No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Comments on `Deterministic Chaos: The Science and the Fiction' by D. Ruelle
Proceedings of the Royal Society A
Abstract
The results of several works concerning estimates of correlation dimension have recently been criticized on the basis of a new upper bound on such estimates (i.e. 2 log10 N, where N is the number of data). It is shown here that this bound is not valid for correlation dimension estimates in general, and furthermore it is too weak to be useful for regimes where it does hold. This casts the indicated criticism into question.
... A further development of Taken's theorem to noisy time series data has been provided by Stark (1999), but prior information about the noise is required. The disadvantages of the employment of reconstructed state vectors are obvious: The time series data is never noiseless, nor is the noise defined, and the sufficient amount of data for the application of such an algorithm is <iattractor < 21ogioN (Ruelle, 1990;Essex and Nerenberger, 1991;Eckmann and Ruelle, 1992) which means that in case of (]%APrERj. iNTRoi%%:ncw 9 (^attractor = », the number of 6 component vectors of a time series must be of order 10^/6. ...
... Another critical issue of statistical descriptive variables, like the dimension (pointwise dimension, information dimension, Hausdorff dimension, etc.) or the Lyapunov exponents from the time series, is the required amount of data. There are quite different suggestions by (Essex and Nerenberger, 1991;Nerenberg and Essex, 1990;Ruelle, 1990;Smith, 1988;Wolf, Swift, Swinney and Vastano, 1985) but unfortunately, they are at odds over the required amount of data. (Ruelle, 1990;Essex and Nerenberger, 1991;Eckmann and Ruelle, 1992) provide estimates for a set of N data points in the time series, with the dimension estimate Jattractor with ^attractor <: 21c%gioJ\f-(3 1:10) ...
... There are quite different suggestions by (Essex and Nerenberger, 1991;Nerenberg and Essex, 1990;Ruelle, 1990;Smith, 1988;Wolf, Swift, Swinney and Vastano, 1985) but unfortunately, they are at odds over the required amount of data. (Ruelle, 1990;Essex and Nerenberger, 1991;Eckmann and Ruelle, 1992) provide estimates for a set of N data points in the time series, with the dimension estimate Jattractor with ^attractor <: 21c%gioJ\f-(3 1:10) ...
p>As an attempt to overcome the unfortunate division between data and physical modelling, this thesis is devoted to the development of a framework which allows the derivation of a physical model that at the same time includes errors and uncertainties about this system which are as unspecified with respect to their statistical properties as possible. The work can be summarized as: 1. Grey box modelling : An appropriate semi-physical model of a ballistic missile with very limited knowledge about the aerodynamical properties of the airframe is being developed. It is shown which kind of dynamics the missile may exhibit and with which methods these dynamics can be analyzed. It is demonstrated that the missile may also show chaotic behaviour, the methods to measure this type of dynamics are presented. The lack of data points may, however, prevent the applicability of these methods which therefore suggests the theoretical framework of an: 2. Endo-observer for linear systems: It is proved that real-life observers require a framework such as a stochastic one for incorporating their uncertainties into a physical model. It is shown how minimally specified uncertainties can enter the dynamics of a free particle. The dynamics turn out to be similar to those given by the Schrödinger equation. 3. Endo-observer for the nonlinear state space: The concepts of physical dynamics with uncertainties which have been developed for the free particle are exhaustively re-developed for the case when the variance of the uncertainties remains within an arbitrary but fixed bound defined prior to the experiment. The dynamics of a nonlinear vector field with uncertainties are expressed in a form which is very much similar to the Schrödinger equation with a 'momentum' consisting of the 'mechanical momentum' from which a 'electromagnetical momentum' is subtracted which in this case here is the vector field of the state space. 4. endo-obsserver with bias: It is argued that an observer which has to select permanently among all available variable and parameter values has to favour the more probable ones rather than the less probable ones. The dynamics then lead to the well known nonlinear Schrödinger equation.</p
... Furthermore, its value must be consistent with the amount of data and number of average cycles being used. There is still considerable argument on the effect of the amount of data as outlined by Essex and Nerenberg (1991). For the data in this paper, the effect of the number of points used has been examined. ...
... Length of the time series. The amount of data required has been under debate (Essex and Nerenberg, 1991). Therefore, the effects of the amount of data has been examined. ...
... In the analysis data presented 5000 points or larger are always used to calculate the correlation dimension. Calculating the points required using the method of Essex and Nerenberg (1991) yields about 18,000 points. ...
This paper presents an application of “chaos” or “deterministic chaos” analysis to the performance of a gas-solid fluidized bed. From the point of view of the still developing field of chaos the significance of this analysis is the constancy of the correlation dimension and the Lyapunov exponent over a range of operating conditions. From the point of view of fluidized bed theory these quantitative numbers provide a method of characterizing performance. In addition, in the latter case, the presence of the chaotic attractor indicates several general properties of this dynamical system.
