Phase Space Reconstruction Method Research Articles

Chaos in Time Series of Sakarya River Daily Flow Rate

In this study, possible low dimensional chaotic behavior of Sakarya river flow rates is investigated via nonlinear time series techniques. To reveal the chaotic dynamics, the maximal positive Lyapunov exponent is calculated from the reconstructed phase space, which is obtained using the phase space reconstruction method. The method reconstructs a phase space from the scalar time series, which depicts the real system’s invariants Positive values, because the Lyapunov exponent values calculated using the appropriate software program indicate possibility of chaotic behavior. Analyzed data involve the monthly average flow rates of eleven main branches of Sakarya River through the years 1960-2000.

The Chaotic Attractor Analysis of DJIA Based on Manifold Embedding and Laplacian Eigenmaps

By using the techniques of Manifold Embedding and Laplacian Eigenmaps, a novel strategy has been proposed in this paper to detect the chaos of Dow Jones Industrial Average. Firstly, the chaotic attractor of financial time series is assumed to lie on a low-dimensional manifold that is embedded into a high-dimensional Euclidean space. Then, an improved phase space reconstruction method and a nonlinear dimensionality reduction method are introduced to help reveal the structure of the chaotic attractor. Next, the empirical study on the financial time series of Dow Jones Industrial Average shows that there exists an attractor which lies on a manifold constructed by the time sequence of Moving average convergence divergence; finally, Determinism Test, Poincaré section, and translation analysis are used as test approaches to prove both whether it is a chaos and how it works.

Mid- and long-term runoff predictions by an improved phase-space reconstruction model

In recent years, the phase-space reconstruction method has usually been used for mid- and long-term runoff predictions. However, the traditional phase-space reconstruction method is still needs to be improved. Using the genetic algorithm to improve the phase-space reconstruction method, a new nonlinear model of monthly runoff is constructed. The new model does not rely heavily on embedding dimensions. Recognizing that the rainfall–runoff process is complex, affected by a number of factors, more variables (e.g. temperature and rainfall) are incorporated in the model. In order to detect the possible presence of chaos in the runoff dynamics, chaotic characteristics of the model are also analyzed, which shows the model can represent the nonlinear and chaotic characteristics of the runoff. The model is tested for its forecasting performance in four types of experiments using data from six hydrological stations on the Yellow River and the Yangtze River. Results show that the medium-and long-term runoff is satisfactorily forecasted at the hydrological stations. Not only is the forecasting trend accurate, but also the mean absolute percentage error is no more than 15%. Moreover, the forecast results of wet years and dry years are both good, which means that the improved model can overcome the traditional ‘‘wet years and dry years predictability barrier,’’ to some extent. The model forecasts for different regions are all good, showing the universality of the approach. Compared with selected conceptual and empirical methods, the model exhibits greater reliability and stability in the long-term runoff prediction. Our study provides a new thinking for research on the association between the monthly runoff and other hydrological factors, and also provides a new method for the prediction of the monthly runoff.

A hybrid wind speed forecasting model based on phase space reconstruction theory and Markov model: A case study of wind farms in northwest China

Forecasting wind speed is a crucial issue in the wind power industry, and wind energy is gradually coming to be viewed as one of the most promising alternative energy sources because of its cleanliness and renewability for large-scale commercial production. With the aim of developing accurate tools for forecasting wind speed, this paper presents a novel hybrid intelligent forecasting model based on Least Square Support Vector Machine and the Markov model. Prior to conducting the wind speed forecasting, the original wind speed series is processed using the C–C method, which is a phase space reconstruction method, to automatically determine the input form. Subsequently, the LSSVM (least squares support vector machine) model, which is optimized using the PSOGSA (partical swarm optimization combined with gravitational search algorithm) algorithm, is employed to forecast wind speed. Additionally, the Markov model is developed to make error corrections with the state ranges determined using the FCM (fuzzy C-means) model. The proposed model takes the average forecasting values between the outputs generated by LSSVM and error corrected LSSVM model as the final forecasting results. The simulation results indicate that the proposed model can outperform the other models discussed in this paper with respect to forecasting accuracy and can be a suitable tool for forecasting wind speeds.

Research on Fractal Features in Milling ZrO2 Ceramics

In order to study the fractal features of the tool profile of worn surface and cutting force during the process of cutting hard brittle materials such as ceramics, the milling experiments of ZrO2 were carried out. The images of worn surface and cutting force were collected and preprocessed with digital image processing technology and phase space reconstruction method. Then the fractal dimensions were calculated and the change regularity was analyzed. The research findings show that the fractal dimension of worn surface is affected by wear rates and machining time, and can be different while the wear loss is same. The fractal dimension of cutting force oscillates with a high-low-high general trend.

