Phase Space Reconstruction Method Research Articles

Distinction of Chaos from Randomness Is Not Possible from the Degree Distribution of the Visibility and Phase Space Reconstruction Graphs.

We investigate whether it is possible to distinguish chaotic time series from random time series using network theory. In this perspective, we selected four methods to generate graphs from time series: the natural, the horizontal, the limited penetrable horizontal visibility graph, and the phase space reconstruction method. These methods claim that the distinction of chaos from randomness is possible by studying the degree distribution of the generated graphs. We evaluated these methods by computing the results for chaotic time series from the 2D Torus Automorphisms, the chaotic Lorenz system, and a random sequence derived from the normal distribution. Although the results confirm previous studies, we found that the distinction of chaos from randomness is not generally possible in the context of the above methodologies.

CNN-LSTM based incremental attention mechanism enabled phase-space reconstruction for chaotic time series prediction

To improve the prediction accuracy of chaotic time series and reconstruct a more reasonable phase space structure of the prediction network, we propose a convoluted neural network-long short-term memory (CNN-LSTM) prediction model based on the incremental attention mechanism. Firstly, a traversal search is conducted through the traversal layer for finite parameters in the phase space. Then, an incremental attention layer is utilized for parameter judgment based on the dimension weight criteria (DWC). The phase space parameters that best meet DWC are selected and fed into the input layer. Finally, the constructed CNN-LSTM network extracts spatiotemporal features and provides the final prediction results. The model is verified using Logistic, Lorenz, and sunspot chaotic time series, and the performance is compared from the two dimensions of prediction accuracy and network phase space structure. Additionally, the CNN-LSTM network based on incremental attention is compared with LSTM, CNN, recurrent neural network (RNN), and support vector regression (SVR) for prediction accuracy. The experiment results indicate that the proposed composite network model possesses enhanced capability in extracting temporal features and achieves higher prediction accuracy. Also, the algorithm to estimate the phase space parameter is compared with the traditional CAO, false nearest neighbor, and C-C, three typical methods for determining the chaotic phase space parameters. The experiments reveal that the phase space parameter estimation algorithm based on the incremental attention mechanism is superior in prediction accuracy compared with the traditional phase space reconstruction method in five networks, CNN-LSTM, LSTM, CNN, RNN, and SVR.

Алгоритм динамического распределения и балансировки нагрузки в распределенных облачных вычислениях

A mathematical model and algorithm of a two-level load management system for virtual clusters of a data processing center (data center) have been developed. At the first management level, virtual machines (VMs) are assigned to physical servers. At the same time, a greedy algorithm is used with restrictions on the time of searching for acceptable load distribution alternatives. The second level of management is implemented taking into account the chaotic structure of network traffic between the data center and users. Checking for the randomness of a time series of information traffic is carried out using Lyapunov exponents. The predictive model of the load intensity is implemented using the method of phase space reconstruction based on a set of values of a one-dimensional time series. When constructing a reconstructed phase space attractor, the time delay value is selected from the condition of reaching the zero value of the autocorrelation function, and the dimension of the embedding is determined by the angle of inclination of the straight line approximating the dependence of the value of the correlation integral on the radius of a given threshold point. The Tayler window is used to exclude correlated points in the numerical series. The criterion for evaluating the effectiveness of the developed algorithm is an integral indicator of the deviation of the load of each server from a given level. The proposed model can be used to build a data center load balancing system in conditions of its nonlinear nature.

Exploring EEG Emotion Recognition through Complex Networks: Insights from the Visibility Graph of Ordinal Patterns