... Estimates of the correlation dimension D of a time series are commonly used for quantifying the chaotic complexity of a physical system such as the atmosphere (e.g., Grassberger and Procaccia 1983a,b;Eckmann and Ruelle 1985;Fraedrich 1986;Hense 1987;Henderson and Wells 1988;Tsonis and Elsner 1988;Fraedrich and Leslie 1989;Fraedrich et al. 1990;Baker and Gollub 1990;Sharifi et al. 1990;Yang 1991;Tsonis 1992;Zeng et al. 1993;Poveda-Jaramillo and Puente 1993;Orcutt and Arritt 1995;Weber et al. 1995). However, owing to the presence of both intrinsic and extrinsic sources of error, the merits of this procedure have been debated considerably in the literature (e.g., Nicolis 1984, 1987;Grassberger 1986Grassberger , 1987Smith et al. 1986;Theiler 1986Theiler , 1988Procaccia 1988;Smith 1988;Tsonis and Elsner 1988, 1989Pool 1989;Nerenberg and Essex 1990;Ruelle 1990;Essex and Nerenberg 1991;Zeng et al. 1992;Islam et al. 1993;Tsonis et al. 1993;Weber et al. 1995). ...
... These errors arise from such things as time series collection and preprocessing, embedding and model reconstruction, as well as the specific dimension algorithm used. Under time series collection, we include the frequency, number, continuity, and duration of the measurements Nicolis 1984, 1987;Grassberger 1986Grassberger , 1987Fraedrich 1986;Smith et al. 1986;Essex et al. 1987;Smith 1988;Keppene and Nicolis 1989;Ramsey and Yuan 1989;Fraedrich et al. 1990;Nerenberg and Essex 1990;Ruelle 1990;Zeng et al. 1992;Poveda-Jaramillo and Puente 1993;Tsonis et al. 1993;Orcutt and Arritt 1995;Weber et al. 1995) and contamination by noise (Franaszek 1984;Ben-Mizrachi et al. 1984;Theiler 1986;Simm et al. 1987;Osborne and Provenzale 1989;Theiler 1991); under preprocessing, we include low-or high-pass filtering (Hense 1987;Theiler 1991), interpolation and trend removal, and principal component analysis (Albano et al. 1988); under model reconstruction, we include the choice of variable or variables (Lorenz 1991;Islam et al. 1993;Poveda-Jaramillo and Puente 1993), embedding dimension (Mañé 1981;Takens 1981;Grassberger and Procaccia 1983a,b;Keppene and Nicolis 1989;Nerenberg and Essex 1990;Yang 1991;Islam et al. 1993;Poveda-Jaramillo and Puente 1993;Tsonis et al. 1993), and phase lag (Broomhead and King 1986;Fraser and Swinney 1986;Albano et al. 1991; Thomson and Henderson 1992;Tziperman et al. 1995); and under the dimension algorithm used, we include those proposed or used by Kaplan and Yorke (1979), Russell et al. (1980), Takens (1981Takens ( , 1985, Greenside et al. (1982), Grassberger and Procaccia (1983a,b), Termonia and Alexandrowicz (1983), Holzfuss and Mayer-Kress (1986), Theiler (1987), Henderson and Wells (1988), Kember and Fowler (1992), Orcutt and Arritt (1995), Weber et al. (1995), and us in this paper. Yet another source of error is lacunarity of the correlation integral (Theiler 1988), which causes the Takens (1985) estimator to fail, though possibly only to a small degree. ...
The correlation dimension D is commonly used to quantify the chaotic structure of atmospheric time series. The standard algorithm for estimating the value of D is based on finding the slope of the curve obtained by plotting In C(r) versus In r, where C(r) is the correlation integral and r is the distance between points on the attractor. An alternative, probabilistic method proposed by Takens is extended and tested here. This method is based on finding the sample means of the random variable (r/ρ) p[ln(r/ρ)] k, expressed as the conditional expected value E((r/ρ) p[ln(r/ρ)] k : r < ρ), for p and k nonnegative numbers. The sensitivity of the slope method and of the extended estimators D pk(ρ) for approximating D is studied in detail for three ad hoc correlation integrals and for integer values of p and k. The first two integrals represent the effects of noise or undersampling at small distances and the third captures periodic lacunarity, which occurs by definition when the ratio C(xρ)/C(ρ) fails to converge as ρ approaches zero. All the extended estimators give results that are superior to that produced by the most commonly used slope method. Moreover, the various estimators exhibit much different behavior in the two ad hoc cases: noise-contaminated signals are best diagnosed using D 11(ρ), and lacunar signals are best studied using D 0k(ρ), with k as large as possible in magnitude. Therefore, by using a wide range of values of p and k, one can infer whether degradation arising from noise or arising from lacunarity is more pronounced in the time series being studied, and hence, one can decide which of the estimates most efficiently approximates the correlation dimension for the series. These ideas are applied to relatively coarse samplings of the Hénon, Lorenz convection, and Lorenz climate attractors that in each case are obtained by calculating the distances between pairs of points on two trajectories. As expected from previous studies, lacunarity apparently dominates the Hénon results, with the best estimate of D, D = 1.20 ± 0.01, given by the case D 03(ρ). In contrast, undersampling or noise apparently affects the Lorenz convection and climate attractor results. The best estimates of D are given by the estimator D 11(ρ) in both cases. The dimension of the convection attractor is D = 2.06 ± 0.005, and that of the climate attractor is D = 14.9 ± 0.1. Finally, lagged and embedded time series for the Lorenz convection attractor are studied to identify embedding dimension signatures when model reconstruction is employed. In the last part of this study, the above results are used to help identify the best possible estimate of the correlation dimension for a low-frequency boundary layer time series of low-level horizontal winds. To obtain such an estimate, Lorenz notes that an optimally coupled time series must be extracted from the data and then lagged and embedded appropriately. The specific kinetic energy appears to be more closely coupled to the underlying low-frequency attractor, and so more nearly optimal, than is either individual wind component. When several estimates are considered, this kinetic energy series exhibits the same qualitative behavior as does the lagged and embedded Lorenz convective system time series. The series is either noise contaminated or under-sampled, a result that is not surprising given the small number of time series points used, for which the best estimate is given by D 11(ρ). The obtained boundary layer time series dimension estimate, 3.9 ± 0.1, is similar to the values obtained by some other investigators who have analyzed higher-frequency boundary layer time series. Although this time series does not contain as many points as might be required to accurately estimate the dimension of the underlying attractor, it does illustrate the requirement that in any estimate of the correlation dimension, a function of the measured variables must be chosen that is strongly coupled to the attractor.