Harmonic signal detection method from strong chaotic background based on optimal filter

It is of great significance to study the weak harmonic signal detection from strong chaotic background. Current detection methods mainly use the chaotic phase space reconstruction method based on Takens theory, among which the neural network method has attracted the most attention. However, these methods require high signal-to-interference-plus-noise ratio (SINR) and are sensitive to Gaussian white noise, etc. Noticing the fact that the second-order statistical properties of chaotic signals are stationary, we propose a harmonic signal detection method from strong chaotic background based on optimal filter. We first construct a data matrix, whose rows are the detection signal and reference signals. The reference signals only contain chaotic interference. Then we calculate the one-dimensional fast Fourier transformation of the data matrix to make each column of the matrix form a frequency channel. The harmonic signal can be detected by searching each frequency channel in the frequency domain, thus the signal detection problem is converted into an optimization problem. Further, we use the optimization theory to design a filter such that it can maintain the gain of the signal from the current frequency channel and suppress signals from other frequency channels as far as possible. Finally, the harmonic signal can be obtained by calculating the output SINR of each frequency channel. In order to reduce the calculation, we can further design a local region optimal filter. We choose part of frequency channels to constitute a local area, thus the dimension of the chaotic interference covariance matrix is greatly reduced. Theoretically speaking, the more the number of auxiliary frequency channels, the better the detection results are. However, in the practical application, choosing two channels on the left and right side of current channel each can obtain a very good detection effect. After obtaining the chaotic interference covariance matrix, we can further achieve the output SINR of each frequency channel. Compared with the traditional methods, the proposed method has the following advantages: 1) it can detect a weak harmonic signal under lower SINR; 2) it can detect a greater range of signal amplitude; 3) it is robust against white Gaussian noise. The simulation results with taking Lorenz system as the strong chaotic background show that the proposed method still has a very good detection effect when SINR =-81.03 dB, and the stronger the harmonic signal, the better the detection effect is, while the neural network method can work under the condition of SINR higher than -67.03 dB; the proposed method still can correctly detect the target signal in the case that the SNR is as low as -20 dB, but the neural network method has a poor detection effect under the same condition.

An immunity-based time series prediction approach and its application for network security situation

To effectively improve the prediction precision of network security situation and prevent the large-scale network security attacks, an immunity-based time series prediction approach for network security situation (ITSPA) is proposed. In ITSPA, the concepts and formal definitions of antigen, antibody and affinity used for predicting network security situation are given; and meanwhile, the mathematical models of antibody evolution operators used for establishing the prediction model of network security situation are shown. For the time series of network security situation, its chaotic characteristics are analyzed and the corresponding sample space is reconstructed by phase space reconstruction method; then, the corresponding prediction model is constructed by artificial immune mechanism; finally, this prediction model is used for predicting the time series of network security situation. To demonstrate the predicting effectiveness of ITSPA, four typical time series (namely real-time network probe situation, real-time network situation, short-term network probe situation and short-term network situation) obtained from DARPA 1999 data set and long-term network security situation time series obtained from HoneyNet Project data set are used for simulating experiments. The experimental results show that ITSPA is an effective prediction approach for the time series of network security situation.

Data-Driven Based Temperature Prediction of Ferroalloy Electric Furnace Smelting Process

Aiming at the features of nonlinearity, time delay, high dimension, big noise and others in the electric furnace smelting process, and the smelting process parameters some control process needs cannot be obtained by sensors, so the online parameter prediction method is of great significance for dynamic optimization control of system. In this paper, by using chaos theory and phase space reconstruction method, based on the idea of data-driven modeling, the temperature prediction model of crucible in the smelting process of ferroalloy electric furnace is established based on multivariate time series. Experiments show that the established data-driven prediction model can well predict the temperature change in the electric furnace smelting process, the root mean square errors of prediction are respectively less than 0.0715,and this has good guidance for practical production.

Classification of normal and epileptic seizure EEG signals using wavelet transform, phase-space reconstruction, and Euclidean distance

This paper proposes new combined methods to classify normal and epileptic seizure EEG signals using wavelet transform (WT), phase-space reconstruction (PSR), and Euclidean distance (ED) based on a neural network with weighted fuzzy membership functions (NEWFM). WT, PSR, ED, and statistical methods that include frequency distributions and variation, were implemented to extract 24 initial features to use as inputs. Of the 24 initial features, 4 minimum features with the highest accuracy were selected using a non-overlap area distribution measurement method supported by the NEWFM. These 4 minimum features were used as inputs for the NEWFM and this resulted in performance sensitivity, specificity, and accuracy of 96.33%, 100%, and 98.17%, respectively. In addition, the area under Receiver Operating Characteristic (ROC) curve was used to measure the performances of NEWFM both without and with feature selections.