The construction of complex networks from electroencephalography (EEG) proves to be an effective method for representing emotion patterns in affection computing as it offers rich spatiotemporal EEG features associated with brain emotions. In this paper, we propose a novel method for constructing complex networks from EEG signals for emotion recognition, which begins with phase space reconstruction to obtain ordinal patterns and subsequently forms a graph network representation from the sequence of ordinal patterns based on the visibility graph method, named ComNet-PSR-VG. For the proposed ComNet-PSR-VG, the initial step involves mapping EEG signals into a series of ordinal partitions using phase space reconstruction, generating a sequence of ordinal patterns. These ordinal patterns are then quantified to form a symbolized new sequence. Subsequently, the resulting symbolized sequence of ordinal patterns is transformed into a graph network using the visibility graph method. Two types of network node measures, average node degree (AND) and node degree entropy (NDE), are extracted from the graph networks as the inputs of machine learning for EEG emotion recognition. To evaluate the effectiveness of the proposed construction method of complex networks based on the visibility graph of ordinal patterns, comparative experiments are conducted using two types of simulated signals (random and Lorenz signals). Subsequently, EEG emotion recognition is performed on the SEED EEG emotion dataset. The experimental results show that, with AND as the feature, our proposed method is 4.88% higher than the existing visibility graph method and 12.23% higher than the phase space reconstruction method. These findings indicate that our proposed novel method for constructing complex networks from EEG signals not only achieves effective emotional EEG pattern recognition but also exhibits the potential for extension to other EEG pattern learning tasks, suggesting broad adaptability and application potential for our method.

Joint Analysis and Morphological Characterization of HFSWR Echo Properties during Severe Typhoon Muifa

Investigating the dynamic evolution process of the ocean and ionosphere in sudden sea conditions poses a challenging problem. To address this objective, this study utilizes actual data from high-frequency surface wave radar (HFSWR) to analyze, validate, summarize, and characterize the echo properties of the ocean and ionosphere during the severe Typhoon Muifa. By employing the short-time Fourier transform (STFT) method, the HFSWR ocean and ionosphere echoes stimulated by typhoon-induced gravity waves are observed, and the joint gravity wave features of the ocean and ionosphere echoes at different time scales are extracted. Additionally, the phase-space reconstruction method is employed to characterize the dynamical evolution of the joint gravity wave features in higher-dimensional space. Furthermore, the chaotic dynamics behavior of the joint gravity wave features is analyzed using the largest Lyapunov exponents. By combining the gravity wave features with chaotic dynamics, this study introduces a method to characterize the joint gravity wave features. The extraction of joint gravity wave features in HFSWR echoes stimulated by typhoons, along with the construction of a chaotic characterization scheme for the gravity wave features, provides an innovative approach and a solid technical foundation for studying the ocean and ionosphere using HFSWR under sudden sea conditions.

Standardized Construction Indicator Assessment and Practice of Digital Government for Chaos Computing Theory

Abstract In this paper, the maximum Lyapunov exponent and chaotic phase space reconstruction method of the chaotic computing model are studied, the embedding dimension is obtained by using the saturated correlation dimension method, and the fractal scale of the chaotic model is determined by the DFA method. Based on the relevant policy documents, an index system for assessing the level of government digital construction is constructed, and the weights of each index are calculated. The constructed index system is used to analyze the digital construction level and digital service quality of the Chinese government, and the evolutionary characteristics of the standardized construction of China’s digital government are analyzed based on the chaotic computing model. The results show that the total score of each provincial government in the sample reaches above 90, only one provincial government has a total score below 80, and the average score of the total score reaches 85.72, which indicates that most of the provincial governments in China have a good level of digital construction. The purpose of this study is to promote and optimize the digital construction of provincial governments and improve their digital construction level, which is of greater practical significance.

A novel resonance attractor evaluation method for traveling wave vibration of thin-walled gears: Modeling and experiments

Thin-walled gears have high power density and are the main gear type in aeronautical applications, but threatened by the traveling wave vibration (TWV). In this paper, a novel resonance attractor evaluation (RAE) method is proposed to describe this dynamics characteristic for the complex thin-walled gear drivetrain. The unsteady responses of the drivetrain are discretized into different time-series subsets, and each subset is reconstructed into a high-dimensional mapping phase space based on the average mutual information and the false nearest neighbor theory. Finally, the TWV is identified from the evolution of attractor features—which are quantified by indicators: the boundary radius and the phase point expansion rate. To verify the proposed method, a high-speed transmission experiment involving thin-walled bevel gears is designed, and the RAE model is established based on the proposed methodology and compared with the experimental results. Results show that this model accurately identifies resonance frequencies and describes rotation speed bands that excite the TWV. The proposed methodology provides a theoretical basis for introducing the phase space reconstruction method into the TWV study and for the safe operation of aero-engine gears.

