounded we also consider some necessary and sufficient conditions for laws of large numbers for l r fuzzy random numbers and generalize earlier results of fuller fuzzy sets and systems 45 1992 299 303 triesch fuzzy sets and systems 58 1993 339 342 and hong and lee fuzzy sets and systems 121 2001 537 543

approximation dimension reduction and nonconvex optimization using linear superpositions of gaussians this paper concerns neural network approaches to function approximation and optimization using linear superposition of gaussians or what are popularly known as radial basis function rbf networks the problem of function approximation is one of estimating an underlying function f given samples of the form y sub i x sub i i 1 2 n with y sub i f x sub i when the dimension of the input is high and the number of samples small estimation of the function becomes difficult due to the sparsity of samples in local regions the authors find that this problem of high dimensionality can be overcome to some extent by using linear transformations of the input in the gaussian kernels such transformations induce intrinsic dimension reduction and can be exploited for identifying key factors of the input and for the phase space reconstruction of dynamical systems without explicitly computing the dimension and delay they present a generalization that uses multiple linear projections onto scalars and successive rbf networks mlprbf that estimate the function based on these scaler values they derive some key properties of rbf networks that provide suitable grounds for implementing efficient search strategies for nonconvex optimization within the same framework

singular value decomposition in attractor reconstruction pitfalls and precautions applicability of singular value decomposition for reconstructing the strange attractor from one dimensional chaotic time series proposed by broomhead and king is extensively tested and discussed previously published doubts about its reliability are confirmed singular value decomposition by nature a linear method can bring distorted and misleading results when nonlinear structures are studied

state space reconstruction in the presence of noise takens theorem demonstrates that in the absence of noise a multidimensional state space can be reconstructed from a scalar time series this theorem gives little guidance however about practical considerations for reconstructing a good state space we extend takens treatment applying statistical methods to incorporate the effects of observational noise and estimation error we define the distortion matrix which is proportional to the conditional covariance of a state given a series of noisy measurements and the noise amplification which is proportional to root mean square time series prediction errors with an ideal model we derive explicit formulae for these quantities and we prove that in the low noise limit minimizing the distortion is equivalent to minimizing the noise amplification we identify several different scaling regimes for distortion and noise amplification and derive asymptotic scaling laws when the dimension and lyapunov exponents are sufficiently large these scaling laws show that no matter how the state space is reconstructed there is an explosion in the noise amplification from a practical point of view determinism is lost and the time series is effectively a random process in the low noise large data limit we show that the technique of local singular value decomposition is an optimal coordinate transformation in the sense that it achieves the minimum distortion in a state space of the lowest possible dimension however in numerical experiments we find that estimation error complicates this issue for local approximation methods we analyze the effect of reconstruction on estimation error derive a scaling law and suggest an algorithm for reducing estimation errors

a note to the t sum of l r fuzzy numbers abstract we generalize the results of hong and hwang 1994 for the membership function of finite infinite sum of l r fuzzy numbers where the sum is based on some continuous archimedean t norm t

a law of large numbers for fuzzy numbers abstract we study the problem if 1 2 are fuzzy numbers with modal values m1 m2 then what is the strongest t norm for which where mn m1 mn n the arithmetic mean 1 n n i is defined via sup t norm convolution and nes denotes necessity

a note on product sum of l r fuzzy numbers abstract triesch 1993 provided a partial answer to fuller s 1991 question about the membership function of the finite sum defined via the sup product norm convolution of l r fuzzy numbers in this short note we prove the other half

tree structured nonlinear signal modeling and prediction we develop a regression tree approach to identification and prediction of signals that evolve according to an unknown nonlinear state space model in this approach a tree is recursively constructed that partitions the p dimensional state space into a collection of piecewise homogeneous regions utilizing a 2 sup p ary splitting rule with an entropy based node impurity criterion on this partition the joint density of the state is approximately piecewise constant leading to a nonlinear predictor that nearly attains minimum mean square error this process decomposition is closely related to a generalized version of the thresholded ar signal model art which we call piecewise constant ar pcar we illustrate the method for two cases where classical linear prediction is ineffective a chaotic double scroll signal measured at the output of a chua type electronic circuit and a second order art model we show that the prediction errors are comparable with the nearest neighbor approach to nonlinear prediction but with greatly reduced complexity

