on consistent nonparametric order determination and chaos summary we give a brief introduction to deterministic chaos and a link between chaotic deterministic models and stochastic time series models we argue that it is often natural to determine the embedding dimension in a noisy environment first in any systematic study of chaos setting the stochastic models within the framework of non linear autoregression we introduce the notion of a generalized partial autocorrelation and an order we approach the estimation of the embedding dimension via order determination of an unknown non linear autoregression by cross validation and give justification by proving its consistency under global boundedness as a by product we provide a theoretical justification of the final prediction error approach of auestad and tj0stheim some illustrations based on the henon map and several real data sets are given the bias of the residual sum of squares as essentially a noise variance estimator is quantified

nonlinear analysis of river flow time sequences within the field of chaos theory several methods for the analysis of complex dynamical systems have recently been proposed in light of these ideas we study the dynamics which control the behavior over time of river flow investigating the existence of a low dimension deterministic component the present article follows the research undertaken in the work of porporato and ridolfi 1996a in which some clues as to the existence of chaos were collected particular emphasis is given here to the problem of noise and to nonlinear prediction with regard to the latter the benefits obtainable by means of the interpolation of the available time series are reported and the remarkable predictive results attained with this nonlinear method are shown

nonlinear analysis of irregular animal vocalizations animal vocalizations range from almost periodic vocal fold vibration to completely atonal turbulent noise between these two extremes a variety of nonlinear dynamics such as limit cycles subharmonics biphonation and chaotic episodes have been recently observed these observations imply possible functional roles of nonlinear dynamics in animal acoustic communication nonlinear dynamics may also provide insight into the degree to which detailed features of vocalizations are under close neural control as opposed to more directly reflecting biomechanical properties of the vibrating vocal folds themselves so far nonlinear dynamical structures of animal voices have been mainly studied with spectrograms in this study the deterministic versus stochastic dvs prediction technique was used to quantify the amount of nonlinearity in three animal vocalizations macaque screams piglet screams and dog barks results showed that in vocalizations with pronounced harmonic components adult macaque screams certain piglet screams and dog barks deterministic nonlinear prediction was clearly more powerful than stochastic linear prediction the difference termed low dimensional nonlinearity measure lnm indicates the presence of a low dimensional attractor in highly irregular signals such as juvenile macaque screams piglet screams and some dog barks the detectable amount of nonlinearity was comparatively small analyzing 120 samples of dog barks it was further shown that the harmonic to noise ratio hnr was positively correlated with lnm it is concluded that nonlinear analysis is primarily useful in animal vocalizations with strong harmonic components including subharmonics and biphonation or low dimensional chaos

a unified approach to dynamic length algorithms for adaptive linear equalizers the number of taps is an important factor which affects nearly all the performance and complexity measures of adaptive equalizers recently dynamic length algorithms for the adaptive finite impulse response fir filters have attracted considerable attention and several methods have been proposed nevertheless the relationship among these techniques is still unclear which makes the performance comparison remaining difficult in this paper a unified explanation of these techniques is presented which summarizes all proposed algorithms and provides a platform for their performance comparison moreover a new type of dynamic length algorithms the varying threshold dynamic length algorithm vt dla is proposed simulation results show that the performance of the new algorithm outperforms that of existing algorithms

finite size lyapunov exponent review on applications in dynamical systems the growth of infinitesimal perturbations is well characterized by the lyapunov exponents in many situations of interest however important phenomena involve finite amplitude perturbations which are ruled by nonlinear dynamics out of tangent space and thus cannot be captured by the standard lyapunov exponents we review the application of the finite size lyapunov exponent fsle for the characterization of non infinitesimal perturbations in a variety of systems in particular we illustrate their usage in the context of predictability of systems with multiple spatio temporal scales of geophysical relevance in the characterization of nonlinear instabilities and in some aspects of transport in fluid flows we also discuss the application of the fsle to more general aspects such as chaos noise detection and coarse grained descriptions of signals this article is part of a special issue of journal of physics a mathematical and theoretical devoted to lyapunov analysis from dynamical systems theory to applications