... Defining the fractal dimension of the attractor as D 0 and the correlation dimension as D 2 , Smith (1988) suggested that the minimum number of data points in the time series should be in the order of 42 D 0 , while Ruelle (1990) and Essex and Nerenberg (1991) suggested that 10 D 2 /2 and 2 D 2 (D 2 + 1) are required for a reliable reconstruction of an attractor. ...
Analysis methods that have been developed in the field of nonlinear dynamics have provided valuable insights into the physics of turbulent flows, although their application to open flows is less well explored. The nonlinear dynamics of a turbulent jet with a low-to-moderate Reynolds number is investigated by using the single-trajectory framework and ensemble framework. We have used Lyapunov exponents to calculate the spectra of scaling indices of the attractor. First, we evaluated the frameworks on two theoretical models, one with a stationary attractor (Lorenz-63) and the other with time-varying characteristics (Lorenz-84). Theoretical studies showed that in dynamical systems with a stable attractor, both frameworks estimated the same largest Lyapunov exponent. The ensemble framework enables us to resolve the unsteady characteristics of a time-varying strange attractor. Second, we applied both frameworks to time-resolved planar velocity fields in a turbulent jet at local Reynolds numbers (Reδ) of 3000 and 5000. Time-resolved particle image velocimetry was utilized to measure streamwise and transverse velocity components. Results support the presence of a low-dimensional attractor in the reconstructed phase space with a chaotic characteristic. Despite considerable changes in the dynamics for the higher Reynolds number case, the system’s fractal dimension did not change significantly. We have used Lagrangian Coherent Structures (LCSs) to study the relationship between changes in the Lyapunov exponent with flow topological features. Results suggest that holes in the stable LCSs provide a path for the entrainment of the coflow, which is shown to be one of the main contributors to high Lyapunov exponents.
... Kapitaniak (1993) In this regard, we point out that it was argued (Smith (1988); Ruelle (1990); Eckmann and Ruelle (1992)) that a series with 10 5 elements does not allow to accurately resolve higher dimensions than 5. While there have been serious discussions on this "pessimistic view" (Essex and Nerenberg (1991)), also more optimistic estimates (Nerenberg and Essex (1990)) agree that the required amount of data to reliably cover high-dimensional systems quickly grows beyond any limits of experimental or computational feasibility (c.f. ...
We investigate recurrent patterns in a lab-scale fluidized bed consisting of Np ≈ 57 000 glass particles by the means of computational fluid dynamics and the discrete element method (CFD-DEM). We generalize single-point measures for recurrence quantification analysis (RQA) to spatially extended particle density fields and compare the embedding dimensions necessary to unambiguously identify similar states. In accordance with many previous studies, we find locally very low-dimensional behavior (Dloc ≈ 5). Globally, the correlation dimension, i.e. the number of effective degrees of freedom, increases to Dglobal ≈ 18, which is still several orders of magnitude smaller than the 6Np microscopic translational degrees of freedom and explains the observed, characteristic structures. Distance plots reveal a combination of fast, recurrent motion interrupted by rare events like large eruptions. The temporal evolution of the average nearest-neighbor distance shows a rapid initial drop and follows a power law related to the correlation dimension for longer times. Given a sufficient tolerance, most states reappear after a few seconds, which indicates why in recent, small-scale recurrence CFD (rCFD) simulations, the complicated long-term bed motion could be approximated with information from short-term studies within the scope of current computation techniques. From the perspective of numerical efficiency, the presented analysis allows to determine the shortest time series from which a significant amount of information on the underlying system may be obtained.
... Estimates of the correlation dimension d of a time series are commonly used for quantifying the chaotic complexity of a physical system (e.g., Grassberger and Procaccia, 1983a, b;Eckmann and Ruelle, 1985;Henderson and Wells, 1988;Baker and Gollub, 1990;Tsonis, 1992). However, owing to the presence of both intrinsic and extrinsic sources of error, the merits of this procedure have been debated considerably in the literature (e.g., Nicolis, 1984, 1987;Grassberger, 1986Grassberger, , 1987Smith et al., 1986;Theiler, 1986Theiler, , 1988Procaccia, 1988;Smith, 1988;Tsonis and Eisner, 1988Nerenberg and Essex, 1990;Ruelle, 1990;Essex and Nerenberg, 1991). ...