Parameters determination method of phase-space reconstruction based on differential entropy ratio and RBF neural network

Phase space reconstruction is the first step of recognizing the chaotic time series. On the basis of differential entropy ratio method, the embedding dimension mopt and time delay τ are optimal for the state space reconstruction could be determined. But they are not the optimal parameters accepted for prediction. This study proposes an improved method based on the differential entropy ratio and Radial Basis Function (RBF) neural network to estimate the embedding dimension m and the time delay τ, which have both optimal characteristics of the state space reconstruction and the prediction. Simulating experiments of Lorenz system and Doffing system show that the original phase space could be reconstructed from the time series effectively, and both the prediction accuracy and prediction length are improved greatly.

A new method based on PSR and EA-GMDH for host load prediction in cloud computing system

Host load prediction is one of the most effective measures for improving resource utilization in cloud computing systems. Due to the drastic fluctuation of the host load in the Cloud, accurately predicting the host load remains a challenge. In this paper, we propose a new prediction method that combines the Phase Space Reconstruction method and the Group Method of Data Handling based on an Evolutionary Algorithm. The performance of our proposed method is evaluated using two real-world load traces. The first is the load trace in a traditional distributed system, whereas the second is in a Google data center. The results show that the proposed method achieves a better prediction performance than some state-of-the-art methods.

Chaotic Characteristics Identification on Terminal Departing Passenger Traffic Time Series

On the basis of actual survey of departing passengers to reach terminal, chaotic time series prediction theory was adopted for data analysis in this paper. In order to find out the self-similarity of time series, this paper divided passenger traffic into two kinds: holiday traffic and non-holiday traffic by changing interval scale. The optimal delay time and the best embedding dimension had been calculated by using time series phase space reconstruction method. To confirm whether the time series have chaotic characteristics or not, it took the largest Lyapunov exponent as determining criterion.Then the optimal time intervals of passenger traffic time series with chaotic character were determined. The study provides a theoretical basis for the application of chaos theory in passenger traffic forecast.

Non-linear dynamics of lower leg muscle surface electromyogram during repeated plantar flexion

This study investigated non-linear behaviour of gastrocnemius medialis (calf muscle) surface electromyogram (EMG) collected during repeated plantar flexion in both fatigue and non-fatigue conditions. Data were collected in the beginning and immediately after the participant reported fatigue. The study used the phase space reconstruction method to estimate the largest Lyapunov exponents and Kaplan–Yorke dimension of EMG signals. Positive Lyapunov exponent and fractional Kaplan–Yorke dimension values confirmed the presence of chaotic behaviour in the EMG readings. A paired sample t-test showed that the mean largest Lyapunov exponent following fatigue onset was significantly higher than before fatigue (p < 0.05) for right, left and combined right and left gastrocnemius muscles. A logistic regression model revealed the largest Lyapunov exponent as a significant predictor of fatigue onset with a high odds ratio. However, these measures were not found to be good predictors of postural stability as measured by the centre of pressure data.

Structural Damage Detection of the Simple Beam Based on Responses Phase Space

In this paper, the technology of attractor phase space in chaotic theory is introduced and applied in the structural damage detection. Firstly the phase plane is constructed with the displacement and acceleration responses. Using the changes of phase plane topology of intact and damaged responses, a new damage index is extracted, and the structural damage existence and severity are identified successfully. Since some of the state variables can not be measured, a method of phase space reconstruction is proposed using single dynamic response. The dynamic responses are directly displayed into phase space, realizing transforming the signals from time domain to space domain. Then using the reconstructed phase space, the damage is diagnosed. The results indicate that the phase space reconstruction method has good robustness to noise, and higher sensitivity compared with traditional modal-based methods. The phase space reconstruction method can calculate the value of the damage index using single dynamic response, so that a single sensor can monitor structural damage existence and severity.

ROTATING SYNCHRONIZATION IN THREE-DIMENSIONAL CHAOTIC SYSTEMS

Projective synchronization investigates the synchronization of systems evolve in same orientation, however, in practice, the situation of same orientation is only minority, and the majority is different orientation. This paper investigates the latter, proposes the concept of rotating synchronization, and verifies its necessity and feasibility via theoretical analysis and numerical simulations. Three conclusions were elicited: first, in three-dimensional space, two arbitrary nonlinear chaotic systems who evolve in different orientation can realize synchronization at end; second, projective synchronization is a special case of rotating synchronization, so, the application fields of rotating synchronization is more broadly than that of the former; third, the overall evolving information can be reflected by single state variable's evolving, it has self-similarity, this is the same as the basic idea of phase space reconstruction method, it indicates that we got the same result from different approach, so, our method and the phase space reconstruction method are verified each other.