the minimum description length principle in coding and modeling we review the principles of minimum description length and stochastic complexity as used in data compression and statistical modeling stochastic complexity is formulated as the solution to optimum universal coding problems extending shannon s basic source coding theorem the normalized maximized likelihood mixture and predictive codings are each shown to achieve the stochastic complexity to within asymptotically vanishing terms we assess the performance of the minimum description length criterion both from the vantage point of quality of data compression and accuracy of statistical inference context tree modeling density estimation and model selection in gaussian linear regression serve as examples

a universal data compression system a universal data compression algorithm is described which is capable of compressing long strings generated by a finitely generated source with a near optimum per symbol length without prior knowledge of the source this class of sources may be viewed as a generalization of markov sources to random fields moreover the algorithm does not require a working storage much larger than that needed to describe the source generating parameters

nilpotent t norm based sum of fuzzy intervals in this paper the sum of l r fuzzy intervals based on a given nilpotent triangular norm t norm t is studied by modifying markova s results which gives the necessary and sufficient condition ensuring an analytical form of the output sum to the form with arbitrary spreads we generalize the results of hong and hwang for the membership function of the sum of l r fuzzy intervals where the sum is based on some archimedean t norm t

on the role of pattern matching in information theory in this paper the role of pattern matching in information theory is motivated and discussed we describe the relationship between a pattern s recurrence time and its probability under the data generating stochastic source we show how this relationship has led to great advances in universal data compression we then describe nonasymptotic uniform bounds on the performance of data compression algorithms in cases where the size of the training data that is available to the encoder is not large enough so as to yield the asymptotic compression the shannon entropy we then discuss applications of pattern matching and universal compression to universal prediction classification and entropy estimation

existence and multiplicity results for boundary value problems connected with the discrete p ź laplacian on weighted finite graphs we use the direct variational method the ekeland variational principle the mountain pass geometry and karush kuhn tucker theorem in order to investigate existence and multiplicity results for boundary value problems connected with the discrete p laplacian on weighted finite graphs several auxiliary inequalities for the discrete p laplacian on finite graphs are also derived positive solutions are considered

information and entropy in strange attractors a technique for analyzing time series data from experiments is presented that provides estimates of four basic characteristics of a system 1 the measure theoretic entropy 2 the accuracy of the measurements 3 the number of measurements necessary to specify a system state and 4 the best delay time t to use in order to construct phase space portraits by the method of delays these characteristics are obtained by separating the entropy of measurements into a part due to noise and parts due to deterministic effects for the technique to work the noise associated with each measurement must be independent of the noise associated with all other measurements an algorithm for implementing the analysis is presented with three examples

estimating a generating partition from observed time series symbolic shadowing we propose a deterministic algorithm for approximating a generating partition from a time series using tessellations using data generated by henon and ikeda maps we demonstrate that the proposed method produces partitions that uniquely encode all the periodic points up to some order and provide good estimates of the metric and topological entropies the algorithm gives useful results even with a short noisy time series

classification of heart rate data using artificial neural network and fuzzy equivalence relation the electrocardiogram is a representative signal containing information about the condition of the heart the shape and size of the p qrs t wave the time intervals between its various peaks etc may contain useful information about the nature of disease afflicting the heart however these subtle details cannot be directly monitored by the human observer besides since bio signals are highly subjective the symptoms may appear at random in the time scale therefore the signal parameters extracted and analysed using computers are highly useful in diagnostics this paper deals with the classification of certain diseases using artificial neural network ann and fuzzy equivalence relations the heart rate variability is used as the base signal from which certain parameters are extracted and presented to the ann for classification the same data is also used for fuzzy equivalence classifier the feedforward architecture ann classifier is seen to be correct in about 85 of the test cases and the fuzzy classifier yields correct classification in over 90 of the cases

on the convergence of t sum of l r fuzzy numbers abstract this paper presents the membership function of infinite or finite sum defined by the sup t norm convolution of l r fuzzy numbers under the conditions of the convexity of additive generators and the concavity of l r as an application we shall calculate the membership function of the limit distribution of the hamacher sum hr sum for 0 r 2 which generalizes fuller s results fuzzy sets and systems 41 1991 83 87 42 1991 205 212 in the case r 0 1 2

