Determining the minimum embedding dimension for state space reconstruction through recurrence networks

In this paper we suggest that an extension of a procedure recently proposed by Wayland et al. [Phys. Rev. Lett. 70, 580 (1993)] for recognizing determinism in an autonomous time series can also be used as a diagnostic for determining an appropriate embedding dimension for driven ("input-output") systems. We compare the results of this extension to the results produced by the extensions to the method of false nearest neighbors put forward by Rhodes and Morari [Proceedings of the American Control Conference, Seattle, edited by The American Automatic Control Council (IEEE, Piscataway, 1995)] and the method of averaged false nearest neighbors by Cao et al. [Int. J. Bifurcation Chaos 8, 1491 (1998)].

Parallel implementations of the False Nearest Neighbors method to study the behavior of dynamical models

The False Nearest Neighbors (FNN) method is particularly relevant in several fields of science and engineering (medicine, economics, oceanography, biological systems, etc.). In some of these applications, it is important to give results within a reasonable time scale, so the execution time of the FNN method has to be reduced. This paper describes two parallel implementations of the FNN method based on the distribution of embedding dimensions for distributed memory architectures. A "Single-Program, Multiple Data" (SPMD) paradigm is employed using a simple data decomposition approach where each processor runs the same program but acts on a different subset of the data. The computationally intensive part of the method lies mainly in the neighbor search and this task is therefore parallelized and executed using 4 to 64 processors. The accuracy and performance of the two parallel approaches are then assessed and compared to the best sequential implementation of the FNN method which appears in the TISEAN project. The results indicate that the two parallel approaches, when the method is run using 64 processors on the MareNostrum supercomputer, are between 17 and 37 times faster than the sequential one. Efficiency is between 26% and 59%.

Analysis of the global warming dynamics from temperature time series

Thread-based implementations of the false nearest neighbors method

Fault diagnosis of key components of high-speed train is an important issue to guarantee safety operation of high-speed train. In this paper, chaotic method based on reconstructed phase space is introduced for analyzing fault information of high-speed train accounting for nonlinearity, aperiodicity and nonrandomness of vibration signal of high-speed train. Firstly, the optimal delay time and the minimum embedding dimension are determined by mutual information method and Cao's method respectively in order to reconstruct phase space of vibration signals of high-speed train. And then the pseudo-phase trajectories and correlation dimension are analyzed to determine the fault information and failure states of critical components. The analysis shows the vibration signals of high-speed train have chaotic motion characteristics, and their attractors in reconstructed phase space have the similar movement structure. In addition, the areas of pseudo-phase trajectories change gradually with the damping changing of the secondary lateral damper on high-speed train. The results demonstrate that chaotic method is an effective method for analyzing fault information of high-speed train.

Correlation dimension based lossless compression of EEG signals

Appendix I - The Fractal Dimension from the Experimental Pipe-Vibration Signal

A new nonlinear method is proposed for predicting the mine water inrush. According to the fractal theory, the phase space of time series obtained from the mine water inrush is reconstructed. And the minimum embedding dimension is determined with Cao method, then the minimum embedding dimension is used for the input node of the support vector machines. The prediction model of time series is established based on the support vector machines. The mine water inrush of new well in Fang Shan mine is predicted with this method. And the prediction results show that they coincide with the observing values better. Different dimensions is chosen to predict the mine water inrush, and when the minimum embedding dimension is four, its prediction precision is the highest. Comparing with the predicting results of three methods including least square method, index fitting method and SVM method, the prediction precision of SVM is the best. It proves that the method put forward in this paper is reliable to predict the mine water inrush.

