on the descending central sequence of absolute galois groups let p be an odd prime number and f a field containing a primitive p th root of unity we prove a new restriction on the group theoretic structure of the absolute galois group g f of f namely the third subgroup g f 3 in the descending p central sequence of g f is the intersection of all open normal subgroups n such that g f n is 1 bbb z p 2 or the extra special group m p 3 of order p 3 and exponent p 2

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

filtering and frequency interpretations of singular spectrum analysis new filtering and spectral interpretations of singular spectrum analysis ssa are provided it is shown that the variables reconstructed from diagonal averaging of reduced rank approximations to the trajectory matrix can be obtained from a noncausal convolution filter with zero phase characteristics the reconstructed variables are readily constructed using a two pass filtering algorithm that is well known in the signal processing literature when the number of rows in the trajectory matrix is much larger than number of columns many results reported in the signal processing literature can be used to derive the properties of the resulting filters and their spectra new features of the reconstructed series are revealed using these results two examples are used to illustrate the results derived in this paper

an analytic approach to practical state space reconstruction we study the three standard methods for reconstructing a state space from a time series delays derivatives and principal components we derive a closed form solution to principal component analysis in the limit of small window widths this solution explains the relationship between delays derivatives and principal components it shows how the singular spectrum scales with dimension and delay time and it explains why the eigenvectors resemble the legendre polynomials most importantly the solution allows us to derive a guideline for choosing a good window width unlike previous suggestions this guideline is based on first principles and simple quantities we argue that discrete legendre polynomials provide a quick and not so dirty substitute for principal component analysis and that they are a good practical method for state space reconstruction

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

a note on the gamma test analysis of noisy input output data and noisy time series abstract in a smooth input output process y f x if the input data x r d is noise free and only the output data y is corrupted by noise then a near optimal smooth model g will be a close approximation to f however as previously observed for example in h kantz t schreiber nonlinear time series analysis 2nd ed cambridge univ press 2004 if the input data is also corrupted by noise then this is no longer the case with noise on the inputs the best predictive smooth model based on noisy data need not be an approximation to the actual underlying process rather the best predictive model depends on both the underlying process and the noise a corollary of this observation is that one cannot readily infer the nature of a process from noisy data since almost all data has associated noise this conclusion has some unsettling implications in this note we show how these effects can be quantified using the gamma test in particular we examine the gamma test analysis of noisy time series data we show that the noise level on the best predictive smooth model based on the noisy data can be much higher than the noise level on individual time series measurements and we give an upper bound for the first in terms of the second

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

comparative study of embedding methods embedding experimental data is a common first step in many forms of dynamical analysis the choice of appropriate embedding parameters dimension and lag is crucial to the success of the subsequent analysis we argue here that the optimal embedding of a time series cannot be determined by criteria based solely on the time series itself therefore we base our analysis on an examination of systems that have explicit analytic representations a comparison of analytically obtained results with those obtained by an examination of the corresponding time series provides a means of assessing the comparative success of different embedding criteria the assessment also includes measures of robustness to noise the limitations of this study are explicitly delineated while bearing these limitations in mind we conclude that for the examples considered here the best identification of the embedding dimension was achieved with a global false nearest neighbors argument and the best value of lag was identified by the mutual information function

sampling period statistical complexity and chaotic attractors abstract we analyze the statistical complexity measure vs entropy plane representation of sampled chaotic attractors as a function of the sampling period and show that if the bandt and pompe procedure is used to assign a probability distribution function pdf to the pertinent time series the statistical complexity measure scm attains a definite maximum for a specific sampling period t m on the contrary the usual histogram approach for assigning pdfs to a time series leads to essentially constant scm values for any sampling period the significance of t m is further investigated by comparing it with typical times found in the literature for the two main reconstruction processes the takens one in a delay time embedding on one hand and the exact nyquist shannon reconstruction on the other one it is shown that t m is compatible with those times recommended as adequate delay ones in takens reconstruction the reported results correspond to three representative chaotic systems having correlation dimension 2 d 2 3 one recent experiment confirms the analysis presented here

