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181
Interdisciplinary application of nonlinear time series methods
 Phys. Reports
, 1998
"... This paper reports on the application to field measurements of time series methods developed on the basis of the theory of deterministic chaos. The major difficulties are pointed out that arise when the data cannot be assumed to be purely deterministic and the potential that remains in this situatio ..."
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Cited by 88 (4 self)
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This paper reports on the application to field measurements of time series methods developed on the basis of the theory of deterministic chaos. The major difficulties are pointed out that arise when the data cannot be assumed to be purely deterministic and the potential that remains in this situation is discussed. For signals with weakly nonlinear structure, the presence of nonlinearity in a general sense has to be inferred statistically. The paper reviews the relevant methods and discusses the implications for deterministic modeling. Most field measurements yield nonstationary time series, which poses a severe problem for their analysis. Recent progress in the detection and understanding of nonstationarity is reported. If a clear signature of approximate determinism is found, the notions of phase space, attractors, invariant manifolds etc. provide a convenient framework for time series analysis. Although the results have to be interpreted with great care, superior performance can be achieved for typical signal processing tasks. In particular, prediction and filtering of signals are discussed, as well as the classification of system states by means of time series recordings.
Chaotic Invariants of Lagrangian Particle Trajectories for Anomaly Detection in Crowded Scenes
"... A novel method for crowd flow modeling and anomaly detection is proposed for both coherent and incoherent scenes. The novelty is revealed in three aspects. First, it is a unique utilization of particle trajectories for modeling crowded scenes, in which we propose new and efficient representative tra ..."
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Cited by 35 (4 self)
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A novel method for crowd flow modeling and anomaly detection is proposed for both coherent and incoherent scenes. The novelty is revealed in three aspects. First, it is a unique utilization of particle trajectories for modeling crowded scenes, in which we propose new and efficient representative trajectories for modeling arbitrarily complicated crowd flows. Second, chaotic dynamics are introduced into the crowd context to characterize complicated crowd motions by regulating a set of chaotic invariant features, which are reliably computed and used for detecting anomalies. Third, a probabilistic framework for anomaly detection and localization is formulated. The overall workflow begins with particle advection based on optical flow. Then particle trajectories are clustered to obtain representative trajectories for a crowd flow. Next, the chaotic dynamics of all representative trajectories are extracted and quantified using chaotic invariants known as maximal Lyapunov exponent and correlation dimension. Probabilistic model is learned from these chaotic feature set, and finally, a maximum likelihood estimation criterion is adopted to identify a query video of a scene as normal or abnormal. Furthermore, an effective anomaly localization algorithm is designed to locate the position and size of an anomaly. Experiments are conducted on known crowd data set, and results show that our method achieves higher accuracy in anomaly detection and can effectively localize anomalies. 1. Introduction and Related
Local dynamic stability versus kinematic variability of continuous overground and treadmill walking
 Journal of Biomechanical Engineering
, 2001
"... This study quantified the relationships between local dynamic stability and variability during continuous overground and treadmill walking. Stridetostride standard deviations were computed from temporal and kinematic data. Maximum finitetime Lyapunov exponents were estimated to quantify local dyn ..."
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Cited by 32 (2 self)
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This study quantified the relationships between local dynamic stability and variability during continuous overground and treadmill walking. Stridetostride standard deviations were computed from temporal and kinematic data. Maximum finitetime Lyapunov exponents were estimated to quantify local dynamic stability. Local stability of gait kinematics was shown to be achieved over multiple consecutive strides. Traditional measures of variability poorly predicted local stability. Treadmill walking was associated with significant changes in both variability and local stability. Thus, motorized treadmills may produce misleading or erroneous results in situations where changes in neuromuscular control are likely to affect the variability and/or stability of locomotion. �DOI: 10.1115/1.1336798� 1
Testing for chaos in deterministic systems with noise. Physica D: Nonlinear Phenomena
, 2005
"... Recently, we introduced a new test for distinguishing regular from chaotic dynamics in deterministic dynamical systems and argued that the test had certain advantages over the traditional test for chaos using the maximal Lyapunov exponent. In this paper, we investigate the capability of the test to ..."
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Cited by 24 (6 self)
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Recently, we introduced a new test for distinguishing regular from chaotic dynamics in deterministic dynamical systems and argued that the test had certain advantages over the traditional test for chaos using the maximal Lyapunov exponent. In this paper, we investigate the capability of the test to cope with moderate amounts of noisy data. Comparisons are made between an improved version of our test and both the “tangent space ” and “direct method ” for computing the maximal Lyapunov exponent. The evidence of numerical experiments, ranging from the logistic map to an eightdimensional Lorenz system of differential equations (the Lorenz 96 system), suggests that our method is superior to tangent space methods and that it compares very favourably with direct methods.
Towards LongTerm Prediction
, 2000
"... This paper describes a simple method of obtaining longerterm predictions from a nonlinear timeseries, assuming one already has a reasonably good shortterm predictor. The usefulness of the technique is that it eliminates, to some extent, the systematic errors of the iterated shortterm predictor. ..."
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Cited by 16 (4 self)
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This paper describes a simple method of obtaining longerterm predictions from a nonlinear timeseries, assuming one already has a reasonably good shortterm predictor. The usefulness of the technique is that it eliminates, to some extent, the systematic errors of the iterated shortterm predictor. The technique we describe also provides an indication of the prediction horizon. We consider systems with both observational and dynamic noise and analyse a number of artificial and experimental systems obtaining consistent results. We also compare this method of longerterm prediction with ensemble prediction.