periodic and neumann problems for discrete p laplacian abstract using critical point theory we obtain the existence of solutions for some periodic boundary value problems involving the discrete p laplacian these extend and improve known results for similar problems with discrete p laplacian similar results for neumann problems are also provided as applications we prove upper and lower solutions theorems for both considered cases

on computation of the compositional rule of inference under triangular norms abstract this paper is devoted to the derivation of exact calculation formulas for the compositional rule of inference under archimedean t norms when both the observation and the relation parts are given by hellendoorn s function 6

the context tree weighting method basic properties describes a sequential universal data compression procedure for binary tree sources that performs the double mixture using a context tree this method weights in an efficient recursive way the coding distributions corresponding to all bounded memory tree sources and achieves a desirable coding distribution for tree sources with an unknown model and unknown parameters computational and storage complexity of the proposed procedure are both linear in the source sequence length the authors derive a natural upper bound on the cumulative redundancy of the method for individual sequences the three terms in this bound can be identified as coding parameter and model redundancy the bound holds for all source sequence lengths not only for asymptotically large lengths the analysis that leads to this bound is based on standard techniques and turns out to be extremely simple the upper bound on the redundancy shows that the proposed context tree weighting procedure is optimal in the sense that it achieves the rissanen 1984 lower bound

the permutation entropy rate equals the metric entropy rate for ergodic information sources and ergodic dynamical systems abstract permutation entropy quantifies the diversity of possible orderings of the values a random or deterministic system can take as shannon entropy quantifies the diversity of values we show that the metric and permutation entropy rates measures of new disorder per new observed value are equal for ergodic finite alphabet information sources discrete time stationary stochastic processes with this result we then prove that the same holds for deterministic dynamical systems defined by ergodic maps on n dimensional intervals this result generalizes a previous one for piecewise monotone interval maps on the real line c bandt g keller b pompe entropy of interval maps via permutations nonlinearity 15 2002 1595 1602 at the expense of requiring ergodicity and using a definition of permutation entropy rate differing modestly in the order of two limits the case of non ergodic finite alphabet sources is also studied and an inequality developed finally the equality of permutation and metric entropy rates is extended to ergodic non discrete information sources when entropy is replaced by differential entropy in the usual way

existence of solutions for p x laplacian dirichlet problem this paper presents several sufficient conditions for the existence of solutions for the dirichlet problem of p x laplacian div u p x 2 u f x u x u 0 x especially an existence criterion for infinite many pairs of solutions for the problem is obtained the discussion is based on the theory of the spaces lp x and w01 p x

quantization and the method of k means asymptotic results from the statistical theory of k means clustering are applied to problems of vector quantization the behavior of quantizers constructed from long training sequences of data is analyzed by relating it to the consistency problem for k means

universal modeling and coding the problems arising in the modeling and coding of strings for compression purposes are discussed the notion of an information source that simplifies and sharpens the traditional one is axiomatized and adaptive and nonadaptive models are defined with a measure of complexity assigned to the models a fundamental theorem is proved which states that models that use any kind of alphabet extension are inferior to the best models using no alphabet extensions at all a general class of so called first in first out fifo arithmetic codes is described which require no alphabet extension devices and which therefore can be used in conjunction with the best models because the coding parameters are the probabilities that define the model their design is easy and the application of the code is straightforward even with adaptively changing source models

necessary and sufficient conditions for the equality of the interactive and non interactive sums of two fuzzy numbers this study aims to answer the open question proposed by carlsson et al 2004 4 which asks whether it is possible to find the properties needed by a joint possibility distribution function such that the interactive sum with respect to this function and the standard sum of two fuzzy numbers will coincide we provide the necessary and sufficient conditions such that this equality would hold in addition we find the necessary and sufficient conditions such that the interactive and standard sums will coincide at some point in the domain in order to obtain an interactive addition that does not coincide with the standard addition we show that we need to search for a joint possibility distribution that coincides as little as possible with the independent joint possibility distribution throughout the so called diagonal of the two fuzzy numbers as simple consequences we deduce similar results for triangular norm based additions of fuzzy numbers

complete and flexible replacement of chaotic uncertainty with transmitted information natural chaos can be described as an information source emitting symbolic sequences with positive entropy we use two algorithmic techniques from data compression in a nonstandard way along with a control scheme to replace the natural uncertainty in chaotic systems with an arbitrary digital message unlike previous targeting based control the controlled deterministic transmission appears statistically identical to natural chaos with a message modulated on it at the intrinsic kolmogorov sinai information generation rate of the chaotic oscillator thus chaotic communication by targeting need not consume any additional channel capacity beyond that required by the message itself and the message bearing signal may appear identical to the uncontrolled oscillator we also demonstrate control and data transmission at the channel capacity of the oscillator the maximum possible data rate compatible with the grammar