multifractal detrended cross correlation analysis for two nonstationary signals we propose a method called multifractal detrended cross correlation analysis to investigate the multifractal behaviors in the power law cross correlations between two time series or higher dimensional quantities recorded simultaneously which can be applied to diverse complex systems such as turbulence finance ecology physiology geophysics and so on the method is validated with cross correlated one and two dimensional binomial measures and multifractal random walks as an example we illustrate the method by analyzing two financial time series

reconstructing function fields from rational quotients of mod ell galois groups in this paper we develop the main step in the global theory for the mod ell analogue of bogomolov s program in birational anabelian geometry for higher dimensional function fields over algebraically closed fields more precisely we show how to reconstruct a function field k of transcendence degree ge 5 over an algebraically closed field up to inseparable extensions from the mod ell abelian by central galois group of k endowed with the collection of mod ell rational quotients

distortions of reconstruction for chaotic attractors abstract we consider the problem of quality evaluation for delay reconstruction of chaotic attractors from scalar time series the proposed approach is based upon local linear analysis of the reconstruction mapping it is shown that some linear distortions in principle could be corrected but usually nonlinear distortions make it dificult to efficiently apply this technique the new algorithms of estimating the scale of nonlinear distortion and irrelevance time upper limit for the reconstruction window from a time series are proposed both of them do not require pricise knowledge of minimal embedding dimension for the attractor studied

a fast general purpose algorithm for the computation of auto and cross correlation integrals from single channel data abstract we developed an optimized algorithm that allows computation of auto and cross correlation integrals from single channel time series without restricting the range of hypersphere radii and embedding dimensions optimization was achieved by eliminating any multiple computation of subsets entering the distance function in the time delay reconstruction of the phase space repeatedly for increasing embedding dimensions this is most effective when using the maximum norm compared to more naive implementations an improvement of 2 3 was achieved depending on the type of workstation the operating system and the compiler an additional optimization for 80 86 assembly language allows torun the algorithm on a standard personal computer as fast as on a workstation in contrast to other implementations the execution speed of this algorithm is nearly unaffected by the type of underlying data thus it is optimal for real time analyses

estimating the embedding dimension abstract well reconstructed dynamics should be described as a mapping we propose to search for the lowest embedding dimension for which the property of a continuous mapping is satisfied the images of close points are close we demonstrate the effectiveness of the approach by finding the lowest embedding dimension of several chaotic systems embedded by the method of delays

dimension measurement on high dimensional systems abstract using experimental data from a variety of laboratory electrical systems the upper limits on dimension measurement are examined and it is shown that contrary to standard scaling estimates it can be possible to reliably measure correlation dimensions larger than 10 the practical implications of the constraints on such a measurement are examined including the role of statistical errors geometrical artifacts dimension algorithm metric choice and the process power spectrum these ideas are applied to electrical 1 f noise from both macroscopic and microscopic sources to show that it cannot be explained by a deterministic model with fewer than roughly 20 degrees of freedom

direct parameterization of postural stability during quiet upright stance effects of age and altered sensory conditions the purpose of this study was to determine the relationship between measures of local dynamic stability lds during upright stance and both descriptive measures of postural sway and a scaling index derived from detrended fluctuation analysis center of pressure cop time series were obtained from healthy participants 16 young and 16 older during upright quiet stance vision and somatosensation were altered by eye closure and standing on a compliant surface respectively a non linear time series analysis method was used to compute three lds parameters from the cop data a which was defined as the cop excursion range in state space and s and l which were defined as the divergence rates over short and long term timescales respectively lds parameters descriptive cop measures and had generally consistent sensitivities to age and or altered sensory conditions age x sensory condition interactions however had distinct effects on lds parameters compared to the other cop based measures older individuals exhibited faster divergence rates while having similar magnitudes of a compared to young individuals these results suggest that older individuals stiffen the musculoskeletal system via increased muscle activity perhaps as an age related postural adaptation in addition correlations between lds parameters and other cop measures were relatively small r2 0 29 hence lds parameters a s and l provide distinct information on postural control and stability supplementing other cop based measures

