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346
Determining Lyapunov Exponents from a Time Series
 Physica
, 1985
"... We present the first algorithms that allow the estimation of nonnegative 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 n ..."
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Cited by 495 (1 self)
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We present the first algorithms that allow the estimation of nonnegative 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 longterm 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 BelousovZhabotinskii reaction and CouetteTaylor flow. Contents 1.
A practical method for calculating largest Lyapunov exponents from small data sets
 PHYSICA D
, 1993
"... 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 statespace trajectories and estimate the amount of chaos in a system. We present a new m ..."
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Cited by 181 (0 self)
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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 statespace 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.
Estimating fractal dimension
 Journal of the Optical Society of America A
, 1990
"... Fractals arise from a variety of sources and have been observed in nature and on computer screens. One of the exceptional characteristics of fractals is that they can be described by a noninteger dimension. The geometry of fractals and the mathematics of fractal dimension have provided useful tools ..."
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Cited by 123 (4 self)
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Fractals arise from a variety of sources and have been observed in nature and on computer screens. One of the exceptional characteristics of fractals is that they can be described by a noninteger dimension. The geometry of fractals and the mathematics of fractal dimension have provided useful tools for a variety of scientific disciplines, among which is chaos. Chaotic dynamical systems exhibit trajectories in their phase space that converge to a strange attractor. The fractal dimension of this attractor counts the effective number of degrees of freedom in the dynamical system and thus quantifies its complexity. In recent years, numerical methods have been developed for estimating the dimension directly from the observed behavior of the physical system. The purpose of this paper is to survey briefly the kinds of fractals that appear in scientific research, to discuss the application of fractals to nonlinear dynamical systems, and finally to review more comprehensively the state of the art in numerical methods for estimating the fractal dimension of a strange attractor. Confusion is a word we have invented for an order which is not understood.Henry Miller, "Interlude," Tropic of Capricorn Numerical coincidence is a common path to intellectual perdition in our quest for meaning. We delight in catalogs of disparate items united by the same number, and often feel in our gut that some unity must underlie it all.
Generalized information potential criterion for adaptive system training
 IEEE Trans. Neural Networks
, 2002
"... Abstract—We have recently proposed the quadratic Renyi’s error entropy as an alternative cost function for supervised adaptive system training. An entropy criterion instructs the minimization of the average information content of the error signal rather than merely trying to minimize its energy. In ..."
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Cited by 58 (28 self)
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Abstract—We have recently proposed the quadratic Renyi’s error entropy as an alternative cost function for supervised adaptive system training. An entropy criterion instructs the minimization of the average information content of the error signal rather than merely trying to minimize its energy. In this paper, we propose a generalization of the error entropy criterion that enables the use of any order of Renyi’s entropy and any suitable kernel function in density estimation. It is shown that the proposed entropy estimator preserves the global minimum of actual entropy. The equivalence between global optimization by convolution smoothing and the convolution by the kernel in Parzen windowing is also discussed. Simulation results are presented for timeseries prediction and classification where experimental demonstration of all the theoretical concepts is presented. Index Terms—Minimum error entropy, Parzen windowing, Renyi’s entropy, supervised training.
Comparative study of stock trend prediction using time delay, recurrent and probabilistic neural networks
 IEEE TRANSACTIONS ON NEURAL NETWORKS
, 1998
"... Three networks are compared for low false alarm stock trend predictions. Shortterm trends, particularly attractive for neural network analysis, can be used profitably in scenarios such as option trading, but only with significant risk. Therefore, we focus on limiting false alarms, which improves ..."
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Cited by 54 (0 self)
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Three networks are compared for low false alarm stock trend predictions. Shortterm trends, particularly attractive for neural network analysis, can be used profitably in scenarios such as option trading, but only with significant risk. Therefore, we focus on limiting false alarms, which improves the risk/reward ratio by preventing losses. To predict stock trends, we exploit time delay, recurrent, and probabilistic neural networks (TDNN, RNN, and PNN, respectively), utilizing conjugate gradient and multistream extended Kalman filter training for TDNN and RNN. We also discuss different predictability analysis techniques and perform an analysis of predictability based on a history of daily closing price. Our results indicate that all the networks are feasible, the primary preference being one of convenience.
