nonlinear dynamics delay times and embedding windows in order to construct an embedding of a nonlinear time series one must choose an appropriate delay time d often d is estimated using the autocorrelation function however this does not treat the nonlinearity appropriately and it may yield an incorrect value for d on the other hand the correct value of d can be found from the mutual information but this process is rather cumbersome computationally here we suggest a simpler method for estimating d using the correlation integral we call this the c c method and we test it on several nonlinear time series obtaining estimates of d in agreement with those obtained using the mutual information furthermore some researchers have suggested that one should not choose a fixed delay time d independent of the embedding dimension m but rather one should choose an appropriate value for the delay time window w d m 1 which is the total time spanned by the components of each embedded point unfortunately w cannot be estimated using the autocorrelation function or the mutual information and no standard procedure for estimating w has emerged however we show that the c c method can also be used to estimate w basically w is the optimal time for independence of the data while d is the first locally optimal time as tests we apply the c c method to the lorenz system a three dimensional irrational torus the rossler system and the rabinovich fabrikant system we also demonstrate the robustness of this method to the presence of noise c 1999 elsevier science b v all rights reserved

dynamic analysis of traffic time series at different temporal scales a complex networks approach abstract the analysis of dynamics in traffic flow is an important step to achieve advanced traffic management and control in intelligent transportation system its complexity and periodicity are definitely two fundamental properties in traffic dynamics in this study we first measure the complexity of traffic flow data by lempel ziv algorithm at different temporal scales and the data are collected from loop detectors on freeway second to obtain more insight into the complexity and periodicity in traffic time series we then construct complex networks from traffic time series by considering each day as a cycle and each cycle as a single node the optimal threshold value of complex networks is estimated by the distribution of density and its derivative in addition the complex networks are subsequently analyzed in terms of some statistical properties such as average path length clustering coefficient density average degree and betweenness finally take 2 min aggregation data as example we use the correlation coefficient matrix adjacent matrix and closeness to exploit the periodicity of weekdays and weekends in traffic flow data the findings in this paper indicate that complex network is a practical tool for exploring dynamics in traffic time series

financial asset returns direction of change forecasting and volatility dynamics we consider three sets of phenomena that feature prominently in the financial economics literature 1 conditional mean dependence or lack thereof in asset returns 2 dependence and hence forecastability in asset return signs and 3 dependence and hence forecastability in asset return volatilities we show that they are very much interrelated and explore the relationships in detail among other things we show that 1 volatility dependence produces sign dependence so long as expected returns are nonzero so that one should expect sign dependence given the overwhelming evidence of volatility dependence 2 it is statistically possible to have sign dependence without conditional mean dependence 3 sign dependence is not likely to be found via analysis of sign autocorrelations runs tests or traditional market timing tests because of the special nonlinear nature of sign dependence so that traditional market timing tests are best viewed as tests for sign dependence arising from variation in expected returns rather than from variation in volatility or higher moments 4 sign dependence is not likely to be found in very high frequency e g daily or very low frequency e g annual returns instead it is more likely to be found at intermediate return horizons and 5 the link between volatility dependence and sign dependence remains intact in conditionally non gaussian environments for example with time varying conditional skewness and or kurtosis

characterization of chaotic dynamic behavior in the gas liquid slug flow using directed weighted complex network analysis abstract understanding the nonlinear and complex dynamics underlying the gas liquid slug flow is a significant but challenging problem we systematically carried out gas liquid two phase flow experiments for measuring the time series of flow signals which is studied in terms of the mapping from time series to complex networks in particular we construct directed weighted complex networks dwcn from time series and then associate different aspects of chaotic dynamics with the topological indices of the dwcn and further demonstrate that the dwcn can be exploited to detect unstable periodic orbits of low periods examples using time series from classical chaotic systems are provided to demonstrate the effectiveness of our approach we construct and analyze numbers of dwcns for different gas liquid flow patterns and find that our approach can quantitatively distinguish different experimental gas liquid flow patterns furthermore the dwcn analysis indicates that slug flow shows obvious chaotic behavior and its unstable periodic orbits reflect the intermittent quasi periodic oscillation behavior between liquid slug and large gas slug these interesting and significant findings suggest that the directed weighted complex network can potentially be a powerful tool for uncovering the underlying dynamics leading to the formation of the gas liquid slug flow