Retracted: Recognition of Continuous Music Segments Based on the Phase Space Reconstruction Method.

[This retracts the article DOI: 10.1155/2022/4099505.].

Electric Bus Pedal Misapplication Detection Based on Phase Space Reconstruction Method.

Due to the environmental protection of electric buses, they are gradually replacing traditional fuel buses. Several previous studies have found that accidents related to electric vehicles are linked to Unintended Acceleration (UA), which is mostly caused by the driver pressing the wrong pedal. Therefore, this study proposed a Model for Detecting Pedal Misapplication in Electric Buses (MDPMEB). In this work, natural driving experiments for urban electric buses and pedal misapplication simulation experiments were carried out in a closed field; furthermore, a phase space reconstruction method was introduced, based on chaos theory, to map sequence data to a high-dimensional space in order to produce normal braking and pedal misapplication image datasets. Based on these findings, a modified Swin Transformer network was built. To prevent the model from overfitting when considering small sample data and to improve the generalization ability of the model, it was pre-trained using a publicly available dataset; moreover, the weights of the prior knowledge model were loaded into the model for training. The proposed model was also compared to machine learning and Convolutional Neural Networks (CNN) algorithms. This study showed that this model was able to detect normal braking and pedal misapplication behavior accurately and quickly, and the accuracy rate on the test dataset is 97.58%, which is 9.17% and 4.5% higher than the machine learning algorithm and CNN algorithm, respectively.

Multistep Wind Power Prediction Using Time-Varying Filtered Empirical Modal Decomposition and Improved Adaptive Sparrow Search Algorithm-Optimized Phase Space Reconstruction–Echo State Network

Accurate wind power prediction is vital for improving grid stability. In order to improve the accuracy of wind power prediction, in this study, a hybrid prediction model combining time-varying filtered empirical modal decomposition (TVFEMD), improved adaptive sparrow search algorithm (IASSA)-optimized phase space reconstruction (PSR) and echo state network (ESN) methods was proposed. First, the wind power data were decomposed into a set of subsequences by using TVFEMD. Next, PSR was used to construct the corresponding phase space matrix for sequences, which were then divided into training sets, validation sets, and testing sets. Then, ESN was used for subsequence prediction. Finally, the predicted values of all the subseries were used to determine the final predicted power. To enhance the model performance, the sparrow search algorithm was improved in terms of the discoverer position update strategy, the follower position update strategy, and the population structure. IASSA was employed to synchronously optimize multiple parameters of PSR-ESN. The results revealed that the proposed model has higher applicability and prediction accuracy than existing models.

Machine learning models combined with wavelet transform and phase space reconstruction for groundwater level forecasting

Reliable forecasting of groundwater levels plays an important role in water resource management and the prevention of environmental problems. This study aims to introduce a novel hybrid model combining machine learning with wavelet transform (WT) and phase space reconstruction (PSR) for groundwater level forecasting. The development of the proposed hybrid model consists of two levels. Firstly, a popular data-driven nonlinear artificial neural network (ANN) model was selected as the base model. Thereafter, the WT is used to decompose the groundwater level time series. In addition, the PSR method is further adopted to select the suitable sub-series components resulting from the WT decomposition processes. At the first level, the hybrid model only combines the WT with ANN (WT-ANN). At the second level, a two-step preprocessing process, WT coupled with PSR, is implemented before the ANN, namely WT-PSR-ANN. Meanwhile, three mother wavelets (Daubechies, Symlets, and Dmeyer) are selected to further investigate the effect of wavelet type. Finally, the model's performance is evaluated using visual analysis and statistical metrics (root means square error, correlation coefficient, and the Nash-Sutcliffe efficiency coefficient). The results show that the WT-ANN model outperformed the vanilla ANN model. Furthermore, the WT-PSR-ANN model achieved the best accuracy for all selected mother wavelets. The most suitable mother wavelet is specific to each well, and there is no one-size-fits-all answer. This study highlights the effectiveness of PSR combined with WT as a data preprocessing step when the ANN model is applied for groundwater level forecasting.