a universal finite memory source an irreducible parameterization for a finite memory source is constructed in the form of a tree machine a universal information source for the set of finite memory sources is constructed by a predictive modification of an earlier studied algorithm context it is shown that this universal source incorporates any minimal data generating tree machine in an asymptotically optimal manner in the following sense the negative logarithm of the probability it assigns to any long typical sequence generated by any tree machine approaches that assigned by the tree machine at the best possible rate

symbolic dynamics of noisy chaos abstract one model of randomness observed in physical systems is that low dimensional deterministic chaotic attractors underly the observations a phenomenological theory of chaotic dynamics requires an accounting of the information flow from the observed system to the observer the amount of information available in observations and just how this information affects predictions of the system s future behavior in an effort to develop such a description we discuss the information theory of highly discretized observations of random behavior metric entropy and topological entropy are well defined invariant measures of such an attractor s level of chaos and are computable using symbolic dynamics real physical systems that display low dimensional dynamics are however inevitably coupled to high dimensional randomless e g thermal noise we investigate the effects of such fluctuations coupled to deterministic chaotic systems in particular the metric entropy s response to the fluctuations we find that the entropy increases with a power law in the noise level and that the convergence of the entropy and the effect of fluctuations can be cast as a scaling theory we also argue that in addition to the metric entropy there is a second scaling invariant quantity that characterizes a deterministic system with added fluctuations i0 the maximum average information obtainable about the initial condition that produces a particular sequence of measurements or symbols

observation of a strange attractor abstract phase space portraits have been constructed and analyzed for noisy nonperiodic data obtained in an experiment on a nonequilibrium homogeneous chemical reaction the phase space trajectories define a limit set that is an attractor following a perturbation the trajectory quickly returns to the attracting set this attracting set is shown to be strange nearby trajectories separate exponentially on the average moreover the poincare sections exhibit the stretching and folding that is characteristic of strange attractors

t norm based addition of fuzzy intervals abstract the aim of this paper is to provide new results regarding the effective practical computation of t norm based additions of lr fuzzy intervals

shape preserving additions of fuzzy intervals abstract triangular norm based additions preserving the lr shape of lr fuzzy intervals are studied the connection between t norm based additions preserving the linearity of fuzzy intervals and additions preserving lr shape for invertible shapes l and r is given a possibility of controlling the growth of the spread during the calculations with lr fuzzy intervals is shown

on the compositional rule of inference under triangular norms abstract the aim of this paper is to provide a close upper bound of the membership function for the compositional rule of inference under archimedean t norm where both the observation and the relation parts are given by hellendoorn s function 1980 in particular if the left and right spreads of the observation part is the same as those of the relation part then this upper bound is the exact membership function which generalizes the earlier result by fuller and zimmermann 1982 in that assumption of twice differentiability is deleted

memory flow structures in lorenz chaos memory flow structures of a coupled three element model here a lorenz chaos have been investigated although in the models of coupled two element systems a bidirectional connectivity of information transference is held and the memory flows of one channel exhibit a nearly out of phase relation it is found that in this lorenz chaos one additional unidirectional connectivity can also occur and the nearly out of phase relation in the coupled two element models will be replaced by an antiphaselike relation in lorenz chaos

shape preserving multiplications of fuzzy numbers abstract in this note we show that t w the weakest t norm is the only t norm t which induces a shape preserving multiplications of lr fuzzy numbers

t sum of l r fuzzy numbers abstract the sum of l r fuzzy numbers based on a given archimedian continuous t norm t is studied a series of improving sufficient conditions of several authors ensuring an analytical form of the output sum is completed by a sufficient and necessary condition i e the best sufficient condition some applications are indicated