... For example, Tsonis criterion defines minimal length as ≥ 10 2+0,4 2 , where 2 is assumed correlation dimension of reconstructed attractor [29]. According to other criteria, > 10 ( 2 +2)/2 [30], ≥ 2 2 ( 2 +1) [31], or ≥ 10 2 /2 [32]. Finally, there is an opinion that the minimal series length depends not only on dimension parameters but also on process autocorrelation, and ≥ 2 /2 (where is autocorrelation time, i.e., number of steps till its first zero or minimum, and is embedding dimension) [33]. ...
We propose the method to compute the nonlinear parameters of heart rhythm (correlation dimension D 2 and correlation entropy K 2) using 5-minute ECG recordings preferred for screening of population. Conversion of RR intervals' time series into continuous function x(t) allows getting the new time series with different sampling rate dt. It has been shown that for all dt (250, 200, 125, and 100 ms) the cross-plots of D 2 and K 2 against embedding dimension m for phase-space reconstruction start to level off at m = 9. The sample size N at different sampling rates varied from 1200 at dt = 250 ms to 3000 at dt = 100 ms. Along with, the D 2 and K 2 means were not statistically different; that is, the sampling rate did not influence the results. We tested the feasibility of the method in two models: nonlinear heart rhythm dynamics in different states of autonomous nervous system and age-related characteristics of nonlinear parameters. According to the acquired data, the heart rhythm is more complex in childhood and adolescence with more influential parasympathetic influence against the background of elevated activity of sympathetic autonomous nervous system.
... While some of these studies found a strange attractor with a finite dimension, ranging from 3 to 13, others did not find signs of deterministic chaos (the same problem appears for other natural systems: see, e.g., the study by Carbonell et al. 1994 for the search for deterministic chaos in solar activity). In particular, the positive results were criticized by Grassberger (1986) and Ruelle (1990) [in turn criticized in an analytical study by Essex and Nerenberg (1991)], who addressed the problem of sparse data as well as the errors of the estimates. An argument against the characterization of deterministic versus stochastic variability through the determination of a finite correlation dimension was suggested by Osborne and Provenzale (1989) [subsequently criticized in an analytical study by Theiler (1991)], who showed that a finite dimension can be found in systems that are not deterministic but that possess random, self-similar time series that exhibit fractional Brownian motion (Mandelbrot and VanNess 1968). ...
Northern Hemisphere stratospheric variability is investigated with respect to chaotic behavior using time series from three different variables extracted from four different reanalysis products and two numerical model runs with different forcing. The time series show red spectra at all frequencies and the probability distribution functions show persistent deviations from a Gaussian distribution. An exception is given by the numerical model forced with perpetual winter conditions-a case that shows more variability and follows a Gaussian distribution, suggesting that the deviation from Gaussianity found in the observations is due to the transition between summer and winter variability. To search for the presence of a chaotic attractor the correlation dimension and entropy, the Lyapunov spectrum, and the associated Kaplan-Yorke dimension are estimated. A finite value of the dimensions can be computed for each variable and data product, with the correlation dimension ranging between 3.0 and 4.0 and the Kaplan-Yorke dimension between 3.3 and 5.5. The correlation entropy varies between 0.6 and 1.1. The model runs show similar values for the correlation and Lyapunov dimensions for both the seasonally forced run and the perpetual-winter run, suggesting that the structure of a possible chaotic attractor is not determined by the seasonality in the forcing, but must be given by other mechanisms.
... Thus typical examples use 1.5 x 10 4 -2.5 X 10 4 (Grassberger and Procaccia, 1983b) and 0.8 x 10 4 -30 X 10 4 data points (Atten et alii, 1984). Thus there seems to be no agreeel upon rule to determine the amount of data required to estimate dimensions with confidence but it appears that at least a few thousanel points for low dimensional attractors are neeeleel (Theiler, 1986;Havstad and Ehlers, 1989;Ruelle, 1990;Essex and Nerenberg, 1991). In particular, N > lO D d 2 has been quoteel in (Ding et alii, 1993). ...
The relevance of nonlinear dynamics and chaos in science and engineering cannot be overemphasized. Unfortunately, the enormous wealth of techniques for the analysis of linear systems which is currently available is to tally inadequate for handling nonlinear systems in a system atic, consistent and global way. This paper presents a brief introduction to some of the main concepts and tools used in the analysis of nonlinear dynamical system and chaos which have been currently used in the literature. The main objec tive is to present a readable introduction to the subject and provide several references for further reading. A number of well known and well documented nonlinear models are also included. Such models can be used as benchmarks not only for testing some of the tools described in this paper, but also for developing and troubleshooting other algorithms in the comprehensive fields of identification, analysis and con trol of nonlinear dynamical systems. Some of these aspects will be addressed in a companion paper which follows.