exceptional events as evidence for determinism abstract by focussing attention on close returns of a trajectory to itself the existence of deterministic dynamics underlying a time series can be detected even in very short data sets this provides a practical means of detecting determinism in moderate dimensional e g 7 noisy systems or low dimensional systems with large lyapunov exponents such as computer random number generators

using surrogate data to detect nonlinearity in time series we address the issue of reliably discriminating between chaos and noise from a time series in particular we are interested in avoiding claims of chaos when simpler models such as linearly correlated noise can explain the data we take a statistical approach and use a form of bootstrapping to detect nonlinearity by showing that a given linear model is unlikely to have produced the data our method requires the careful statement of a null hypothesis which characterizes a candidate linear process the generation of an ensemble of surrogate data sets which are similar to the original time series but consistent with the null hypothesis and the computation of a discriminating statistic for the original and for each of the surrogate data sets the idea is to test the original time series against the null hypothesis by checking whether the discriminating statistic computed for the original time series differs significantly from the statistics computed for each of the surrogate sets we present algorithms for generating surrogate data under various null hypotheses and we show the results of numerical experiments on artificial data using correlation dimension lyapunov exponent and forecasting error as discriminating statistics finally we consider a number of experimental timemore series including sunspots electroencephalogram eeg signals and fluid convection and evaluate the statistical significance of the evidence for nonlinear structure in each case 52 refs 10 figs less

measurement of the lyapunov spectrum from a chaotic time series the exponential divergence or convergence of nearby trajectories lyapunov exponents is conceptually the most basic indicator of deterministic chaos we propose a new method to determine the spectrum of several lyapunov exponents including positive zero and even negative ones from the observed time series of a single variable we have applied the method to various known model systems and also to the rayleigh b enard experiment and have elucidated the dependence of the lyapunov exponents on the rayleigh number

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

a full scale approximation of covariance functions for large spatial data sets summary gaussian process models have been widely used in spatial statistics but face tremendous computational challenges for very large data sets the model fitting and spatial prediction of such models typically require o n3 operations for a data set of size n various approximations of the covariance functions have been introduced to reduce the computational cost however most existing approximations cannot simultaneously capture both the large and the small scale spatial dependence a new approximation scheme is developed to provide a high quality approximation to the covariance function at both the large and the small spatial scales the new approximation is the summation of two parts a reduced rank covariance and a compactly supported covariance obtained by tapering the covariance of the residual of the reduced rank approximation whereas the former part mainly captures the large scale spatial variation the latter part captures the small scale local variation that is unexplained by the former part by combining the reduced rank representation and sparse matrix techniques our approach allows for efficient computation for maximum likelihood estimation spatial prediction and bayesian inference we illustrate the new approach with simulated and real data sets

nonlinear independent component analysis and multivariate time series analysis abstract we derive an information theory based unsupervised learning paradigm for nonlinear independent component analysis nica with neural networks we demonstrate that under the constraint of bounded and invertible output transfer functions the two main goals of unsupervised learning redundancy reduction and maximization of the transmitted information between input and output infomax principle are equivalent no assumptions are made concerning the kind of input and output distributions i e the kind of nonlinearity of correlations an adapted version of the general nica network is used for the modeling of multivariate time series by unsupervised learning given time series of various observables of a dynamical system our net learns their evolution in time by extracting statistical dependencies between past and present elements of the time series multivariate modeling is obtained by making present value of each time series statistically independent not only from their own past but also from the past of the other series therefore in contrast to univariate methods the information lying in the couplings between the observables is also used and a detection of higher order cross correlations is possible we apply our method to time series of the two dimensional henon map and to experimental time series obtained from the measurements of axial velocities in different locations in weakly turbulent taylor couette flow

finite correlation dimension for stochastic systems with power law spectra abstract we discuss a counter example to the traditional view that stochastic processes lead to a non convergence of the correlation dimension in computed or measured time series specifically we show that a simple class of colored random noises characterized by a power law power spectrum have a finite and predictable value for the correlation dimension these results have implications on the experimental study of deterministic chaos as they indicate that the soie observation of a finite fractal dimension from the analysis of a time series is not sufficient to infer the presence of a strange attractor in the system dynamics we demonstrate that the types of random noises considered herein may be given an interpretation in terms of their fractal properties the consequent exploitation of the non gaussian behavior of these random noises leads us to the introduction of a new time series analysis method which we call multivariate scaling analysis we apply this approach to characterize several global properties of random noise