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

characterization of time dynamical evolution of electroencephalographic epileptic records abstract since traditional electrical brain signal analysis is mostly qualitative the development of new quantitative methods is crucial for restricting the subjectivity in the study of brain signals these methods are particularly fruitful when they are strongly correlated with intuitive physical concepts that allow a better understanding of the brain dynamics the processing of information by the brain is reflected in dynamical changes of the electrical activity in time frequency and space therefore the concomitant studies require methods capable of describing the qualitative variation of the signal in both time and frequency the entropy defined from the wavelet functions is a measure of the order disorder degree present in a time series in consequence this entropy evaluates over eeg time series gives information about the underlying dynamical process in the brain more specifically of the synchrony of the group cells involved in the different neural responses the total wavelet entropy results independent of the signal energy and becomes a good tool for detecting dynamical changes in the system behavior in addition the total wavelet entropy has advantages over the lyapunov exponents because it is parameter free and independent of the stationarity of the time series in this work we compared the results of the time evolution of the chaoticity lyapunov exponent as a function of time with the corresponding time evolution of the total wavelet entropy in two different eeg records one provide by depth electrodes and other by scalp ones

stabilizing embedology geometry preserving delay coordinate maps delay coordinate mapping is an effective and widely used technique for reconstructing and analyzing the dynamics of a nonlinear system based on time series outputs the efficacy of delay coordinate mapping has long been supported by takens embedding theorem which guarantees that delay coordinate maps use the time series output to provide a reconstruction of the hidden state space that is a one to one embedding of the system s attractor while this topological guarantee ensures that distinct points in the reconstruction correspond to distinct points in the original state space it does not characterize the quality of this embedding or illuminate how the specific parameters affect the reconstruction in this paper we extend takens result by establishing conditions under which delay coordinate mapping is guaranteed to provide a stable embedding of a system s attractor beyond only preserving the attractor topology a stable embedding preserves the attractor geometry by ensuring that distances between points in the state space are approximately preserved in particular we find that delay coordinate mapping stably embeds an attractor of a dynamical system if the stable rank of the system is large enough to be proportional to the dimension of the attractor the stable rank reflects the relation between the sampling interval and the number of delays in delay coordinate mapping our theoretical findings give guidance to choosing system parameters echoing the tradeoff between irrelevancy and redundancy that has been heuristically investigated in the literature our initial result is stated for attractors that are smooth submanifolds of euclidean space with extensions provided for the case of strange attractors

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

topology from time series we describe methods for the study of topological properties of the invariant manifolds of experimental dynamical systems we explain how to compute such invariants as the euler characteristic and betti numbers using time series data and suggest a number of potential applications

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

multifractal fourier detrended cross correlation analysis of traffic signals abstract multifractal detrended cross correlation analysis mf dxa has been developed to detect the long range power law cross correlation of considered signals in the presence of non stationarity however crossovers arising from extrinsic periodic trends make the scaling behavior difficult to analyze we introduce a fourier filtering method to eliminate the trend effects and systematically investigate the multifractal cross correlation of simulated and real traffic signals the crossover locations are found approximately corresponding to the periods of underlying trend traffic velocity on one road and flows on adjacent roads show strong cross correlation they also present weak multifractality after periodic trends are removed the traffic velocity and flow are cross correlated in opposite directions which is accordant to their actual evolution

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

reliable detection of nonlinearity in experimental time series with strong periodic components abstract testing with phase randomised surrogate signals has been used extensively to search for interesting nonlinear dynamical structure in experimental time series in this paper we argue that in the case of experimental time series with strong periodic components the method of phase randomised surrogate data may not be particularly suitable to test for nonlinearity since construction of such surrogates by fft requires a time series whose length is a power of 2 we demonstrate that in the case of nearly periodic signals this approach will almost always produce spurious detection of nonlinearity this error can be fixed by adjusting the length of the time series such that it becomes an integer multiple of the dominant periodicity the resulting time series will not be a power of 2 and requires the use of a dft to generate surrogate data dft based surrogates no longer detect spurious nonlinearity but cannot be used to detect periodic nonlinearity we propose a new test nonlinear cross prediction nlcp which avoids some of the problems associated with phase randomised surrogate data and which allows reliable detection of both periodic and aperiodic nonlinearity in the test the original data are used to construct a nonlinear model to predict the original data set as well as amplitude inverted and time reversed versions of the original data lower predictability of the amplitude inverted or time reversed copies reflect respectively an asymmetric amplitude distribution and time irreversibility both of these indicate nonlinearity in the data set