KPCA Denoising and its Application in Machinery Fault Diagnosis

This paper proposes a kernel principal component analysis (KPCA)-based denoising method for removing the noise from vibration signal. Firstly, one-dimensional time series is expanded to multidimensional time series by the phase space reconstruction method. Then, KPCA is performed on the multidimensional time series. The first kernel principal component is the denoised signal. A rolling bearing denoising example verify the effectiveness of the proposed method

Dynamics of hourly sea level at Hillarys Boat Harbour, Western Australia: a chaos theory perspective

Water level forecasting using recorded time series can provide a local modelling capability to facilitate local proactive management practices. To this end, hourly sea water level time series are investigated. The records collected at the Hillarys Boat Harbour, Western Australia, are investigated over the period of 2000 and 2002. Two modelling techniques are employed: low-dimensional dynamic model, known as the deterministic chaos theory, and genetic programming, GP. The phase space, which describes the evolution of the behaviour of a nonlinear system in time, was reconstructed using the delay-embedding theorem suggested by Takens. The presence of chaotic signals in the data was identified by the phase space reconstruction and correlation dimension methods, and also the predictability into the future was calculated by the largest Lyapunov exponent to be 437 h or 18 days into the future. The intercomparison of results of the local prediction and GP models shows that for this site-specific dataset, the local prediction model has a slight edge over GP. However, rather than recommending one technique over another, the paper promotes a pluralistic modelling culture, whereby different techniques should be tested to gain a specific insight from each of the models. This would enable a consensus to be drawn from a set of results rather than ignoring the individual insights provided by each model.

Learning a stick-balancing task involves task-specific coupling between posture and hand displacements

Theories of motor learning argue that the acquisition of novel motor skills requires a task-specific organization of sensory and motor subsystems. We examined task-specific coupling between motor subsystems as subjects learned a novel stick-balancing task. We focused on learning-induced changes in finger movements and body sway and investigated the effect of practice on their coupling. Eight subjects practiced balancing a cylindrical wooden stick for 30min a day during a 20day learning period. Finger movements and center of pressure trajectories were recorded in every fifth practice session (4 in total) using a ten camera VICON motion capture system interfaced with two force platforms. Motor learning was quantified using average balancing trial lengths, which increased with practice and confirmed that subjects learned the task. Nonlinear time series and phase space reconstruction methods were subsequently used to investigate changes in the spatiotemporal properties of finger movements, body sway and their progressive coupling. Systematic increases in subsystem coupling were observed despite reduced autocorrelation and differences in the temporal properties of center of pressure and finger trajectories. The average duration of these coupled trajectories increased systematically across the learning period. In short, the abrupt transition between coupled and decoupled subsystem dynamics suggested that stick balancing is regulated by a hierarchical control mechanism that switches from collective to independent control of the finger and center of pressure. In addition to traditional measures of motor performance, dynamical analyses revealed changes in motor subsystem organization that occurred when subjects learned a novel stick-balancing task.

Phase space reconstruction of chaotic dynamical system based on wavelet decomposition

In view of the disadvantages of the traditional phase space reconstruction method, this paper presents the method of phase space reconstruction based on the wavelet decomposition and indicates that the wavelet decomposition of chaotic dynamical system is essentially a projection of chaotic attractor on the axes of space opened by the wavelet filter vectors, which corresponds to the time-delayed embedding method of phase space reconstruction proposed by Packard and Takens. The experimental results show that, the structure of dynamical trajectory of chaotic system on the wavelet space is much similar to the original system, and the nonlinear invariants such as correlation dimension, Lyapunov exponent and Kolmogorov entropy are still reserved. It demonstrates that wavelet decomposition is effective for characterizing chaotic dynamical system.

Chaotic Time Series Prediction Using Immune Optimization Theory

To solve chaotic time series prediction problem, a novel Prediction approach for chaotic time series based on Immune Optimization Theory (PIOT) is proposed. In PIOT, the concepts and formal definitions of antigen, antibody and affinity being used for time series prediction are given, and the mathematical models of immune optimization operators being used for establishing time series prediction model are exhibited. Chaotic time series is analyzed and corresponding sample space is reconstructed by phase space reconstruction method; then, the prediction model of chaotic time series is constructed by immune optimization theory; finally, using this prediction model to forecast chaotic time series. To demonstrate the effectiveness of PIOT, the three typical chaotic nonlinear time series are generated by nonlinear dynamics systems that are Lorenz, Mackey–Glass and Henon, respectively, and are used for simulating prediction. The simulation results show that PIOT is a feasible and effective prediction method, and meanwhile provides a novel prediction approach for chaotic time series.