Inatorial forecasting method considering macro and micro characteristics of chaotic traffic flow

Traffic flow prediction is an effective strategy to assess traffic conditions and alleviate traffic congestion. Influenced by external non-stationary factors and road network structure, traffic flow sequences have macro spatiotemporal characteristics and micro chaotic characteristics. The key to improving the model prediction accuracy is to fully extract the macro and micro characteristics of traffic flow time sequences. However, traditional prediction model by only considers time features of traffic data, ignoring spatial characteristics and nonlinear characteristics of the data itself, resulting in poor model prediction performance. In view of this, this research proposes an intelligent combination prediction model taking into account the macro and micro features of chaotic traffic data. Firstly, to address the problem of time-consuming and inefficient multivariate phase space reconstruction by iterating nodes one by one, an improved multivariate phase space reconstruction method is proposed by filtering global representative nodes to effectively realize the high-dimensional mapping of chaotic traffic flow. Secondly, to address the problem that the traditional combinatorial model is difficult to adequately learn the macro and micro characteristics of chaotic traffic data, a combination of convolutional neural network (CNN) and convolutional long short-term memory (ConvLSTM) is utilized for capturing nonlinear features of traffic flow more comprehensively. Finally, to overcome the challenge that the combined model performance degrades due to subjective empirical determined network parameters, an improved lightweight particle swarm is proposed for improving prediction accuracy by optimizing model hyperparameters. In this paper, two highway datasets collected by the Caltrans Performance Measurement System (PeMS) are taken as the research objects, and the experimental results from multiple perspectives show that the comprehensive performance of the method proposed in this research is superior to those of the prevalent methods.

A Dam Deformation Residual Correction Method for High Arch Dams Using Phase Space Reconstruction and an Optimized Long Short-Term Memory Network

Dam safety is an important basic part of national water network security. Building a dam deformation prediction model based on monitoring data is crucial to ensure dam safety. However, traditional statistical regression methods have shortcomings, such as a weak nonlinear fitting ability when constructing dam deformation monitoring and prediction models. The residual part of the statistical regression results usually contains parts that cannot be effectively explained by the linear regression method, that is usually highly variable and noisy. In this study, the phase space reconstruction method is used to smooth the residual term of the statistical regression model to eliminate noise interference. On this basis, an improved long short-term memory (LSTM) neural network is used to learn the nonlinearity contained in the residual term of the linear regression. Considering the impact of parameter selection on model performance, the gray wolf optimization (GWO) algorithm is used to determine the optimal parameters of the model for better performance. A high arch dam is used as a case study, with multiple measuring points used as research objects. The experimental results show that the phase space reconstruction can effectively smooth the high-frequency components in the residual term and remove noise interference. In addition, the GWO algorithm can effectively determine the hyperparameters of the LSTM network, thereby constructing a residual prediction model with high prediction accuracy. The combination of statistical models and deep learning prediction methods can effectively improve the model prediction performance while preserving the model interpretability and transparency.

Load Forecasting Combining Phase Space Reconstruction and Stacking Ensemble Learning

Electric power load forecasting used to rely on some key information such as weather or socioeconomic development data. However, it is usually difficult to obtain accurate data on all the contributing factors. A rolling load prediction method based on phase space reconstruction of chaos theory and stacking ensemble learning is proposed. First, we adopted phase space reconstruction method to process the historical load series to avoid collecting some key data. The rolling forecasting method is used to predict a forecast length in advance, and the test data of learning model is obtained by shifting a forecast length of training data. And then we build a stacking ensemble learning model for load forecasting giving full play to the advantage of various algorithms such as Random Forest, Adaptive Boosting, Gradient Boosting Regression, Decision Tree, eXtreme Gradient Boosting and Long Short Term Memory, etc. Finally, the PJM power grid data set of the United States is used to verify the effectiveness of the proposed method, and the performance of the ensemble model is evaluated. Results show that comparing with single model forecasting, the proposed stacking ensemble learning method based on multi model fusion has higher forecasting accuracy.