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

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

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

minute ahead stock price forecasting based on singular spectrum analysis and support vector regression abstract time series modeling and forecasting is an essential and hard task in financial engineering and optimization various models have been proposed in the literature and tested on daily data however a limited attention has been given to intraday data in this regard the current work presents a model for intraday stock price prediction that uses singular spectrum analysis ssa and support vector regression svr coupled with particle swarm optimization pso in particular the ssa decomposes stock price time series into a small number of independent components used as predictors the svr is applied to the task of forecasting and pso is employed to optimize svr parameters the performance of our proposed model is compared to the performance of four models widely used in financial prediction the wavelet transform wt coupled with feedforward neural network ffnn autoregressive moving average arma process polynomial regression polyreg and naive model finally the mean absolute error mae mean absolute percentage error mape and the root mean of squared errors rmse are used as main performance metrics by applying all models to six intraday stock price time series the forecasting results from simulations show that the presented ssa pso svr largely outperforms the conventional wt ffnn arma polynomial regression and naive model in terms of mae mape and rmse therefore our proposed predictive system ssa pso svr shows evident potential for noisy financial time series analysis and forecasting

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

on the non equivalence of observables in phase space reconstructions from recorded time series in practical problems of phase space reconstruction it is usually the case that the reconstruction is much easier using a particular recorded scalar variable this seems to contradict the general belief that all variables of a dynamical system are equivalent in phase space reconstruction problems this paper will argue that in many cases the choice of a particular scalar time series from which to reconstruct the original dynamics could be critical it is argued that different dynamical variables do not provide the same level of information observability of the underlying dynamics and as a consequence the quality of a global reconstruction critically depends on the recorded variable examples in which the choice of observables is critical are discussed and the level of information contained in a given variable is quantified in the case where the original system is known a clear example of such a situation arises in the r ossler system for which the performance of a global vector field reconstruction technique is investigated using time series of variables x y or z taken one at a time

using higher order correlations to define an embedding window abstract correlation functions of up to fourth order are investigated numerically in search of a reliable window for embedding or reconstructing chaotic flows by the method of time delays a number of these higher order correlations display coincident extrema the time at which this coincidence occurs is shown to be a reasonable embedding window for calculating the correlation dimension of a number of well known chaotic attractors

dimensions and entropies of strange attractors from a fluctuating dynamics approach abstract it is shown that the fluctuations in the divergence of near by trajectories on strictly deterministic strange attractors can be modelled by stochastic concepts in particular we propose kramers moyal type equations for correlation functions between points on the attractor the drift terms are the lyapunov exponents the diffusion terms depend on the above fluctuations from this we obtain bounds on generalized dimensions and entropies numerical results show that in nearly all studied cases henon map zaslavskii map mackey glass eq the attractors are fractal measures in the sense of farmer information dimension hausdorff dimension metric entropy topological entropy

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

bursting as a source for predictability in biological neural network activity abstract the role of bursting as a unit of neural information has received considerable support in the recent years experimental evidence shows that in many different neural systems e g visual cortex or hippocampus bursting is essential for coding and processing we have recently demonstrated menendez de la prida et al 1996 the spontaneous presence of bursts in in vitro hippocampal slices from newborn animals providing a good system to investigate bursting dynamics in physiological conditions here we analyze the interspike intervals isis of five intracellularly recorded cells from immature hippocampal networks first we test the time series against poisson processes typical of pure random behavior using the kolmogorov smirnov test only 2 5 records strongly deviate from poisson process nonlinear diction tests are then applied to compare original series with its gaussian scaled random phase surrogates and signs of short time predictability are observed 1 5 this predictability is originated by the intrinsic structure of bursts in an otherwise purely random process and can be removed completely by eliminating the bursts from the original time series here we introduce this method of eliminating bursts to get insight into the nonlinear dynamics of firing also the interburst intervals are indistinguishable from pure noise the analysis of unstable periodicities within the bursts in the original isis shows that signs of nonlinearities can be statistically differentiated from their surrogate realizations pierson moss method we discuss the computational implication of these results

