Embedding Phase Space Research Articles

A complex systems analysis of stick-slip dynamics of a laboratory fault

We study the stick-slip behavior of a granular bed of photoelastic disks sheared by a rough slider pulled along the surface. Time series of a proxy for granular friction are examined using complex systems methods to characterize the observed stick-slip dynamics of this laboratory fault. Nonlinear surrogate time series methods show that the stick-slip behavior appears more complex than a periodic dynamics description. Phase space embedding methods show that the dynamics can be locally captured within a four to six dimensional subspace. These slider time series also provide an experimental test for recent complex network methods. Phase space networks, constructed by connecting nearby phase space points, proved useful in capturing the key features of the dynamics. In particular, network communities could be associated to slip events and the ranking of small network subgraphs exhibited a heretofore unreported ordering.

基于Lyapunov-Elman的网络控制系统时延预测方法

The time-delay in the networked control system is an important parameter affecting the performance of a system. A maximum Lyapunov exponent and Elman neural network prediction method is presented according to the time-delay prediction in the networked control system based on the Internet. Firstly, time-delay sequences are reconstructed by phase space reconstruction, embedding dimension and time delayed variable is obtained. Then one step time-delay is forecasted by maximum Lyapunov exponent and Elman neural network. A final time delay prediction value is obtained after two prediction methods are multiplied by different weight coefficients. Finally, based on the weight coefficient optimization, this paper presents an improved free search algorithm, whose convergence accuracy and speed are superior to those of the standard free search algorithm. Simulation results show that, time-delay prediction method based on maximum Lyapunov exponent and Elman neural network in the paper has higher prediction accuracy and smaller prediction error than other prediction methods.

Chaotic time series prediction of E-nose sensor drift in embedded phase space

Chemical sensor drift shows a chaotic behavior and unpredictability in long-term observation which makes it difficult to construct an appropriate sensor drift treatment. The main purpose of this paper is to study a new methodology for chaotic time series modeling of chemical sensor observations in embedded phase space. This method realizes a long-term prediction of sensor baseline and drift based on phase space reconstruction (PSR) and radial basis function (RBF) neural network. PSR can memory all of the properties of a chaotic attractor and clearly show the motion trace of a time series, thus PSR makes the long-term drift prediction using RBF neural network possible. Experimental observation data of three metal oxide semiconductor sensors in a year demonstrate the obvious chaotic behavior through the Lyapunov exponents. Results demonstrate that the proposed model can make long-term and accurate prediction of chemical sensor baseline and drift time series.

Estimating the largest Lyapunov exponent and noise level from chaotic time series

A novel method for estimating simultaneously the largest Lyapunov exponent (LLE) and noise level (NL) from a noisy chaotic time series is presented in this paper. We research the influence of noise on the average distance of different pairs of points in an embedding phase space and provide a rescaled formula for calculating the LLE when the time series is contaminated with noise. Our algorithm is proposed based on this formula and the invariant of the LLE in different dimensional embedding phase spaces. With numerical simulation, we find that the proposed method provides a reasonable estimate of the LLE and NL when the NL is less than 10% of the signal content. The comparison with Kantz algorithm shows that our method gives more accurate results of the LLE for the noisy time series. Furthermore, our method is not sensitive to the distribution of the noise.

Noise-level estimation of noisy chaotic time series based on the invariant of the largest Lyapunov exponent

A novel method of estimating the noise level from a noisy chaotic time series based on the invariant of the largest Lyapunov exponent is presented in this paper. The influence of noise on the distance between two points in an embedding phase space is considered, and then based on the invariant of the largest Lyapunov exponent in a different dimensional embedding phase space, the algorithm is proposed to estimate the noise level. Simulation results show that the estimated values of noise level agree well with the true values when the noise level is less than 10%. And this method is not sensitive to the distribution of noise. Therefore, the method is useful for estimating the noise level of noisy chaotic time series.