On the Existence of Low-Dimensional Chaos of the Tropical Cyclone Intensity in an Idealized Axisymmetric Simulation

Abstract This study examines the potential limit in predicting tropical cyclone (TC) intensity under idealized conditions. Using the phase-space reconstruction method for TC intensity time series obtained from the CM1 idealized simulations, it is found that CM1 axisymmetric dynamics contain low-dimensional chaos at the maximum intensity equilibrium. Examination of several attractor invariants including the largest Lyapunov exponent, the Sugihara–May correlation, and the correlation dimension captures a consistent range of the chaotic attractor dimension between 4 and 5 for TC intensity at the maximum intensity equilibrium. In addition, the intensity error doubling time estimated from the largest Lyapunov exponent is roughly 1–3 h, which accords with the decay time obtained from the Sugihara–May correlation. Furthermore, the findings in this study reveal a relatively short TC intensity predictability limit for CM1, which is ∼3–9 h based on the maximum tangential wind but noticeably longer for the minimum central pressure (∼12–18 h) after reaching the mature stage. So long as the traditional metrics for TC intensity such as the maximum surface wind or the minimum central pressure is used for intensity forecast, our results support that TC intensity forecast errors will not be reduced indefinitely in any model, even in the absence of all model and observational errors. As such, the future improvement of TC intensity forecast should be based on different metrics beyond the absolute intensity errors that are currently used in real-time intensity verification. Significance Statement Using the phase-space reconstruction method for tropical cyclone (TC) intensity time series obtained from idealized axisymmetric simulations, we show that TC axisymmetric dynamics in CM1 possesses low-dimensional chaos at the maximum intensity equilibrium. This low-dimensional dynamics explains the long tradition of representing TC intensity by a few measures as in the current practice. The chaotic property of CM1 axisymmetric dynamics also suggests a relatively short predictability range for TC intensity at the maximum intensity equilibrium. The potential existence of low-dimensional chaos for TC intensity in CM1 idealized simulations as found in this study supports the use of different intensity verification metrics beyond the traditional absolute intensity errors currently used in operational model evaluation.

Phase space reconstruction, geometric filtering based Fisher discriminant analysis and minimum distance to the Riemannian means algorithm for epileptic seizure classification

BackgroundMost clinicians observe the changes of the electroencephalograms (EEGs) via the visual inspection of the recorded signals of the patients. However, the visual inspection is time consuming. Also, it is subjective and empirical. Therefore, designing an accurate and an automatic epilepsy classification system has a profound significance for reducing the workload of the medical officers. MethodFirst, this paper applies the phase space reconstruction (PSR) method for constructing the covariance matrix for characterizing the chaotic states of the EEGs. The covariance matrices are symmetric positive definite (SPD) matrices used as the feature descriptors of the EEGs. This set of the covariance matrices is in fact a kind of the Riemannian manifold. Denote this Riemannian manifold as M. Secondly, the tangent space at the Riemannian geometric mean is computed via the logarithmic mapping operator. Denote this tangent space as TM. This tangent space TM is a kind of the Euclidean space containing a point in the Riemannian manifold M. That is, the SPD matrices in the Riemannian manifold M are mapped to the matrices in the tangent space TM at the Riemannian geometric mean via the logarithmic mapping operator. Then, these matrices in the tangent space TM are further mapped to the tangent feature vectors. The set of the tangent feature vectors is a kind of the vector space. Denote this vector space as VT. Thirdly, the transformation matrix based on the Fisher discrimination analysis (FDA) is used to map the high dimensional vectors in the vector space VT to the low dimensional vectors in another vector space. Then, the discriminative vectors are projected back to the Riemannian manifold M via the exponential mapping operator. Finally, the minimum distance to the Riemannian means (MDRM) algorithm is applied for performing the epileptic seizure classification. ResultsThe computer numerical simulations are conducted based on three well known EEGs epileptic seizure datasets. Overall, our proposed method achieves the best performances for three datasets compared to the state of the art methods. Also, the computational complexity of our proposed method is low. ConclusionOur proposed method is effective and robust for performing the seizure classification. Also, our proposed method can be implemented in the real time.