some extensions of radial basis functions and their applications in artificial intelligence abstract radial basis functions have recently found interesting applications in artificial intelligence and in particular in the problem of learning to perform a particular task from a set of examples however in many practical cases the radial basis functions method cannot be applied in a straight forward manner because it does not take into account some features that are typical of the problem of learning from examples in this paper we show some extensions of the standard theory introduced in order to deal with a large number of examples and with problems in which different variables play very different roles we present some examples and also point out some open problems

topological properties and fractal analysis of a recurrence network constructed from fractional brownian motions many studies have shown that we can gain additional information on time series by investigating their accompanying complex networks in this work we investigate the fundamental topological and fractal properties of recurrence networks constructed from fractional brownian motions fbms first our results indicate that the constructed recurrence networks have exponential degree distributions the average degree exponent increases first and then decreases with the increase of hurst index h of the associated fbms the relationship between h and can be represented by a cubic polynomial function we next focus on the motif rank distribution of recurrence networks so that we can better understand networks at the local structure level we find the interesting superfamily phenomenon i e the recurrence networks with the same motif rank pattern being grouped into two superfamilies last we numerically analyze the fractal and multifractal properties of recurrence networks we find that the average fractal dimension db of recurrence networks decreases with the hurst index h of the associated fbms and their dependence approximately satisfies the linear formula db 2 h which means that the fractal dimension of the associated recurrence network is close to that of the graph of the fbm moreover our numerical results of multifractal analysis show that the multifractality exists in these recurrence networks and the multifractality of these networks becomes stronger at first and then weaker when the hurst index of the associated time series becomes larger from 0 4 to 0 95 in particular the recurrence network with the hurst index h 0 5 possesses the strongest multifractality in addition the dependence relationships of the average information dimension d 1 and the average correlation dimension d 2 on the hurst index h can also be fitted well with linear functions our results strongly suggest that the recurrence network inherits the basic characteristic and the fractal nature of the associated fbm series

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

statistical analysis of geopotential height gh timeseries based on tsallis non extensive statistical mechanics abstract in this paper we perform statistical analysis of time series deriving from earth s climate the time series are concerned with geopotential height gh and correspond to temporal and spatial components of the global distribution of month average values during the period 1948 2012 the analysis is based on tsallis non extensive statistical mechanics and in particular on the estimation of tsallis q triplet namely q stat q sens q rel the reconstructed phase space and the estimation of correlation dimension and the hurst exponent of rescaled range analysis r s the deviation of tsallis q triplet from unity indicates non gaussian tsallis q gaussian non extensive character with heavy tails probability density functions pdfs multifractal behavior and long range dependences for all timeseries considered also noticeable differences of the q triplet estimation found in the timeseries at distinct local or temporal regions moreover in the reconstructive phase space revealed a lower dimensional fractal set in the gh dynamical phase space strong self organization and the estimation of hurst exponent indicated multifractality non gaussianity and persistence the analysis is giving significant information identifying and characterizing the dynamical characteristics of the earth s climate

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

finite tripod variants of i om on ihara s question oda matsumoto conjecture in this note we introduce and prove a wide generalization and sharpening of ihara s question oda matsumoto conjecture for short i om that leads to a quite concrete topological combinatorial description of absolute galois groups in particular of mathrm gal mathbb q mathrm aut overline mathbb q as envisioned by grothendieck in his esquisse d un programme