recognizing chaotic time waveforms in terms of a parametrized family of nonlinear predictors abstract consider a chaotic dynamical system which exhibits a variety of chaotic time waveforms with a change in the bifurcation parameters this paper presents an algorithm for estimating the underlying bifurcation parameters of the chaotic time waveforms in experimental situation in which no a priori analytical knowledge of the dynamical system is available first we construct qualitatively similar parametrized family of nonlinear predictors only from several sets of chaotic time waveforms qualitatively similar parametrized family means that the family of nonlinear predictors exhibits qualitatively similar bifurcation phenomena as the original chaotic time waveforms are then characterized in terms of the qualitatively similar bifurcation parameters of the nonlinear predictors we call the characterization of chaotic time waveforms in terms of the underlying bifurcation parameters chaotic time waveform recognition several numerical experiments using the rossler equations show the efficiency of the algorithm the effect of observational noise included in chaotic time waveforms is also considered

positive solutions for discrete boundary value problems involving the p laplacian with potential terms in this paper we discuss the existence and the uniqueness of discrete boundary value problems involving the p laplacian with potential terms to this end we deal with the strong comparison principle and the dirichlet principle for the p laplacian with potential terms on weighted graphs with boundary moreover we provide a lower bound of the potential terms which guarantees the existence of solutions of the boundary value problem for the operator finally we also discuss inverse conductivity and potential problems for the operator

the dimension of chaotic attractors dimension is perhaps the most basic property of an attractor in this paper we discuss a variety of different definitions of dimension compute their values for a typical example and review previous work on the dimension of chaotic attractors the relevant definitions of dimension are of two general types those that depend only on metric properties and those that depend on the frequency with which a typical trajectory visits different regions of the attractor both our example and the previous work that we review support the conclusion that all of the frequency dependent dimensions take on the same value which we call the dimension of the natural measure and all of the metric dimensions take on a common value which we call the fractal dimension furthermore the dimension of the natural measure is typically equal to the lyapunov dimension which is defined in terms of lyapunov numbers and thus is usually far easier to calculate than any other definition because it is computable and more physically relevant we feel that the dimension of the natural measure is more important than the fractal dimension

t sum of bell shaped fuzzy intervals the usual arithmetic operations on real numbers can be extended to arithmetical operations on fuzzy intervals by means of zadeh s extension principle based on a t norm t a t norm is called consistent with respect to a class of fuzzy intervals for some arithmetic operation if this arithmetic operation is closed for this class it is important to know which t norms are consistent with a particular type of fuzzy intervals recently dombi and gyorbiro proved that addition is closed if the dombi t norm is used with two bell shaped fuzzy intervals a result proved by mesiar on a strict t norm based shape preserving additions of lr fuzzy intervals with unbounded support is recalled as applications we define a broader class of bell shaped fuzzy intervals then we study t norms which are consistent with these particular types of fuzzy intervals dombi and gyorbiro s results are special cases of the results described in this paper

optimal reconstruction of dynamical systems a noise amplification approach in this work we propose an objective function to guide the search for a state space reconstruction of a dynamical system from a time series of measurements these statistics can be evaluated on any reconstructed attractor thereby allowing a direct comparison among different approaches uniform or nonuniform delay vectors pca legendre coordinates etc it can also be used to select the most appropriate parameters of a reconstruction strategy in the case of delay coordinates this translates into finding the optimal delay time and embedding dimension from the absolute minimum of the advocated cost function its definition is based on theoretical arguments on noise amplification the complexity of the reconstructed attractor and a direct measure of local stretch which constitutes an irrelevance measure the proposed method is demonstrated on synthetic and experimental time series

statistical validation of mutual information calculations comparison of alternative numerical algorithms given two time series x and y their mutual information i x y i y x is the average number of bits of x that can be predicted by measuring y and vice versa in the analysis of observational data calculation of mutual information occurs in three contexts identification of nonlinear correlation determination of an optimal sampling interval particularly when embedding data and in the investigation of causal relationships with directed mutual information in this contribution a minimum description length argument is used to determine the optimal number of elements to use when characterizing the distributions of x and y however even when using partitions of the x and y axis indicated by minimum description length mutual information calculations performed with a uniform partition of the xy plane can give misleading results this motivated the construction of an algorithm for calculating mutual information that uses an adaptive partition this algorithm also incorporates an explicit test of the statistical independence of x and y in a calculation that returns an assessment of the corresponding null hypothesis the previously published fraser swinney algorithm for calculating mutual information includes a sophisticated procedure for local adaptive control of the partitioning process when the fraser and swinney algorithm and the algorithm constructed here are compared they give very similar numerical results less than 4 difference in a typical application detailed comparisons are possible when x and y are correlated jointly gaussian distributed because an analytic expression for i x y can be derived for that case based on these tests three conclusions can be drawn first the algorithm constructed here has an advantage over the fraser swinney algorithm in providing an explicit calculation of the probability of the null hypothesis that x and y are independent second the fraser swinney algorithm is marginally the more accurate of the two algorithms when large data sets are used with smaller data sets however the fraser swinney algorithm reports structures that disappear when more data are available third the algorithm constructed here requires about 0 5 of the computation time required by the fraser swinney algorithm