nonlinear time series forecasting of time delay neural network embedded with bayesian regularization abstract based on the idea of nonlinear prediction of phase space reconstruction this paper presents a time delay bp neural network model whose generalization capability is improved by bayesian regularization furthermore the model is applied to forecast the imp and exp trades in one industry the results show that the improved model has excellent generalization capabilities which not only learns of the historical curve but efficiently predicts the trend of business comparing with other forecasts we draw a conclusion that nonlinear forecast can not only focus on data combination and precision improvement but also vividly reflect the nonlinear characteristic of the forecasting system while analyzing the forecasting precision of the model we give a model judgment by calculating the nonlinear characteristic value of the combined serial and original serial which proves that the forecasting model can reasonably catch the dynamic characteristic of the nonlinear system which produces the origin serial

modeling uncertainty in multi criteria decision analysis abstract his paper provides a review of multiple criteria decision analysis mcda for cases where attribute evaluations are uncertain the main aim is to identify different tools which can be used to represent uncertain evaluations and to broadly survey the available decision models that can be used to support uncertain decision making the review includes models using probabilities or probability like quantities explicit risk measures such as quantiles and variances fuzzy numbers and scenarios the practical assessment of uncertain outcomes and preferences associated with these outcomes is also discussed

motif distributions in phase space networks for characterizing experimental two phase flow patterns with chaotic features the dynamics of two phase flows have been a challenging problem in nonlinear dynamics and fluid mechanics we propose a method to characterize and distinguish patterns from inclined water oil flow experiments based on the concept of network motifs that have found great usage in network science and systems biology in particular we construct from measured time series phase space complex networks and then calculate the distribution of a set of distinct network motifs to gain insight we first test the approach using time series from classical chaotic systems and find a universal feature motif distributions from different chaotic systems are generally highly heterogeneous our main finding is that the distributions from experimental two phase flows tend to be heterogeneous as well suggesting the underlying chaotic nature of the flow patterns calculation of the maximal lyapunov exponent provides further support for this motif distributions can thus be a feasible tool to understand the dynamics of realistic two phase flow patterns

outranking under uncertainty using scenarios abstract this paper considers the use of scenarios to treat uncertain attribute evaluations in the outranking methods the scenario based approach allows the decision maker to think deterministically about the problem by attaching causal links to a small number of potential outcomes instead of using probability distributions the scenario approach can be expressed as a simplified version of the comprehensive but practically complex distributive outranking method of d avignon and vincke using a scenario approach has distinct practical advantages but also presents the inherent danger that meaningful information is ignored the extent of this danger is assessed using a simulation experiment where it is found to be of a magnitude that is non trivial but still potentially acceptable for certain decision contexts

state space reconstruction parameters in the analysis of chaotic time series the role of the time window length abstract the most common state space reconstruction method in the analysis of chaotic time series is the method of delays mod many techniques have been suggested to estimate the parameters of mod i e the time delay and the embedding dimension m we discuss the applicability of these techniques with a critical view as to their validity and point out the necessity of determining the overall time window length w for successful embedding emphasis is put on the relation between w and the dynamics of the underlying chaotic system and we suggest to set w p the mean orbital period p is approximated from the oscillations of the time series the procedure is assessed using the correlation dimension for both synthetic and real data for clean synthetic data values of w larger than p always give good results given enough data and thus p can be considered as a lower limit w p for noisy synthetic data and real data an upper limit is reached for w which approaches p for increasing noise amplitude

from time series to complex networks the phase space coarse graining abstract in this paper we present a simple and fast computational method the phase space coarse graining algorithm that converts a time series into a directed and weighted complex network the constructed directed and weighted complex network inherits several properties of the series in its structure thereby periodic series convert into regular networks and random series do so into random networks moreover chaotic series convert into scale free networks it is shown that the phase space coarse graining algorithm allows us to distinguish identify and describe in detail various time series finally we apply the phase space coarse graining algorithm to the practical observations series international gasoline regular spot price series and identify its dynamic characteristics