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

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

reconstructing attractors from filtered time series abstract in this paper we investigate the criteria necessary to reconstruct an attractor from a time series that has been passed through an iir filter to guarantee that the whole of the attractor is reconstructed requires uniform bounds on the contraction rate of the dynamics however we show that if we use a condition that is only based on the average contraction rates the lyapunov exponents then a set of high probability can still be reconstructed we argue that in practice this is all that is necessary

effective scaling regime for compution the correlating dimension from chaotic time series abstract in the analysis of chaotic time series a standard technique is to reconstruct an image of the original dynamical system using delay coordinates if the original dynamical system has an attractor then the correlation dimension d2 of its image in the reconstruction can be estimated using the grassberger procaccia algorithm the quality of the reconstruction can be probed by measuring the length of the linear scaling region used in this estimation in this paper we show that the quality is constrained by both the embedding dimension m and more importantly by the delay time for a given embedding dimension and a finite time series there exists a maximum allowed delay time beyond which the size of the scaling region is no longer reliably discernible we derive an upper bound for this maximum delay time numerical experiments on several model chaotic time series support the theoretical argument they also clearly indicate the different roles played by the embedding dimension and the delay time in the reconstruction as the embedding dimension is increased it is necessary to reduce the delay time substantially to guarantee a reliable estimate of d2 our results imply that it is the delay time itself rather than the total observation time m 1 which plays the most critical role in the determination of the correlation dimension

interpreting supply chain dynamics a quasi chaos perspective abstract chaotic phenomena chaos amplification and other interesting nonlinear behaviors have been observed in supply chain systems chaos can be defined theoretically if the dynamics under study are produced only by deterministic factors however deterministic settings rarely present themselves in reality in fact real data are typically unknown how can the chaos theory and its related methodology be applied in the real world when the demand is stochastic the interpretation and distribution of the lyapunov exponents derived from the effective inventory at different supply chain levels are not similar to those under deterministic demand settings are the observed dynamics of the effective inventory random chaotic or simply quasi chaos in this study we investigate a situation whereby the chaos analysis is applied to a time series as if its underlying structure deterministic or stochastic is unknown the result shows clear distinction in chaos characterization between the two categories of demand process deterministic vs stochastic it also highlights the complexity of the interplay between stochastic demand processes and nonlinear dynamics therefore caution should be exercised in interpreting system dynamics when applying chaos analysis to a system of unknown underlying structure by understanding this delicate interplay decision makers have the better chance to tackle the problem correctly or more effectively at the demand end or the supply end

reliability of recurrence quantification analysis of postural sway data a comparison of two methods to determine recurrence thresholds ageing affects balance control resulting in a greater amplitude of sway and alterations in structure of the sway time series recurrence quantification analysis rqa has been used to determine the structure of center of pressure cop a measure that reflects standing postural control data as a means to quantify how cop repeats itself recurs below a certain threshold this study aimed to determine how the method of threshold determination below which a recurrence is defined affects the within session reliability of rqa in an elderly population within session reliability of rqa of cop motion in the anterior posterior and mediolateral directions was assessed in 267 individuals 65 years old when standing on firm or foam surface with eyes open or closed for each of two recurrence threshold methods one threshold method sets the recurrence threshold level such that the recurrence rate is fixed to 5 the other method sets the recurrence threshold based on 27 of the mean distance between all points from which recurrences are quantified reliability across four 30 s balance trials within each of four balance conditions firm vs foam eyes open vs closed was determined using intra class correlation standard error of measurement and minimal detectable change iccs were better the standard error of measurement and minimal detectable change were smaller when the recurrence threshold was set to 5 using the fixed recurrence threshold fixing recurrence rate improves the within session reliability of rqa and could increase sensitivity to identify fall risk