Chaotic Characterization of Milling Vibration Information

In view of the strong nonlinearity of the signals in the gestation period of milling chatter, and the problem that the traditional time-frequency analysis methods cannot reveal the weak characteristics of the gestation period of chatter well, a chaotic characteristic analysis method of milling vibration information is proposed. The milling force signals of stable milling, chatter gestation and chatter outbreak states are collected through variable working condition milling force measurement experiments, and the chaotic phase space reconstruction method is used to obtain the attractor images of milling force signals in different vibration states. The experiments show that the attractor features in the chatter gestation period are more significant than the traditional time-frequency features, and the chaotic attractor images can better reveal the weak features in the chatter gestation period.

Fault identification of high voltage circuit breaker trip mechanism based on PSR and SVM

AbstractThe trip mechanism is a weakness in circuit breakers. Traditional fault identification based on the coil current is difficult to report early mechanical defects such as coil‐plunger jam. Here, the vibration signal during the trip process was extracted. Based on the coil current signal and vibration signal, the characteristics of the trip mechanism are analyzed. The phase space reconstruction (PSR) method is used to extract features from the vibration signal. Combined with the features from the coil current waveform, the feature set representing the health condition of the trip mechanism is proposed. The fault simulation tests are carried out and the variation of current vibration characteristics under fault conditions is studied. The fault identification model based on a support vector machine (SVM) is proposed and compared with the identification results when features are extracted from a single signal. When the power supply voltage is dispersed, the prediction accuracy of fault identification is 83.3% considering only the features of the current waveform or vibration signal. And the identification accuracy rises to 96.7% while using the feature set of current and vibration signals. On basis of the current signal, the method further combines the vibration signal so that the robustness of defect identification improves.

Data Mining in the Vibration Signal of the Trip Mechanism in Circuit Breakers Based on VMD-PSR

To address the difficulty in characterizing early mechanical faults in the trip mechanism of circuit breakers, a data mining method based on variational mode decomposition (VMD) and phase space reconstruction (PSR) method was proposed. First, the vibration signal in the trip stage was separated from the whole according to the current features. Then, it was decomposed using the VMD algorithm to obtain the intrinsic mode functions (IMFs) and these sub signals were mapped to high-dimensional phase space based on the PSR algorithm. Then, the features of the attractor trail shape and the recurrence plot matrixes were extracted. In order to judge the fault in the trip mechanism, a fault simulation test was carried out and the characteristic under different faults was analyzed. Based on these samples, a fault identification model is established by support vector machine (SVM) and the effectiveness is verified by other test samples. The accuracy of the SVM model is 98%, which is higher than that of the BPNN and KNN clustering models. This research supplements the existing method for condition evaluation of the trip mechanism and can provide a reference for circuit breaker fault diagnosis.

SOC prediction of Volterra adaptive filter based on chaotic time series

This paper presents an SOC (State of Charge) prediction method based on a chaotic time series Volterra adaptive filter. This method first verifies the chaotic characteristics of the SOC time series of lithium-ion batteries and then implements the phase space reconstruction method to reorganize the voltage and current data into multi-dimensional data. Furthermore, we use the Volterra adaptive filter to predict the SOC of the lithium-ion battery pack on the reconstructed multi-dimensional data. This method can effectively solve the problem that the SOC estimation error increases due to the poor real-time performance of battery voltage, current, and other parameters of vehicular lithium-ion battery packs in electric vehicles obtained by the battery management system. According to the simulation and test results, the error of the SOC estimation algorithm studied in this paper is less than 0.6%, which indicates that the Volterra adaptive filter based on chaotic time series comprehensively uses the linear and non-linear parameter characteristics of lithium-ion batteries and recombines the voltage and current data into multi-dimensional data through phase space reconstruction technology. It can reveal the relationship between the battery SOC and other variables better, achieve higher precision lithium-ion battery SOC prediction, and have better realizability.