... Up to now such attempts as a rule have been considerably more successful in the idealized case of two-dimensional turbulence than in real three-dimensional turbulence (see, e.g., [3, 30]). This may be connected with the fact that the strange-attractor scenario of transition to turbulence and the other proposed model scenarios were related to finite-dimensional dynamical systems;they are clearly not universal and their appropriateness for three-dimensional fluid flows is still not clear (see, e.g., [18, 52, 72, 142, 151]). In applications of the finite-dimensional transition-to-chaos models to three-dimensional turbulence usually the simple finite-dimensional approximations are used instead of the full Navier–Stokes equations; however it is unclear to what degree the behavior of such approximate models agrees with the behavior of real fluid flows. ...
A brief, superficial survey of some very personal nominations for highpoints of the last hundred years in turbulence. Some
conclusions can be dimly seen. This field does not appear to have a pyramidal structure, like the best of physics. We have
very few great hypotheses. Most of our experiments are exploratory experiments. What does this mean?
We believe it means that, even after 100 years, turbulence studies are still in their infancy. We are naturalists, observing
butterflies in the wild. We are still discovering how turbulence behaves, in many respects. We do have a crude, practical,
working understanding of many turbulence phenomena but certainly nothing approaching a comprehensive theory, and nothing that
will provide predictions of an accuracy demanded by designers.
... There has been a great deal of discussion in the literature regarding the amount of data required to obtain a meaningful estimate of the characteristics of chaotic dynamical systems. For the correlation exponent, several authors [66][67][68][69] have provided estimates of the minimum "number of data points" required. Unfortunately, it is not easy to determine the number of data points in a time series in this sense. ...
This contribution focuses upon extracting information from dynamic reconstructions of experimental time series data. In addition to the problem of distinguishing between deterministic dynamics and stochastic dynamics, applied questions, such as the detection of parametric drift, are addressed. Nonlinear prediction and dimension algorithms are applied to geophysical laboratory data, and the significance of these results is established by comparison with results from similar surrogate series, generated so as not to contain the property of interest. A global nonlinear predictor is introduced which attempts to correct systematic bias due to the inhomogeneous distribution of data common in strange attractors. Variations in the quality of predictions with location in phase space are examined in order to estimate the uncertainty in a forecast at the time it is made. Finally, the application of these methods to truly stochastic systems is discussed and the distinction between deterministic, stochastic, and low dimensional dynamics is considered.
... Other research found that smaller data size are needed. For instance, the minimum data points for reliable d 2 are 10 d 2 /2 [Ruelle, 1990;Essex and Nerenberg, 1991], or ffiffiffiffiffiffiffiffiffi 27:5 p d 2 [Hong and Hong, 1994] and empirical results of dimension calculations are not substantially altered by going from 3000 or 6000 points to subsets of 500 points [Abraham et al., 1986]. ...
DOI: 10.1029/2007WR006737 In this paper, the accuracy performance of monthly streamflow forecasts is discussed when using data-driven modeling techniques on the streamflow series. A crisp distributed support vectors regression (CDSVR) model was proposed for monthly streamflow prediction in comparison with four other models: autoregressive moving average (ARMA), K-nearest neighbors (KNN), artificial neural networks (ANNs), and crisp distributed artificial neural networks (CDANN). With respect to distributed models of CDSVR and CDANN, the fuzzy C-means (FCM) clustering technique first split the flow data into three subsets (low, medium, and high levels) according to the magnitudes of the data, and then three single SVRs (or ANNs) were fitted to three subsets. This paper gives a detailed analysis on reconstruction of dynamics that was used to identify the configuration of all models except for ARMA. To improve the model performance, the data-preprocessing techniques of singular spectrum analysis (SSA) and/or moving average (MA) were coupled with all five models. Some discussions were presented (1) on the number of neighbors in KNN; (2) on the configuration of ANN; and (3) on the investigation of effects of MA and SSA. Two streamflow series from different locations in China (Xiangjiaba and Danjiangkou) were applied for the analysis of forecasting. Forecasts were conducted at four different horizons (1-, 3-, 6-, and 12-month-ahead forecasts). The results showed that models fed by preprocessed data performed better than models fed by original data, and CDSVR outperformed other models except for at a 6-month-ahead horizon for Danjiangkou. For the perspective of streamflow series, the SSA exhibited better effects on Danjingkou data because its raw discharge series was more complex than the discharge of Xiangjiaba. The MA considerably improved the performance of ANN, CDANN, and CDSVR by adjusting the correlation relationship between input components and output of models. It was also found that the performance of CDSVR deteriorated with the increase of the forecast horizon. Author name used in this publication: K. W. Chau
... 16 Basically, GPA measures the rate of change of the local point densities around the attractor surface as a function of the local radius. However, this algorithm suffers from certain disadvantages: ͑1͒ highly demanding computational requirements, 30,95,96 and ͑2͒ spurious estimates of D 2 due to the presence of measurement noise. For a data length of N, GPA needs a computation on the order of O(N 2 ) for one local center and one embedding dimension. ...