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

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

analysis of nonlinear dynamics and chaos in a fractional order financial system with time delay in this paper a delayed fractional order financial system is proposed and the complex dynamical behaviors of such a system are discussed by numerical simulations a great variety of interesting dynamical behaviors of such a system including single periodic multiple periodic and chaotic motions are displayed in particular the effect of time delay on the chaotic behavior is investigated it is found that an approximate time delay can enhance or suppress the emergence of chaos meanwhile corresponding to different values of delay the lowest orders for chaos to exist in the delayed fractional order financial systems are determined respectively

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

extracting qualitative dynamics from experimental data abstract we consider the notion of qualitative information and the practicalities of extracting it from experimental data our approach based on a theorem of takens draws on ideas from the generalized theory of information known as singular system analysis due to bertero pike and co workers we illustrate our technique with numerical data from the chaotic regime of the lorenz model

a method for stochastic multiple criteria decision making based on pairwise comparisons of alternatives with random evaluations this paper proposes a method for solving stochastic multiple criteria decision making mcdm problems where evaluations of alternatives on considered criteria are random variables with known probability density functions or probability mass functions probabilities on all possible results of pairwise comparisons of alternatives are first calculated using probability theory then all possible results of pairwise comparisons are classified into superior indifferent and inferior ones using a predefined identification rule consequently the probabilities on all possible results of pairwise comparisons are partitioned into superior indifferent and inferior probabilities furthermore based on the derived probabilities an algorithm is developed to rank the alternatives finally a numerical example is used to illustrate the feasibility and validity of the proposed method

rigorous statistical procedures for data from dynamical systems various questions about the invariant measures of a dynamical system can be answered by computations of regular functionals or by ranking methods based on a set of observations this includes symmetry tests and the determination of dimension coefficients the paper contains the theoretical results and several simulations explain the methods

transfer entropy computation using the perron frobenius operator we propose a method for computing the transfer entropy between time series using ulam s approximation of the perron frobenius transfer operator associated with the map generating the dynamics our method differs from standard transfer entropy estimators in that the invariant measure is estimated not directly from the data points but from the invariant distribution of the transfer operator approximated from the data points for sparse time series and low embedding dimension the transfer operator is approximated using a triangulation of the attractor whereas for data rich time series or higher embedding dimension we use a faster grid approach we compare the performance of our methods with existing estimators such as the k nearest neighbors method and kernel density estimation method using coupled instances of well known chaotic systems coupled logistic maps and a coupled rossler lorenz system we find that our estimators are robust against moderate levels of noise for sparse time series with less than 100 observations and low embedding dimension our triangulation estimator shows improved ability to detect coupling directionality relative to standard transfer entropy estimators

reconstruction expansion as a geometry based framework for choosing proper delay times abstract the quality of attractor reconstruction using the method of delays is known to be sensitive to the delay parameter here we develop a new computationally efficient approach to choosing that quantifies reconstruction expansion from the identity line of the embedding space we show that reconstruction expansion is related to the concept of reconstruction signal strength and that increased expansion corresponds to diminished effects of measurement error thus reconstruction expansion represents a simple geometrical framework for choosing furthermore we describe the role of dynamical error in attractor expansion and argue that algorithms for determining should be considered as attempts at estimating an upper bound to the optimal delay

uncovering dynamic behaviors underlying experimental oil water two phase flow based on dynamic segmentation algorithm abstract characterizing complex dynamic behaviors arising from various inclined oil water two phase flow patterns is a challenging problem in the fields of nonlinear dynamics and fluid mechanics we systematically carried out inclined oil water two phase flow experiments for measuring the time series conductance fluctuating signals of different flow patterns we using the dynamic segmentation algorithm incorporating with phase space reconstruction analyze the measured experimental signals to uncover the dynamic behaviors underlying different flow patterns specifically given a time series from a two phase flow we move a sliding pointer over the time series and for each position of the pointer we calculate the dynamic difference measure of the phase space orbits generated from the segment to the left and to the right of the pointer a number of experimental signals under different flow conditions are investigated in order to reveal the dynamical characteristics of inclined oil water flows the results indicate that the heterogeneity of dynamic difference measure series is sensitive to the transition among different flow patterns and the standard deviation of dynamic difference measure series can yield quantitative insights into the nonlinear dynamics of the two phase flow these properties render the dynamic segmentation algorithm based approach particularly useful for uncovering the dynamic behaviors of inclined oil water two phase flows