A New Ensemble Learning Method for Temporal Pattern Identification

In this paper we present a method for identification of temporal patterns predictive of significant events in a dynamic data system. A new hybrid model using Reconstructed Phase Space (MRPS) and Hidden Markov Model (HMM) is applied to identify temporal patterns. This method constructs phase space embedding by using individual embedding of each variable sequences. We also employ Hidden Markov Models (HMM) to the multivariate sequence data to categorize multi-dimensional data into three states, e.g. normal, patterns and events. A support vector machine optimization method is used to search an optimal classifier to identify temporal patterns that are predictive of future events. We performed two experimental applications using chaotic time series and natural gas usage series related to the natural gas usage forecasting problem. Experiments show that the new method significantly outperforms the original RPS framework and neural network method.

Water-level oscillations in the Adriatic Sea as coherent self-oscillations inferred by independent component analysis

We analyze tide gauge records at four stations of the ISPRA network located in the Adriatic Sea basin (Eastern Italy), namely, going from North to South: Trieste, Ancona, Ortona and Otranto. We use linear and nonlinear methods in the frequency and time domains, including spectral and Independent Component Analysis (ICA), inter-times occurrence, and phase space embedding dimension evaluation. We show that four tidal constituents can be extracted by ICA and interpreted as coherent self-sustained oscillations. Finally, we show that these constituents can be reproduced by adopting a simple nonlinear oscillatory model that generalizes classical Andronov oscillator with the inclusion of a time dependent pumping.

Performance improvement of a phase space detection algorithm for electrocardiogram wave morphology classification

An algorithm based on embedding phase space signal was developed by the authors in previous works. The algorithm detects the characteristic points of the waves of a multilead electroacardiogram (ECG). In the present work, the parameters of this algorithm are optimized to improve its performance. The algorithm uses 2 configurable parameters to obtain the phase spacethe dimension of the phase space and the delayand a threshold to select the points of the ECG. By a proper selection of these parameters, the algorithm obtains all the points in the ECG that are similar to a reference one that was selected by the analyst. Several strategies have been developed and incorporated in the phase space algorithm to obtain the optimal values of these parameters based on the sampling rate and the number of leads in the records. The professional only needs to mark the reference point and the associated wave to it, for example, the start and end of a P wave and its peak. The algorithm obtains every P wave of the ECG record and a classification of their morphology using clustering techniques. Moreover, a simple graphical interface has been developed to ease its use. The algorithm was applied to detect the start, peak, and end of the P waves of a collection of ECG records of 6 minutes. Using this information, the algorithm extracts and classifies the P waves by applying clustering techniques to study their variability. The algorithm can also be used online to detect and classify different types of morphologies in any ECG wave. A future use of this algorithm will be the detection of several extracardiac pathologies in the ECG Holter, for example, sleep apnea. (Less)

Synergy of chaos theory and artificial neural networks in chaotic time series forecasting

A unique technique based on chaos theory and artificial neural networks (ANN) is developed to analyse and forecast chaotic time series. An embedding theorem is used to determine the embedding parameters. Accordingly the chaotic time series is reconstructed into phase space points. Based on chaos theory, there exists an unknown mathematical equation which can forecast the future value of the phase space points. Therefore, the embedded phase space points are fed into a neural network and trained. When the unknown phase space is predicted, the future value of time series is obtained accordingly. Two neural network architectures, feedforward and Elman, are utilised in this study. The Mackey-Glass (M-G), logistic and Henon time series are used to validate the performance of the proposed technique. The numerical experimental results confirm that the proposed method can forecast the chaotic time series effectively and accurately when compared with the existing forecasting methods.

Analysis of nonlinear dynamics of vocal folds using high-speed video observation and biomechanical modeling

Primary voice production occurs in the larynx through vibrational movements carried out by vocal folds. However, many problems can affect this complex system resulting in voice disorders. In this context, time–frequency–shape analysis based on embedding phase space plots and nonlinear dynamics methods have been used to evaluate the vocal fold dynamics during phonation. For this purpose, the present work used high-speed video to record the vocal fold movements of three subjects and extract the glottal area time series using an image segmentation algorithm. This signal is used for an optimization method which combines genetic algorithms and a quasi-Newton method to optimize the parameters of a biomechanical model of vocal folds based on lumped elements (masses, springs and dampers). After optimization, this model is capable of simulating the dynamics of recorded vocal folds and their glottal pulse. Bifurcation diagrams and phase space analysis were used to evaluate the behavior of this deterministic system in different circumstances. The results showed that this methodology can be used to extract some physiological parameters of vocal folds and reproduce some complex behaviors of these structures contributing to the scientific and clinical evaluation of voice production.