testing nonlinear markovian hypotheses in dynamical systems abstract we present a statistical approach for detecting the markovian character of dynamical systems by analyzing their flow of information especially in the presence of noise which is mostly the case for real world time series the calculation of the information flow of the underlying system via the concept of symbolic dynamics is rather problematic since one has to use infinitesimal partitions we circumvent this difficulty by measuring the information flow indirectly more precisely we calculate a measure based on higher order cumulants which quantifies the statistical dependencies between the past values of the time series and the point r steps ahead as an extension of theiler s method of surrogate data theiler et al 1992 this cumulant based information flow a function of the look ahead r is used as the discriminating statistic in testing the observed dynamics against a hierarchy of null hypotheses corresponding to nonlinear markov processes of increasing order this procedure is iterative in the sense that whenever a null hypothesis is rejected new data sets can be generated corresponding to better approximations of the original process in terms of information flow since we use higher order cumulants for calculating the discriminating statistic our method is also applicable to small data sets numerical results on artificial and real world examples including non chaotic nonlinear processes autoregressive models and noisy chaos show the effectiveness of our approach

some results on the addition of fuzzy intervals abstract a simple new method of computing t sum of fuzzy intervals having the same results as the sum of fuzzy intervals based on the weakest t norm t w is introduced this work extends that of mesiar fuzzy sets and systems 91 1997 231 237 and markova fuzzy sets and systems 91 1997 253 258

parameter estimations of mutually t related fuzzy variables abstract cai fuzzy sets and systems 55 1993 179 185 discussed parameter estimation methods for normal fuzzy variables in nahmias sense in this paper we investigate more the parameter estimations of general t related not necessarily normal fuzzy variables

a law of large numbers for fuzzy numbers in a banach space abstract we study a law of large numbers for mutually t related fuzzy number in a banach space where t is an archimedean t norm and extend earlier results of fuller and triesch in this area

estimating the errors on measured entropy and mutual information abstract information entropy and the related quantity mutual information are used extensively as measures of complexity and to identify nonlinearity in dynamical systems expressions for the probability distribution of entropies and mutual informations calculated from finite amounts of data exist in the literature but the expressions have seldom been used in the field of nonlinear dynamics in this paper formulae for estimating the errors on observed information entropies and mutual informations are derived using the standard error analysis familiar to physicists their validity is demonstrated by numerical experiment for illustration the formulae are then used to evaluate the errors on the time lagged mutual information of the logistic map

t sum of sigmoid shaped fuzzy intervals the usual arithmetic operations on real numbers can be extended to arithmetical operations on fuzzy intervals by means of zadeh s extension principle based on a t norm t a t norm is called consistent with respect to a class of fuzzy intervals for some arithmetic operation if this arithmetic operation is closed for this class it is important to know which t norms are consistent with particular types of fuzzy intervals recently dombi and gyorbiro j dombi n gyorbiro additions of sigmoid shaped fuzzy intervals using the dombi operator and infinite sum theorems fuzzy sets and systems 157 2006 952 963 proved that addition is closed if the dombi t norm is used with sigmoid shaped fuzzy intervals in this paper we define a broader class of sigmoid shaped fuzzy intervals then we study t norms that are consistent with these particular types of fuzzy intervals dombi and gyorbiro s results are special cases of the results described in this paper

the performance of universal encoding universal coding theory is surveyed from the viewpoint of the interplay between delay and redundancy the price for universality turns out to be acceptably small