The notion that a deterministic nonlinear dynamical system (with relatively few degrees of freedom) can display aperiodic behavior has a strong bearing on sea clutter characterization: random-looking sea clutter may be the outcome of a chaotic process. This new approach envisages deterministic rules for the underlying sea clutter dynamics, in contrast to the stochastic approach where sea clutter is viewed as a random process with a large number of degrees of freedom. In this paper, we demonstrate, convincingly for the first time, the chaotic dynamics of sea clutter. We say so on the basis of results obtained using radar data collected from a series of extensive and thorough experiments, which have been carried out with ground-truthed sea clutter data sets at three different sites. The study includes correlation dimension analysis (based on the maximum likelihood principle) and Lyapunov spectrum analysis. The Lyapunov (Kaplan-Yorke) dimension, which is a byproduct of Lyapunov spectrum analysis, shows that it is indeed a good estimator of the correlation dimension. The Lyapunov spectrum also reveals that sea clutter is produced by a coupled system of nonlinear differential equations of order five or six. (c) 1997 American Institute of Physics.
This section will introduce a variety of approaches to the analysis of data. The primary focus will be on the application of neural network-based techniques to the tasks of prediction, classification, and function approximation. This section will therefore begin by discussing the following neural network functions that are available in Simulnet.
The paper discusses and compares application of several stochastic and deterministic methods for analysis of geological time-series, taking the record of the Pacific core V28-239 as a test object. The stochastic techniques include finite parameter linear models, such as ARMA and ARIMA, while the deterministic chaos approach deals with the Lyapunov exponent and correlation dimension, utilizing the Wolf algorithm and the Grassberger–Procaccia procedure, respectively. Neither stochastic nor deterministic models lead to definitive and unique conclusions. This is true for our results as well as for those of the previous authors obtained on the same dataset. The major limitations are the small size of the sample set and the uncertainties in assigned ages for the values. It may be that the emerging nonlinear stochastic techniques prove to be more suitable for treatment of geological time-series.
Recent developments in non-linear dynamics and complex phenomena have resulted in mathematical tools which are potentially useful in the control and design of bubble column reactors. It is conjectured that analysis of conductivity time series, measured in a bubble column, indicated a low-dimensional chaotic attractor. The cross-sectional average correlation dimension varied with the liquid and the gas flow rate. The cross-sectional average correlation dimension characterized the global bubble column hydrodynamics and, more specifically, the average bubble size.
There have been numerous attempts to detect the presence of deterministic chaos by estimating the correlation dimension. The values of reported correlation dimension for various geophysical time series vary between 1.3 and virtually infinity (i.e., no saturation). It is pointed out that analyzing variables that depend on physical constraints and thresholds, like precipitation, may lead to underestimation of the correlation dimension of the underlying dynamical system.
This paper is concerned with the estimation of dynamical invariants from relatively short and possibly noisy sets of chaotic data. In order to overcome the difficulties associated with the size and quality of the data records, a two-step procedure is investigated. Firstly NARMAX models are fitted to the data. Secondly, such models are used to generate longer and cleaner time sequences from which dynamical invariants such as Lyapunov exponents, correlation dimension, the geometry of the attractors, Poincaré maps and bifurcation diagrams can be estimated with relative ease. An additional advantage of this procedure is that because the models are global and have a simple structure, such models are amenable for analysis. It is shown that the location and stability of the fixed points of the original systems can be analytically recovered from the identified models. A number of examples are included which use the logistic and Hénon maps, Duffing and modified van der Pol oscillators, the Mackey-Glass delay system, Chua’s circuit, the Lorenz and Rössler attractors. The identified models of these systems are provided including discrete multivariable models for Chua’s double scroll, Lorenz and Rössler attractors which are used to reconstruct the trajectories in a three-dimensional state space.
In the chemical literature of recent years, there has been considerable interest in the study of deterministic chaos within the context of nonlinear kinetic schemes. However, dynamical systems theory admits a rather strict definition of ‘‘chaos’’ which has seldom been confirmed in the many cases where it is claimed that chaos exists in coupled chemical kinetic models. In this paper we carry out a systematic study of the dynamical properties of two such model systems, computing Lyapunov exponents, fractal dimensions, and power spectra from the (time) series arising from the associated differential equations. In both cases, the analyses presented here provide strong support for the existence of chaotic dynamics for certain values of the appropriate control parameters. In view of the potential difficulty of resolving stochastic fluctuations from chaotic temporal behavior in experimental situations, it is recommended that, wherever possible, authors report estimates of Lyapunov exponents and fractal dimensions of associated chaotic attractors.
The possible chaotic nature of the turbulence of the atmospheric boundary layer in and above a decidious forest is investigated. In particular, this work considers high resolution temperature and three-dimensional wind speed measurements, gathered at six alternative elevations at Camp Borden, Ontario, Canada (Shawet al., 1988). The goal is to determine whether these time series may be described (individually) by sets of deterministic nonlinear differential equations, such that: (i) the data's intrinsic (and seemingly random) irregularities are captured by suitable low-dimensional fractal sets (strange attractors), and (ii) the equation's lack of knowledge of initial conditions translates into unpredictable behavior (chaos). Analysis indicates that indeed all series exhibit chaotic behavior, with strange attractors whose (correlation) dimensions range from 4 to 7. These results support the existence of a low-dimensional chaotic attractor in the lower atmosphere.