Chaotic time series prediction with residual analysis method using hybrid Elman–NARX neural networks

Residual analysis using hybrid Elman–NARX neural network along with embedding theorem is used to analyze and predict chaotic time series. Using embedding theorem, the embedding parameters are determined and the time series is reconstructed into proper phase space points. The embedded phase space points are fed into an Elman neural network and trained. The residual of predicted time series is analyzed, and it was observed that residuals demonstrate chaotic behaviour. The residuals are considered as a new chaotic time series and reconstructed according to embedding theorem. A new Elman neural network is trained to predict the future value of the residual time series. The residual analysis is repeated several times. Finally, a NARX network is used to capture the relationship among the predicted value of original time series and residuals and original time series. The method is applied to Mackey–Glass and Lorenz equations which produce chaotic time series, and to a real life chaotic time series, Sunspot time series, to evaluate the validity of the proposed technique. Numerical experimental results confirm that the proposed method can predict the chaotic time series more effectively and accurately when compared with the existing prediction methods.

Chaotic operation by a single transistor circuit in the reverse active region

In this paper, we present an externally triggered experimental chaotic circuit with a bipolar junction transistor operating in its reverse active region in order to investigate for possible control features in its output phase portraits. Nonlinear time series modeling techniques are applied to analyze the circuit's output voltage oscillations and reveal the presence of chaos, while the chaos itself is achieved by controlling the amplitude of the applied input signal. The phase space, which describes the behavior evolution of a nonlinear system, is reconstructed using the delay embedding theorem suggested by Takens. The time delay used for this reconstruction is chosen after examining the first minimum of the collected data average mutual information, while the sufficient embedding dimension is estimated using the false nearest-neighbor algorithm which has a value of 5. Also the largest Lyapunov exponent is estimated and found equal to 0.020 48. Finally, the phase space embedding based weight predictor algorithm is employed to make a short-term prediction of the chaotic time series for which the system's governing equations may be unknown.

Traffic Flow Forecasting Based on Chaos Neural Network

Traffic flow forecasting has become an emphasis question for discussion in traffic engineering domain and one kernel study in Intelligent Transportation System. After ensuring the traffic flow has the characteristic of chaos, traffic flow real data have been used to reconstruct phase space. Calculate the saturation phase space embedding dimension and maximal Lyapunov exponent. By above all, a chaos neural network model is constructed, which can make high precision short-term forecast for the nonlinear big-lagged system even by imperfect and variation inputs. At last, a forecasting example provides that the traffic flow forecasting based on chaos neural network is validity and feasibility.

Fractal surface analysis of Zircaloy-4 SEM micrographs using the time-series method

Abstract The SEM microfractographies of Zircaloy-4 are studied by the time-series method. We first develop a computer application which associates a time series to each SEM micrograph. Furthermore, we will apply the phase space embedding technique to reconstruct the attractor and to compute the autocorrelation dimension. Using the fractal analysis technique, the SEM microfractographies of the fracture surface of the Zircaloy-4 samples have been analyzed.

Input space to neural network based load forecasters

The literature has presented a great variety of methodologies for electricity load forecasting. This fact reflects the possible consequences of electricity load forecasting errors, which can cause blackouts with serious social and economic implications. In recent years, papers on this topic have shown the potential of neural network based models. For short term horizons in particular, this capability is explained by the neural networks' flexibility in capturing nonlinear interdependencies between the load and exogenous variables. However, one issue in the development of neural network based electricity load forecasts has not been treated with proper care. Input space representation and complexity control interact strongly, and new solutions for this problem are needed. This paper proposes original automatic procedures for selecting input variables in feedforward neural network based load forecasting models. Filter (model independent) and wrapper (model dependent) approaches have both been investigated in the context of short term load forecasting. Regarding the filter approach, a procedure for identifying relevant subspaces is introduced. For each relevant subspace, interdependences based on a normalized measure of mutual information are estimated to select input variables. Furthermore, phase-space embedding, based on Takens' theorem, is applied in combination with multi-resolution decomposition via wavelet transformation. The results have been compared with a Bayesian wrapper, in which probes have been employed as input references of usefulness.