Abstract. Sivakumar et al. (2000a), by employing the correlation dimension method, provided preliminary evidence of the existence of chaos in the monthly rainfall-runoff process at the Gota basin in Sweden. The present study verifies and supports the earlier results and strengthens such evidence. The study analyses the monthly rainfall, runoff and runoff coefficient series using the nonlinear prediction method, and the presence of chaos is investigated through an inverse approach, i.e. identifying chaos from the results of the prediction. The presence of an optimal embedding dimension (the embedding dimension with the best prediction accuracy) for each of the three series indicates the existence of chaos in the rainfall-runoff process, providing additional support to the results obtained using the correlation dimension method. The reasonably good predictions achieved, particularly for the runoff series, suggest that the dynamics of the rainfall-runoff process could be understood from a chaotic perspective. The predictions are also consistent with the correlation dimension results obtained in the earlier study, i.e. higher prediction accuracy for series with a lower dimension and vice-versa, so that the correlation dimension method can indeed be used as a preliminary indicator of chaos. However, the optimal embedding dimensions obtained from the prediction method are considerably less than the minimum dimensions essential to embed the attractor, as obtained by the correlation dimension method. A possible explanation for this could be the presence of noise in the series, since the effects of noise at higher embedding dimensions could be significantly greater than that at lower embedding dimensions. Keywords: Rainfall-runoff; runoff coefficient; chaos; phase-space; correlation dimension; nonlinear prediction; noise

Realizing Euclidean distance matrices by sphere intersection

A novel non linear signal prediction method is presented using non linear signal analysis and deterministic chaos techniques in combination with Radial Basis Functions (RBF) Neural Networks for diode resonator chaotic circuits, used in industrial processes, as well as for Magnetic Resonance Spectroscopy (MRS) processes. The Time series analysis is performed by the method proposed by Grasberger and Procaccia, involving estimation of the correlation and minimum embedding dimension as well as of the corresponding Kolmogorov entropy. These parameters are used to construct the first stage of a one step / multistep predictor while an RBF Artificial Neural Network (ANN) is involved in the second stage to enhance prediction results. The novelty of the proposed two stage predictor lies on that the RBF ANN is employed as a second order predictor, that is, as an error predictor of the non-linear signal analysis stage application. This novel two stage predictor is evaluated through an extensive experimental study for both resonator circuits for industrial processes as well as for MRS signals in a preliminary stage of analysis. Different types of Neural Networks are compared as well.

A novel non linear signal prediction method is presented using non linear signal analysis and deterministic chaos techniques in combination with neural networks for a diode resonator chaotic circuit. Multisim is used to simulate the circuit and show the presence of chaos. The Time series analysis is performed by the method proposed by Grasberger and Procaccia, involving estimation of the correlation and minimum embedding dimension as well as of the corresponding Kolmogorov entropy. These parameters are used to construct the first stage of a one step / multistep predictor while a back-propagation Artificial Neural Network (ANN) is involved in the second stage to enhance prediction results. The novelty of the proposed two stage predictor lies on that the backpropagation ANN is employed as a second order predictor, that is as an error predictor of the non-linear signal analysis stage application. This novel two stage predictor is evaluated through an extensive experimental study.

The rattleback is a canoe-shaped object with the curious property of spin asymmetry. It provides a prototype of chiral dynamics, wherein lack of mirror-symmetry leads to unconventional dynamics. In this paper dynamics of rattleback are introduced and by applying Kane’s model, it is shown that one can construct a realistic mathematical model by assuming rolling without slipping and employing a torque proportional to angular velocity, in order to provide for energy dissipation. Time series analysis is performed following Grassberger–Procaccia method. The time series corresponds to the dependence of yaw, roll and spin angles on time and their corresponding angular velocities. Finally, rattleback’s strange attractor’s invariant parameters as correlation and minimum embedding dimension, Kolmogorov entropy and Lyapunov exponents, are calculated.

The paper presents the analysis of the Heart Rate Variability (HRV) signal in 19 subjects who recently had a Myocardial Infarction episode (MI). The study follows a nonlinear approach based on the multiparametric analysis of some invariant properties of the dynamical system generating the time series. First the authors reconstruct the system embedding space from the HRV time series. The False Nearest Neighbors (FNN) criterion provides the real embedding dimension value. Results show that through the FNN method it is possible to identify the correct number of LE in the system. Parameter values significantly separate subjects who after MI keep a good performance of the cardiac pump (normal ventricular ejection function, NEF) vs. the group which after MI shows a reduced ventricular ejection fraction (REF).