Differences Between Local and Orbital Dynamic Stability During Human
"... Currently there is no commonly accepted way to define, much less quantify, locomotor stability. In engineering, “orbital stability ” is defined using Floquet multipliers that quantify how purely periodic systems respond to perturbations discretely from one cycle to the next. For aperiodic systems, “ ..."
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Cited by 13 (3 self)
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Currently there is no commonly accepted way to define, much less quantify, locomotor stability. In engineering, “orbital stability ” is defined using Floquet multipliers that quantify how purely periodic systems respond to perturbations discretely from one cycle to the next. For aperiodic systems, “local stability ” is defined by local divergence exponents that quantify how the system responds to very small perturbations continuously in real time. Triaxial trunk accelerations and lower extremity sagittal plane joint angles were recorded from ten young healthy subjects as they walked for 10 min over level ground and on a motorized treadmill at the same speed. Maximum Floquet multipliers (Max FM) were computed at each percent of the gait cycle (from 0 % to 100%) for each time series to quantify the orbital stability of these movements. Analyses of variance comparing Max FM values between walking conditions and correlations between Max FM values and previously published local divergence exponent results were computed. All subjects exhibited orbitally stable walking kinematics (i.e., magnitudes of Max FM �1.0), even though these same kinematics were previously found to be locally unstable. Variations in orbital stability across the gait cycle were generally small and exhibited no
The dynamics of human gait
 European journal of physics
, 2005
"... Abstract We analyse the dynamics of human gait with simple nonlinear time series analysis methods that are appropriate for undergraduate courses. We show that short continuous recordings of the human locomotory apparatus possess properties typical of deterministic chaotic systems. To facilitate int ..."
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Cited by 11 (0 self)
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Abstract We analyse the dynamics of human gait with simple nonlinear time series analysis methods that are appropriate for undergraduate courses. We show that short continuous recordings of the human locomotory apparatus possess properties typical of deterministic chaotic systems. To facilitate interest and enable the reproduction of presented results, as well as to promote applications of nonlinear time series analysis to other experimental systems, we provide userfriendly programs for each implemented method. Thus, we provide new insights into the dynamics of human locomotion, and make an effort to ease the inclusion of nonlinear time series analysis methods into the curriculum at an early stage of the educational process.
Observing and predicting chaotic signals: Is 2% noise too much?
, 1996
"... : We discuss the influence of noise on the analysis of complex time series data. How harmful it is depends on the nature of the noise, the complexity of the signal and on the application in mind. We will give generally valid upper bounds on the feasible noise level for dimension, entropy and Lyapuno ..."
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Cited by 10 (1 self)
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: We discuss the influence of noise on the analysis of complex time series data. How harmful it is depends on the nature of the noise, the complexity of the signal and on the application in mind. We will give generally valid upper bounds on the feasible noise level for dimension, entropy and Lyapunov estimates and lower bounds for the optimal achievable prediction error. We illustrate in a number of examples why it is hard to reach these bounds in practice. We briefly sketch methods to detect, analyze and reduce measurement noise. Contents 1 Introduction 2 2 Measurement error and dynamical noise 2 3 Noise and prediction 3 3.1 Examples with known dynamics : : : : : : : : : : : : : : : : : : : : : : : : : : 4 3.2 Dynamics from a time series : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 4 Noise and scaling 8 4.1 Dimensions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 10 4.2 Entropies : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :...
Noise in Chaotic Data: Diagnosis and Treatment
 CHAOS
, 1995
"... : A prominent limiting factor in the analysis of chaotic time series are measurement errors in the data. We show that this influence can be quite severe, depending on the nature of the noise, the complexity of the signal, and on the application one has in mind. Theoretical considerations yield gener ..."
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Cited by 8 (2 self)
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: A prominent limiting factor in the analysis of chaotic time series are measurement errors in the data. We show that this influence can be quite severe, depending on the nature of the noise, the complexity of the signal, and on the application one has in mind. Theoretical considerations yield general upper bounds on the tolerable noise level for dimension, entropy and Lyapunov estimates. We discuss methods to detect and analyze the noise present in a measured data set. We show how the situation can be improved by nonlinear noise reduction. 1 Introduction Most characteristic quantities of living beings change with time. Observing this time evolution can provide better understanding of the processes at work. Qualitative changes in the dynamics can provide information about the state a subject is in, e.g. if it is healthy or suffers from some disease. Time records of biological or medical phenomena typically show some irregularity; they are neither constant nor strictly periodic. The ...
Kozelova: Cellular automata model of magnetosphericionospheric coupling Burlaga, L. F.: Multifractal structure of the interplanetary magnetic field: Voyager 2 observations near 25
"... Abstract. We propose a cellular automata model (CAM) to describe the substorm activity of the magnetosphericionospheric system. The state of each cell in the model is described by two numbers that correspond to the energy content in a region of the current sheet in the magnetospheric tail and to t ..."
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Cited by 5 (1 self)
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Abstract. We propose a cellular automata model (CAM) to describe the substorm activity of the magnetosphericionospheric system. The state of each cell in the model is described by two numbers that correspond to the energy content in a region of the current sheet in the magnetospheric tail and to the conductivity of the ionospheric domain that is magnetically connected with this region. The driving force of the system is supposed to be provided by the solar wind that is convected along the two boundaries of the system. The energy flux inside is ensured by the penetration of the energy from the solar wind into the array of cells (magnetospheric tail) with a finite velocity. The third boundary (near to the Earth) is closed and the fourth boundary is opened, thereby modeling the flux far away from the tail. The energy dissipation in the system is quite similar to