Study on gastric interdigestive pressure activity based on phase space reconstruction and FastICA algorithm

To investigate the features of the gastric interdigestive pressure activity under normal physiological conditions, we have developed the wireless radiotelemetry capsule based on a telemetry technique. Twelve healthy volunteers participated in this study. Pressure activity data which are an important index of gastric motility can be obtained from the wireless radiotelemetry capsule. But the capsule only records single-dimensional pressure time series which may contain a few interdependent components simultaneously. Automated embedding phase space reconstruction algorithm is employed to reconstruct multi-dimensional phase space. Then the dominant and separated component of the gastric contractions is identified using FastICA algorithm. Finally the use of Hilbert Huang transform method for analyzing the characters of gastric motility is investigated. The results show that the proposed method is an effective approach for the analysis of the gastric pressure series.

Phase Space Analysis: The Equilibrium of the Solar Magnetic Cycle

We present the results of a statistical study of the solar cycle based on the analysis of the superficial toroidal magnetic field component phase space. The magnetic field component used to create the embedded phase space was constructed from monthly sunspot number observations since 1750. The phase space was split into 32 sections (or time instants) and the average values of the orbits on this phase space were calculated (giving the most probable cycle). In this phase space it is shown that the magnetic field on the Sun’s surface evolves through a set of orbits that go around a mean orbit (i.e., the most probable magnetic cycle that we interpret as the equilibrium solution). It follows that the most probable cycle is well represented by a van der Pol oscillator limit curve (equilibrium solution), as can be derived from mean-field dynamo theory. This analysis also retrieves the empirical Gnevyshev – Ohl’s rule between the first and second parts of the solar magnetic cycle. The sunspot number evolution corresponding to the most probable cycle (in phase space) is presented.

Chaotic Time Series Prediction Using a Neuro-Fuzzy System with Time-Delay Coordinates

This paper presents an investigation into the use of the delay coordinate embedding technique in the multi-input- multioutput-adaptive-network-based fuzzy inference system (MANFIS) for chaotic time series prediction. The inputs to the MANFIS are embedded-phase-space (EPS) vectors preprocessed from the time series under test, while the output time series is extracted from the output EPS vectors from the MANFIS. A moving root-mean-square error is used to monitor the error over the prediction horizon and to tune the membership functions in the MANFIS. With the inclusion of the EPS preprocessing step, the prediction performance of the MANFIS is improved significantly. The proposed method has been tested with one periodic function and two chaotic functions including Mackey-Glass chaotic time series and Duffing forced-oscillation system. The prediction performances with and without EPS preprocessing are statistically compared by using the t-test method. The results show that EPS preprocessing can help improve the prediction performance of a MANFIS significantly.

Method for detection of changes in the EEG induced by the presence of sensory stimuli

The onset and offset of sensory stimuli evoke transient changes in the electroencephalogram (EEG) that can be detected by linear and/or nonlinear analysis. However, there is presently no systematic procedure to quantify the brain-electrical-activity correlate of the presence of a stimulus (as opposed to its onset evoked potential). We describe a method for detecting a stimulus-related change in brain electrical activity that persists while the stimulus is present (presence effect). The method, which is based on phase-space embedding of the EEG time series followed by quantitative analysis of the recurrence plot of the embedded signal, was used to demonstrate the occurrence of a presence effect in separate groups of human subjects exposed to sound, a magnetic field, and light. Any form of law-governed dynamical activity induced in the EEG can be detected, particularly activity that is nonlinearly related to the stimulus. Salient mathematical features of the method were reproduced in a model EEG system containing known nonlinear determinism.

Nonlinear noise reduction of chaotic time series based on multidimensional recurrent LS-SVM

In order to resolve the noise reduction in chaotic time series, a novel method based on multidimensional recurrent least squares support vector machine (MDRLS-SVM) is proposed in this paper. Considering the evolvement feature of the chaotic system, we utilize the recurrent version of least squares support vector machines (LS-SVM) to manipulate the iterative problem. From the high-dimensional phase space point of view, the function approximation in the high-dimensional embedding phase space is carried out and the noise reduction achieved simultaneously based on the reconstructed embedding phase theory. We show by means of simulation of Ikeda map that the proposed method is able to provide accurate results in noise reduction of chaotic system.