The series of mean daily temperature of air recorded over a period of 215 years is used for analysing the dimensionality and the predictability of the atmospheric system. The total number of data points of the series is 78527. Other 37 versions of the original series are generated, including seasonally adjusted data, a smoothed series, series without annual course, etc.Modified methods of Grassberger & Procaccia are applied. A procedure for selection of the meaningful scaling region is proposed. Several scaling regions are revealed in the lnC(r) versus Inr diagram. The first one in the range of larger lnr has a gradual slope and the second one in the range of intermediate lnr has a fast slope. Other two regions are settled in the range of small lnr. The results lead us to claim that the series arises from the activity of at least two subsystems. The first subsystem is low-dimensional (d
f
=1.6) and it possesses the potential predictability of several weeks. We suggest that this subsystem is connected with seasonal variability of weather. The second subsystem is high-dimensional (d
f
>17) and its error-doubling time is about 4–7 days.It is found that the predictability differs in dependence on season. The predictability time for summer, winter and the entire year (T
2
4.7 days) is longer than for transitionseasons (T
2
4.0 days for spring,T
2
3.6 days for autumn).The role of random noise and the number of data points are discussed. It is shown that a 15-year-long daily temperature series is not sufficient for reliable estimations based on Grassberger & Procaccia algorithms.
Chaotic time series data are observed routinely in experiments on physical systems and in observations in the field. The authors review developments in the extraction of information of physical importance from such measurements. They discuss methods for (1) separating the signal of physical interest from contamination ("noise reduction"), (2) constructing an appropriate state space or phase space for the data in which the full structure of the strange attractor associated with the chaotic observations is unfolded, (3) evaluating invariant properties of the dynamics such as dimensions, Lyapunov exponents, and topological characteristics, and (4) model making, local and global, for prediction and other goals. They briefly touch on the effects of linearly filtering data before analyzing it as a chaotic time series. Controlling chaotic physical systems and using them to synchronize and possibly communicate between source and receiver is considered. Finally, chaos in space-time systems, that is, the dynamics of fields, is briefly considered. While much is now known about the analysis of observed temporal chaos, spatio-temporal chaotic systems pose new challenges. The emphasis throughout the review is on the tools one now has for the realistic study of measured data in laboratory and field settings. It is the goal of this review to bring these tools into general use among physicists who study classical and semiclassical systems. Much of the progress in studying chaotic systems has rested on computational tools with some underlying rigorous mathematics. Heuristic and intuitive analysis tools guided by this mathematics and realizable on existing computers constitute the core of this review.
Mobile robotics research to date is still largely reliant on trial-and-error procedures, rather than exploiting established theories describing robot–environment interaction in a formal manner, making falsifiable predictions and allowing quantitative descriptions of a robot's behaviour.We argue that quantitative performance measures are the first step towards a theory of robot–environment interaction, and present the theoretical background to such measures, as well as their practical application to mobile robotics research.Results obtained with a Pioneer II mobile robot, executing a number of different tasks in a range of environments, are presented.
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.
Direct estimation of the largest Lyapunov exponent from deterministically chaotic data is well established. From stochastic data a finite, positive value is obtained as well; we show how it is determined by spectral properties of the signal and by computational parameters. Distinction of chaos versus noise is discussed.
Abstract: Data-driven techniques such as Auto-Regressive Moving Average (ARMA) K-Nearest-Neighbors (KNN) and Artificial Neural Networks (ANN) are widely applied to hydrologic time series prediction This paper investigates different data-driven models to determine the optimal approach of predicting monthly streamflow time series Four sets of data from different locations of People s Republic of China (Xiangjiaba Cuntan Manwan and Danjiangkou) are applied for the investigation process Correlation integral and False Nearest Neighbors (FNN) are first employed for Phase Space Reconstruction (PSR) Four models ARMA ANN KNN and Phase Space Reconstruction-based Artificial Neural Networks (ANN-PSR) are then compared by one-month-ahead forecast using Cuntan and Danjiangkou data The KNN model performs the best among the four models but only exhibits weak superiority to ARMA Further analysis demonstrates that a low correlation between model inputs and outputs could be the main reason to restrict the power of ANN A Moving Average Artificial Neural Networks (MA-ANN) using the moving average of streamflow series as inputs is also proposed in this study The results show that the MA-ANN has a significant improvement on the forecast accuracy compared with the original four models This is mainly due to the improvement of correlation between inputs and outputs depending on the moving average operation The optimal memory lengths of the moving average were three and six for Cuntan and Danjiangkou respectively when the optimal model inputs are recognized as the previous twelve months.
Keywords: Hydrologic time series; Auto regressive moving average; K nearest neighbors; Artificial neural networks; Phase space reconstruction; False nearest neighbors; Dynamics of chaos
This entry for the New Palgrave covers developments in nonlinear time series analysis over the last 25 years.
The number and variety of methods used in dynamical analysis has increased dramatically during the last fifteen years, and the limitations of these methods, especially when applied to noisy biological data, are now becoming apparent. Their misapplication can easily produce fallacious results. The purpose of this introduction is to identify promising new methods and to describe safeguards that can be used to protect against false conclusions.