exploring dynamic property of traffic flow time series in multi states based on complex networks phase space reconstruction versus visibility graph abstract a new method based on complex network theory is proposed to analyze traffic flow time series in different states we use the data collected from loop detectors on freeway to establish traffic flow model and classify the flow into three states based on k means method we then introduced two widely used methods to convert time series into networks phase space reconstruction and visibility graph furthermore in phase space reconstruction we discuss how to determine delay time constant and embedding dimension and how to select optimal critical threshold in terms of cumulative degree distribution in the visibility graph we design a method to construct network from multi variables time series based on logical or finally we study and compare the statistic features of the networks converted from original traffic time series in three states based on phase space and visibility by using the degree distribution network structure correlation of the cluster coefficient to betweenness and degree degree correlation

evidence of chaotic behavior in noise from industrial process usually in dealing with the dynamical behavior of noise from an industrial process nfip random or even white noise is assumed to describe it however since white noise is just an ideal model in mathematics and can hardly exist in an industrial process this assumption seriously hinders the successful application of classical control theory in practice for this reason nonlinear time series analysis is used to analyze nfip in an attempt to capture the signature of chaos the chaotic behavior is apparent in nfip due to the presence of finite and non integer correlation dimension and the minimum number of macroscopic variables required to model nfip dynamics is obtained these results indicate that nfip is deterministic and high dimensional chaos but not random or white noise and in the future might lead the way towards the direct solution of prediction and modeling of nfip and a series of changes to the traditional algorithms in the control field may appear

nonlinear similarity analysis for epileptic seizures prediction abstract the prediction of epileptic seizures can promise a new diagnostic application and a novel approach for seizure control this paper proposes an improved dynamical similarity measure to predict epileptic seizures in electroencephalographic eeg first mutual information and cao s method are employed to reconstruct a phase space of preprocessed eeg recordings by using the positive zero crossing method second a gaussian function replaces the heavyside function within correlation integral at calculating a similarity index the crisp boundary of the heavyside function is eliminated because of the gaussian function s smooth boundary third an adaptive detection method based on the similarity index is proposed to predict the epileptic seizures in light of test results of eeg recordings of rats it is found that the new dynamical similarity index is insensitive to the selection of the radius value of gaussian function and the size of segmented eeg recordings comparing with the dynamical similarity index proposed by le van quyen et al anticipating epileptic seizures in real time by a non linear analysis of similarity between eeg recordings neuroreport 10 1999 2149 2155 the tests of twelve rats show the new dynamical similarity index is better to predict the epileptic seizures

characterizing traffic time series based on complex network theory abstract a complex network is a powerful tool to research complex systems traffic flow being one of the most complex systems in this paper we use complex network theory to study traffic time series which provide a new insight into traffic flow analysis firstly the phase space which describes the evolution of the behavior of a nonlinear system is reconstructed using the delay embedding theorem secondly in order to convert the new time series into a complex network the critical threshold is estimated by the characteristics of a complex network which include degree distribution cumulative degree distribution and density and clustering coefficients we find that the degree distribution of associated complex network can be fitted with a gaussian function and the cumulative degree distribution can be fitted with an exponential function density and clustering coefficients are then researched to reflect the change of connections between nodes in complex network and the results are in accordance with the observation of the plot of an adjacent matrix consequently based on complex network analysis the proper range of the critical threshold is determined finally to mine the nodes with the closest relations in a complex network the modularity is calculated with the increase of critical threshold and the community structure is detected according to the optimal modularity the work in our paper provides a new way to understand the dynamics of traffic time series

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

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

value of additional information in multicriterion analysis under uncertainty abstract the expected value of information in classical monocriterion decision analysis has been well covered in the literature one cannot say the same thing about the multicriterion analysis particularly when one is in the presence of multicriterion aggregation procedures based on outranking relations for a ranking problematic the objective of this paper is to try to extend the bayesian approach to a multicriterion analysis in the context of uncertainty after illustrating the a posteriori analysis we shall mention some difficulties associated with the pre a posteriori analysis and the concepts of the expected value of perfect or imperfect information