indications of nonlinearity intraindividual specificity and stability of human eeg the unfolding dimension abstract with the help of a parameter which we call the unfolding dimension ww provide a bivariate representation of dynamical eeg analysis applied to human eeg our approach successfully discriminates surrogate data from raw eeg and similarly shows human eeg to be both intraindividually specific and stable over time the heart of our approach is the grassberger procaccia algorithm for the determination of the correlation dimension d 2 within the context of the well known method of time delays to guarantee the reproducibility of results this algorithm as well as an estimate for the k 2 entropy and the determination of f spectra have been integrated into a computer program which encompasses an operator user independent automatic and reproducible specification of both an optimal time delay for calculating the correlation integral as well as an optimal scalar invariant plateau region for the extraction of d 2 and k 2 our embedding protocol applied to mathematical systems is shown to be consistent with findings based on other algorithms in particular with false nearest neighbor considerations and sauer s minimal embedding criteria particular intelligent features of the automation and optimization logic plus the inclusion of estimates of the inherent systematic error make the proposed algorithm especially appropriate for applications to biological psychological data

a practical method for calculating largest lyapunov exponents from small data sets detecting the presence of chaos in a dynamical system is an important problem that is solved by measuring the largest lyapunov exponent lyapunov exponents quantify the exponential divergence of initially close state space trajectories and estimate the amount of chaos in a system we present a new method for calculating the largest lyapunov exponent from an experimental time series the method follows directly from the definition of the largest lyapunov exponent and is accurate because it takes advantage of all the available data we show that the algorithm is fast easy to implement and robust to changes in the following quantities embedding dimension size of data set reconstruction delay and noise level furthermore one may use the algorithm to calculate simultaneously the correlation dimension thus one sequence of computations will yield an estimate of both the level of chaos and the system complexity

construction and analysis of differential equations from experimental time series of oscillatory systems abstract we propose a novel method to derive an autonomous system of ordinary differential equations from an experimental time series this model describe quantitatively transient behaviour of a system with limit cycle dynamics and is simplified to an evolution equation for the amplitude and frequency the landau equation in addition the model is used for a qualitative prediction of the influence of periodic excitation on the experimental system using the van der pol transformation and floquet theory in particular the instability of the locking solution including the hopf bifurcation the intermittency scenario and the period doubling cascade is considered the method of constructing a differential equation from an experimental time signal is successfully applied to a velocity fluctuation of the laminar periodic wake behind a circular cylinder

an approximate analytic solution of the lorenz equations multiple time perturbation theory is applied to the lorenz equations to obtain approximate solutions these solutions are used to obtain the form of the mapping relating two successive maxima of one of the variables of the equations good agreement is obtained with the numerical results presented by lorenz

assessing gait stability the influence of state space reconstruction on inter and intra day reliability of local dynamic stability during over ground walking estimating local dynamic stability is considered a powerful approach to identify persons with balance impairments its validity has been studied extensively and provides evidence that short term local dynamic stability is related to balance impairments and the risk of falling thus far however this relation has only been proven on group level for clinical use differences on the individual level should also be detectable requiring reliability to be high in the current study reliability of short term local dynamic stability was investigated within and between days participants walked 500 m back and forth on a straight outdoor footpath on 2 non consecutive days and 3d linear accelerations were measured using an accelerometer dynaport minimod the state space was reconstructed using 4 common approaches all based on delay embedding within session intra class correlation coefficients were good 0 70 however between session intra class correlation coefficients were poor to moderate 0 63 and influenced by the reconstruction method the same holds for the smallest detectable difference which ranged from 17 to 46 depending on the state space reconstruction method the best within and between session intra class correlation coefficients and smallest detectable differences were achieved with a state space reconstruction with a fixed time delay and number of embedding dimensions overall due to the influence of biological variation and measurement error the short term local dynamic stability can only be used to detect substantial differences on the individual level

comparison of algorithms calculating optimal embedding parameters for delay time coordinates the success of noise reduction processes and forecasting methods as well as accurate calculations of fractal dimensions and lyapunov exponents from scalar time series depends strongly on the quality of the reconstructed phase space we present a comparison of four methods which simultaneously estimate the two embedding parameters delay time and sufficiently large embedding dimension dime for takens delay time coordinates recently we introduced a global geometrical measure the fill factor and a local dynamical measure the averaged integral local deformation we will discuss these methods briefly and compare them to the topological wavering product in addition a simple algorithm is presented which observes the spreading of trajectories at the transition from dime to dime 1 and gives the opportunity to estimate the correlation entropy k2 we applied these algorithms to experimental data series measured from a rotational taylor couette experiment and verified the quality of the obtained embedding parameters by the estimation of correlation dimensions for different reconstructed attractors