Using the Fractal Dimension to Cluster Datasets
 IN PROCEEDINGS OF THE SIXTH ACM SIGKDD INTERNATIONAL CONFERENCE ON KNOWLEDGE DISCOVERY AND DATA MINING
, 2000
"... Clustering is a widely used knowledge discovery technique. It helps uncovering structures in data that were not previously known. The clustering of large data sets has received a lot of attention in recent years, however, clustering is a still a challenging task since many published algorithms fail ..."
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Cited by 52 (5 self)
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Clustering is a widely used knowledge discovery technique. It helps uncovering structures in data that were not previously known. The clustering of large data sets has received a lot of attention in recent years, however, clustering is a still a challenging task since many published algorithms fail to do well in scaling with the size of the data set and the number of dimensions that describe the points, or in finding arbitrary shapes of clusters, or dealing effectively with the presence of noise. In this paper, we present a new clustering algorithm, based in the fractal properties of the data sets. The new algorithm which we call Fractal Clustering (FC) places points incrementally in the cluster for which the change in the fractal dimension after adding the point is the least. This is a very natural way of clustering points, since points in the same cluster have a great degree of selfsimilarity among them (and much less selfsimilarity with respect to points in other clusters). FC requires one scan of the data, is suspendable at will, providing the best answer possible at that point, and is incremental. We show via experiments that FC effectively deals with large data sets, highdimensionality and noise and is capable of recognizing clusters of arbitrary shape.
Solving Problems with GCMs: General Circulation Models and Their Role in the Climate Modeling Hierarchy
 IN GENERAL CIRCULATION MODEL DEVELOPMENT: PAST, PRESENT AND FUTURE, EDITED BY
, 2000
"... We outline the familiar concept of a hierarchy of models for solving problems in climate dynamics. General circulation models (GCMs) occupy a special position at the apex of this hierarchy, and provide the main link between basic conceptsbest captured by very simple, "toy" modelsand ..."
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Cited by 47 (31 self)
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We outline the familiar concept of a hierarchy of models for solving problems in climate dynamics. General circulation models (GCMs) occupy a special position at the apex of this hierarchy, and provide the main link between basic conceptsbest captured by very simple, "toy" modelsand the incomplete and inaccurate observations of climate variability in space and time. We illustrate this role of GCMs in addressing the problems of climate variability on three time scales: intraseasonal, seasonaltointerannual, and interdecadal. The problems involved require the use of atmospheric, oceanic, and coupled oceanatmosphere GCMs. We emphasize the role of dynamical systems theory in communicating between the rungs of the modeling hierarchytoy models, intermediate ones, and GCMsand between modeling results and observations.
Multifractal Processes
, 1999
"... This paper has two main objectives. First, it develops the multifractal formalism in a context suitable for both, measures and functions, deterministic as well as random, thereby emphasizing an intuitive approach. Second, it carefully discusses several examples, such as the binomial cascades and sel ..."
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Cited by 41 (6 self)
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This paper has two main objectives. First, it develops the multifractal formalism in a context suitable for both, measures and functions, deterministic as well as random, thereby emphasizing an intuitive approach. Second, it carefully discusses several examples, such as the binomial cascades and selfsimilar processes with a special eye on the use of wavelets. Particular attention is given to a novel class of multifractal processes which combine the attractive features of cascades and selfsimilar processes. Statistical properties of estimators as well as modelling issues are addressed.
Local Dynamic Modeling with SelfOrganizing Maps and Applications to Nonlinear System Identification and Control
 Proceedings of the IEEE
, 1998
"... The technique of local linear models is appealing for modeling complex time series due to the weak assumptions required and its intrinsic simplicity. Here, instead of deriving the local models from the data, we propose to estimate them directly from the weights of a self organizing map (SOM), which ..."
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Cited by 40 (7 self)
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The technique of local linear models is appealing for modeling complex time series due to the weak assumptions required and its intrinsic simplicity. Here, instead of deriving the local models from the data, we propose to estimate them directly from the weights of a self organizing map (SOM), which functions as a dynamicpreserving model of the dynamics. We introduce one modification to the Kohonen learning to ensure good representation of the dynamics and use weighted least squares to ensure continuity among the local models. The proposed scheme is tested using synthetic chaotic time series and real world data. The practicality of the method is illustrated in the identification and control of the NASA Langley wind tunnel during aerodynamic tests of model aircrafts. Modeling the dynamics with a SOM leads to a predictive multiple model control strategy (PMMC). Comparison of the new controller against the existing controller in test runs shows the superiority of our method. 1. Introducti...