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

robust space time intermittency and 1 f noise abstract we investigate a type of intermittency that occurs in space as well as time studying a one dimensional lattice of coupled quadratic maps this system naturally forms spatial domains motion of the domain walls causes spatially localized changes from chaotic to almost periodic behavior the almost periodic phases have eigenvalues quite close to one resulting in long lived laminar bursts with a 1 f low frequency spectrum this behavior has aspects of both crutchfield and pomeau manneville intermittency unlike pomeau manneville however the behavior that we observe here is quite robust under changes of parameters

statistical description of chaotic attractors the dimension function a method for the investigation of fractal attractors is developed based on statistical properties of the distributionp n of nearest neighbor distances between points on the attractor a continuous infinity of dimensions called dimension function is defined through the moments ofp n in particular for the case of self similar sets we prove that the dimension function df yields in suitable points capacity information dimension and all other renyi dimensions an algorithm to compute df is derived and applied to several attractors as a consequence an estimate of nonuniformity in dynamical systems can be performed either by direct calculation of the uniformity factor or by comparison among various dimensions finally an analytical study of the distributionp n is carried out in some simple meaningful examples

a nonparametric method of multi step ahead forecasting in diffusion processes abstract this paper provides a nonparametric model of multi step ahead forecasting in diffusion processes the model is constructed from the local linear model with the gaussian kernel the paper provides simulation studies to evaluate its performance of multi step ahead forecasting by comparing with the global linear model showing the better forecasting performance of the nonparametric model than the global linear model the paper also conducts empirical analysis for forecasting using intraday data of the japanese stock price index and the time series of heart rates the result shows the performance of forecasting does not differ much in the japanese stock price index but that the nonparametric model shows significantly better performance in the analysis of the heart rates

small scale turbulence in the plankton low order deterministic chaos or high order stochasticity abstract almost all regions of the oceans are heavily influenced by the effects of physical processes such as turbulence which in turn influence the distribution and ecology of organisms that occupy these regions there is a real need for additional focus placed on the precise knowledge of both physical and biological processes which is often difficult using basic time series analysis in that way we applied nonlinear analysis techniques to high frequency time series of temperature salinity and phytoplankton concentrations recorded in different hydrodynamical regimes related to tidal forcing in a tidally mixed coastal ecosystem techniques devoted to the identification of low order deterministic chaos cannot find evidence of chaos while a lower dimensionality was identified in low hydrodynamic conditions the results rather suggest stochastic time series with many degrees of freedom no obvious attractor in phase space trajectory lle indistinguishable from zero absence of convergence of the correlation integral we then applied to these data specific multifractal analysis techniques and showed that these time series clearly exhibit high order stochasticity in addition the stochastic structure of purely passive scalars i e temperature and salinity remained invariant while the one of phytoplankton biomass must be regarded as highly structured in time by both hydrodynamic and advective processes

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

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

reconstructing attractors from scalar time series a comparison of singular system and redundancy criteria abstract a delay vector phase space reconstruction in which the delay time satisfiesa minimum redundancy criterion is compared with a reconstruction obtained using a singular system approach minimum redundancy produces the better reconstruction the reconstructions are compared using a distortion functional d which measures how well the location of a point in the original phase space can be determined on the basis of its image under the reconstruction process the superiority of the redundancy analysis over the singular system analysis is found to arise from the former s foundation on the notion of general independence as opposed to the latter s foundation on the notion of linear independence