the kolmogorov sinai entropy in the setting of fuzzy sets for image texture analysis and classification the kolmogorov sinai k s entropy is used to quantify the average amount of uncertainty of a dynamical system through a sequence of observations sequence probabilities therefore play a central role for the computation of the entropy rate to determine if the dynamical system under study is deterministically non chaotic deterministically chaotic or random this paper extends the notion of the k s entropy to measure the entropy rate of imprecise systems using sequence membership grades in which the underlying deterministic paradigm is replaced with the degree of fuzziness while constructing sequential probabilities for the calculation of the k s entropy is difficult in practice the estimate of the k s entropy in the setting of fuzzy sets in an image is feasible and can be useful for modeling uncertainty of pixel distributions in images the fuzzy k s entropy is illustrated as an effective feature for image analysis and texture classification highlightsdevelopment of a new method for feature extraction using chaos and fuzzy sets modeling kolmogorov sinai k s entropy with fuzzy logic is feasible fuzzy k s entropy is an effective feature for analysis and classification of textural images

thermodynamic depth of causal states objective complexity via minimal representations thermodynamic depth is an appealing but flawed structural complexity measure it depends on a set of macroscopic states for a system but neither its original introduction by lloyd and pagels nor any follow up work has considered how to select these states depth therefore is at root arbitrary computational mechanics an alternative approach to structural complexity provides a definition for a system close quote s minimal necessary causal states and a procedure for finding them we show that the rate of increase in thermodynamic depth or ital dive is the system close quote s reverse time shannon entropy rate and so depth only measures degrees of macroscopic randomness not structure to fix this we redefine the depth in terms of the causal state representation emdash epsilon machines emdash and show that this representation gives the minimum dive consistent with accurate prediction thus epsilon machines are optimally shallow copyright ital 1999 ital the american physical society

variable exponent linear growth functionals in image restoration we study a functional with variable exponent 1 leq p x leq 2 which provides a model for image denoising enhancement and restoration the diffusion resulting from the proposed model is a combination of total variation tv based regularization and gaussian smoothing the existence uniqueness and long time behavior of the proposed model are established experimental results illustrate the effectiveness of the model in image restoration

on product sum of triangular fuzzy numbers abstract we study the problem if a i i n are fuzzy numbers of triangular form then what is the membership function of the infinite or finite sum a 1 a 2 defined via the sup product norm convolution

detecting switch dynamics in chaotic time waveform using a parametrized family of nonlinear predictors abstract an algorithm is presented for detecting switch dynamics in chaotic time waveform by the switch dynamics we mean that the chaotic time waveform is measured from a dynamical system whose bifurcation parameters are occasionally switched among a set of slightly different parameter values first the switched chaotic time waveform is divided into windows of short term time waveforms from the set of windowed time waveforms qualitatively similar parametrized family of nonlinear predictors is constructed qualitatively similar parametrized family means that the family of nonlinear predictors exhibits qualitatively similar bifurcation phenomena as the original by characterizing the windows of short term chaotic time waveforms in terms of the qualitative parameters of nonlinear predictors switch dynamics of their associated bifurcation parameters are detected for the lorenz equations the rossler equations and the mackey glass equations efficiency of the algorithm is demonstrated in the experiment chaotic time waveforms contaminated with observational noise is considered

the context tree weighting method extensions first we modify the basic binary context tree weighting method such that the past symbols x sub 1 d x sub 2 d x sub 0 are not needed by the encoder and the decoder then we describe how to make the context tree depth d infinite which results in optimal redundancy behavior for all tree sources while the number of records in the context tree is not larger than 2t 1 here t is the length of the source sequence for this extended context tree weighting algorithm we show that with probability one the compression ratio is not larger than the source entropy for source sequence length t spl rarr spl infin for stationary and ergodic sources

on the convergence of product sum series of l r fuzzy numbers we study series a1 a2 an of l r fuzzy numbers where addition is defined via the sup product norm convulation and obtain some convergence theorems which generalize earlier results by fuller cf 2

testing for nonlinearity using redundancies quantitative and qualitative aspects abstract a method for testing nonlinearity in time series is described based on information theoretic functionals redundancies linear and nonlinear forms of which allow either qualitative or after incorporating the surrogate data technique quantitative evaluation of dynamical properties of scrutinized data an interplay of quantitative and qualitative testing on both the linear and nonlinear levels is analyzed and robustness of this combined approach against spurious nonlinearity detection is demonstrated evaluation of redundancies and redundancy based statistics as functions of time lag and embedding dimension can further enhance insight into dynamics of a system under study

characterisation of archimedian t norms and a law of large numbers abstract we study a law of large numbers for mutually t related fuzzy numbers where t is an archimedian t norm and extend earlier results of fuller in this area in particular we show that the class of archimedian t norms can be characterised by the validity of a very general law of large numbers for sequences of l r fuzzy numbers