faint phase synchronization detection through structured orthomax rotations in singular spectrum analysis multivariate singular spectrum analysis m ssa with a structured varimax rotation is a method that allows a deep characterization of phase synchronization ps phenomena in an almost automatic fashion it has been increasingly used in the study of ps in networks of nonlinear real world and numeric systems this paper investigates the impact of the other recently developed structured orthomax rotations on the m ssa ability to characterize ps the results show that by using the structured quartimax rotation a very faint and intermittent ps regime can be detected in contrast with the structured varimax which demands a stronger more consolidated ps regime this is due to the fact that the different rotations do not have the same efficiency in achieving a simple structure of the m ssa eigenvectors nevertheless for well established ps regimes the same robustness of the original m ssa approach against high levels of additive gaussian noise was found for the structured quartimax and biquartimax rotations however for all approaches we found an overshoot of the qualitative range for the ps onset due to noise

the galois action on geometric lattices and the mod ell i om this paper studies the galois action on a special lattice of geometric origin which is related to mod ell abelian by central quotients of geometric fundamental groups of varieties as a consequence we formulate and prove the mod ell abelian by central variant strengthening of a conjecture due to ihara oda matsumoto

distinguishing between low dimensional dynamics and randomness in measured time series the success of current attempts to distinguish between low dimensional chaos and random behavior in a time series of observations is considered first we discuss stationary stochastic processes which produce finite numerical estimates of the correlation dimension and k2 entropy under naive application of correlation integral methods we then consider several straightforward tests to evaluate whether correlation integral methods reflect the global geometry or the local fractal structure of the trajectory this determines whether the methods are applicable to a given series if they are we evaluate the significance of a particular result for example by considering the results of the analysis of stochastic signals with statistical properties similar to those of observed series from the examples considered it is clear that the correlation integral should not be used in isolation but as one of a collection of tools to distinguish chaos from stochasticity

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

flow transitions resembling bifurcations of the logistic map in simulations of the baroclinic rotating annulus we present evidence for a sequence of bifurcations in simulations of the differentially heated baroclinic rotating annulus similar to bifurcations of the logistic map the met office oxford rotating annulus laboratory simulation morals code is used to construct a detailed numerical regime diagram for the annulus and the distribution of regimes in parameter space is discussed the bifurcations are observed in a sequence of runs at high temperature forcing identified by poincare sections of the dominant temperature mode amplitude time series higher order return maps and predictions using quadratic fits to the data are used to verify this result and lyapunov exponents are calculated to identify and quantify the chaotic parts of the sequence

approximating stable and unstable manifolds in experiments we introduce a procedure to reveal invariant stable and unstable manifolds given only experimental data we assume a model is not available and show how coordinate delay embedding coupled with invariant phase space regions can be used to construct stable and unstable manifolds of an embedded saddle we show that the method is able to capture the fine structure of the manifold is independent of dimension and is efficient relative to previous techniques

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

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

feasibility of real time calculation of correlation integral derived statistics applied to eeg time series abstract background this study assessed the feasibility of online calculation of the correlation integral c r aiming to apply c r derived statistics for real time application it is important to reduce calculation time it is shown how our method works for eeg time series methods to achieve online calculation of c r a non randomly subset of inter vector distances was chosen and computer code was optimized the effect of distance exclusion was investigated for both non randomly and randomly chosen subsets a c r derived statistic was computed an effective correlation dimension cd following the grassberger procaccia takens approach results by taking a subset of the maximum possible number of distances c r computation time could be easily 100 times reduced with marginal changes and minor variability in the c r derived statistic cd applied to the eeg cd gives a good indication of the depth of anesthesia conclusions if applied to the eeg apparently a large number of distances can be omitted in the calculation of c r with minimal consequences this outcome confirms hoeffding s theory of u statistics and hence it is expected to occur generally also for non eeg time series