Dynamic systems may be characterized by their fractal dimension. The classical Grassberger-Procaccia algorithm is widely used to analyze time series. However, if this method is used beyond its intrinsic limitations it may cause incorrect classification of systems. We found that a simple deterministic five-dimensional system leads to erroneous dimension values around 5.5 if the following methods are used uncritically: The classical Grassberger-Procaccia algorithm, a pointwise correlation dimension algorithm, and an algorithm for calculation of the information dimension yielded this erroneous result for a wide range of numbers of data points (N=30 000-106) and various delay times. Estimates of dimensions are only reliable if long plateaus of the local slope of the correlation integrals exist for small distances; these were not found in our example. This example suggests that a correlation dimension of 5 is too high to be recognized using even one million noise-free data points.
We perform a detailed analysis of the sunspot number time series to reconstruct the phase space of the underlying dynamical system. The features of this phase space allow us to describe the behavior of the solar cycle in terms of a simple relaxation oscillator in two dimensions. The absence of systematic self-crossings suggests that the complexity of the sunspot time series does not arise as a consequence of chaos. Instead, we show that it can be adequately modeled through the introduction of a stochastic fluctuation in one of the parameters of the dynamic equations.
With the discovery of chaos came the hope of finding simple models that would be capable of explaining complex phenomena. Numerous papers claimed to find low-dimensional chaos in a number of areas ranging from the weather to the stock market. Years later, many of these claims have been disproved and the fantastic hopes pinned on chaos have been toned down as research with more realistic objectives follows. The difficulty in calculating reliable estimates of the correlation dimension and the maximal Lyapunov exponent, two of the hallmarks of chaos, are explored. Given that nonlinear dynamics is a relatively new and growing field of science, the need for statistical testing is greater than ever. Surrogate data provides one possible approach but great care is needed in generating relevant surrogates and in interpreting the results. Examples of misleading applications and challenges for the future of research in nonlinear dynamics are discussed.
In this paper, the correlation dimension has been shown to be
effective in distinguishing between the melting layer and other regions
in a weather phenomenon. Using the correlation integral method of
Grassberger and Procaccia (1983), data collected from airborne radar has
been analyzed. It has been shown that the correlation dimension dips to
5.5 in the melting layer where as it is around 7 in other regions. The
Lyapunov exponent has been found to be positive in these regions
emphasizing the fact that the time series is indeed chaotic. It is
surmised that the chaos in the time series may have had its origin in
the fractal shape of the scatterers in the melting region
A best estimator derived by Takens for estimating the dimension of a strange attractor from a discrete set of points is shown to be sensitive to lacunarity in the fractal set.
A general review of the ideas of chaos is presented. Particular attention is given to the problem of finding out whether or not various time evolutions observed in nature correspond to low-dimensional deterministic dynamics. The 'dimensions' of the order 6 that are obtained are found to be very close to the upper bound 2log(10)N permitted by the Grassberger-Procaccia algorithm (1983).
Recent work has used ideas from the theory of dynamical systems in the
study of climate and weather over timescales ranging from decades to
hundreds of thousands of years1-5. In this study, similar
ideas are applied to weather observations over a time interval of 11
hours. The results suggest the existence of a low-dimensional strange
attractor.
Recent work has highlighted the possibilities of using certain ideas from the theory of dynamical systems for the study of global climate. These ideas include the geometrical notion of correlation or scaling dimension, first used to analyse attractors arising in mathematical and laboratory systems1–4 and later applied in a geophysical context5,6. The conclusion of the original geophysical work5 has been criticized in the light of a reanalysis7 which questions the existence of a low-dimensional climate attractor. Here results of a similar analysis conducted on daily meteorological observations from 1946 to 1982, are announced. This new work overcomes limitations of previous analyses, and supports the existence of such an attractor.
The combined influences of boundary effects at large scales and nonzero nearest neighbor separations at small scales are used to compute intrinsic limits on the minimum size of a data set required for calculation of scaling exponents. A lower bound on the number of points required for a reliable estimation of the correlation exponent is given in terms of the dimension of the object and the desired accuracy. A method of estimating the correlation integral computed from a finite sample of a white noise signal is given.
It is shown that oscillations in the high-order moments of turbulent velocity fields are inherent to the fractal character of intermittent turbulence and are a feature of the lacunarity of fractal sets. Oscillations in simple Cantor sets are described, and a single parameter to measure lacunarity is identified. The connection between oscillations in fractals and in the turbulent velocity correlations is discussed using the phenomenological beta model of intermittent turbulence (Frisch et al., 1978).
Correlation-dimension calculations have been widely undertaken in attempts to understand various physical and other natural phenomena. Some of these studies have perhaps been incautious in their claims. On the other hand, some authors have been sharply critical of such work. These criticisms hinge on doubts about data-set size. In particular, Smith [Phys. Lett. A 133, 283 (1988)] has suggested that 42D points are a minimum number of data points, where D is the embedding dimension. He has suggested that, as a result, correlation-dimension calculations are limited to dimensions less than 5 or 6 even on supercomputers. By implication, many published results are placed into doubt, discouraging people from undertaking such calculations. We review the concept of critical embedding dimension and undertake a detailed analysis of data-size requirements, which culminates in tight error estimates for determination of correlation dimension. We conclude that, while data requirements are still substantial, they are not nearly so extreme as has been suggested, making the determination of correlation dimension from data sets feasible. Previously reported ``doubtful'' calculations of correlation dimension can be reviewed in the light of explicit error estimates.