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

failure of maximum likelihood methods for chaotic dynamical systems the maximum likelihood method is a basic statistical technique for estimating parameters and variables and is the starting point for many more sophisticated methods like bayesian methods this paper shows that maximum likelihood fails to identify the true trajectory of a chaotic dynamical system because there are trajectories that appear to be far more infinitely more likely than truth this failure occurs for unbounded noise and for bounded noise when it is sufficiently large and will almost certainly have consequences for parameter estimation in such systems the reason for the failure is rather simple in chaotic dynamical systems there can be trajectories that are consistently closer to the observations than the true trajectory being observed and hence their likelihood dominates truth the residuals of these truth dominating trajectories are not consistent with the noise distribution they would typically have too small standard deviation and many outliers and hence the situation may be remedied by using methods that examine the distribution of residuals and are not entirely maximum likelihood based

regranulation a granular algorithm enabling communication between granular worlds abstract in this paper we describe a granular algorithm for translating information between two granular worlds represented as fuzzy rulebases these granular worlds are defined on the same universe of discourse but employ different granulations of this universe in order to translate information from one granular world to the other we must regranulate the information so that it matches the information granularity of the target world this is accomplished through the use of a first order interpolation algorithm implemented using linguistic arithmetic a set of elementary granular computing operations we first demonstrate this algorithm by studying the common fuzzy pd rulebase at several different granularities and conclude that the 3 3 granulation may be too coarse for this objective we then examine the question of what the natural granularity of a system might be this is studied through a 10 fold cross validation experiment involving three different granulations of the same underlying mapping for the problem under consideration we find that a 7 7 granulation appears to be the minimum necessary precision

nonparametric entropy estimation for stationary processes and random fields with applications to english text we discuss a family of estimators for the entropy rate of a stationary ergodic process and prove their pointwise and mean consistency under a doeblin type mixing condition the estimators are cesaro averages of longest match lengths and their consistency follows from a generalized ergodic theorem due to maker 1940 we provide examples of their performance on english text and we generalize our results to countable alphabet processes and to random fields

additions of interactive fuzzy numbers this paper provides an account of an approach to modeling unknown data by means of fuzzy set theory and addresses the problem of deriving the uncertainty on a sum of variables whose values lie within fuzzy intervals the first part is an extensive presentation of the theoretical background of the approach the extension principle is stated in terms of possibility of an event the concept of variable interaction is investigated at length section ii gives new results regarding the effective practical computation of additions of fuzzy numbers its originality lies in the introduction of interaction which enables to control the growth of uncertainty in calculations moreover the problem of computing mathematical expectations with fuzzy probabilities is solved the results derived in this paper can easily be used in decision problems where values of parameters or decision variables are not yet precisely fixed or assessed typical applications could be multicriteria optimization and decision making under uncertainty where fuzzy expected utilities can be obtained out of uncompletely assessed probabilities more generally fuzzy arithmetic can be an important tool for sophisticated computationally tractable sensitivity analysis in systems modeling computer aided design and operations research

on shape preserving additions of fuzzy intervals abstract we prove an open problem suggested by mesiar 1997 fuzzy sets and systems86 73 78 concerning continuous t norm based additions preserving the lr fuzzy intervals

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

overview of differential equations with non standard growth abstract differential equations with non standard growth have been a very active field of investigation in recent years in this survey we present an overview of the field as well as several of the most important results we consider both existence and regularity questions finally we provide a comprehensive list of papers published to date

on a resonance problem with the discrete p laplacian on finite graphs abstract in this paper we discuss the existence of a solution for a resonance problem involving the discrete p laplacian using asymptotic behaviors of g at infinity a nontrivial solution is established moreover we also find a necessary condition for asymptotic behavior of g at zero to guarantee the existence of at least two nontrivial solutions

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow

multiple solutions for neumann and periodic problems with singular ϕ laplacian abstract we use the critical point theory for convex lower semicontinuous perturbations of c 1 functionals to establish existence of multiple radial solutions for some one parameter neumann problems involving the operator v div v 1 v 2 similar results for periodic problems are also provided