Multivariate Phase Space Reconstruction Research Articles

A novel method of nonuniform phase space reconstruction for multivariate prediction of daily runoff

Phase space reconstruction is crucial for predicting chaotic hydrological time series. However, traditional multivariate phase space reconstruction methods, such as high-dimensional and low-dimensional phase space reconstruction (HDPSR and LDPSR) neglect variable interdependence, resulting in redundant information in the reconstructed phase space and affecting the model prediction accuracy. To address this limitation, this study proposes a new nonuniform phase space reconstruction (NUPSR) method that combines the minimum redundancy maximum relevance criterion and conditional entropy. Then, using HDPSR, LDPSR, NUPSR, particle swarm optimization (PSO), support vector regression (SVR), and bidirectional long- and short-term memory neural networks (BiLSTM), the models HDPSR-PSO-SVR, LDPSR-PSO-SVR, NUPSR-PSO-SVR, HDPSR-PSO-BiLSTM, LDPSR-PSO-BiLSTM, and NUPSR-PSO-BiLSTM are built to evaluate the performance of the NUPSR method. To demonstrate the effectiveness of the NUPSR method in multistep daily runoff prediction, the models PSO-NUPSR-BiLSTM, LDPSR-PSO-BiLSTM, and HDPSR-PSO-BiLSTM are also built. The multifactor daily runoff data from five hydrological stations in the Weihe River Basin are used to evaluate the prediction performance of all models. The results show that the state variables obtained by NUPSR are more independent and less redundant than those obtained by HDPSR and LDPSR, as evidenced by the embedded state variables and lower embedding dimension of NUPSR and the performance of the model based on NUPSR outperforms based on HDPSR and LDPSR. Particularly, the PSO-NUPSR-BiLSTM model exhibits a slower decrease in prediction accuracy compared to the other two models as the number of prediction steps increases. In summary, the NUPSR method effectively selects and reconstructs predictors, significantly improves the prediction accuracy of the coupled models, and has broad application prospects in multivariate prediction in fields such as hydrology.

Multivariate phase space reconstruction and Riemannian manifold for sleep stage classification

Background and ObjectiveSleep was highly imperative in human daily life. However, an increasing number of people were undergoing sleep deprivation and sleep disorders. Sleep stage classification became a highly essential process in sleep scoring. Nevertheless, the visual scoring of arousals during sleep routinely performed by sleep specialists was a challenging exercise. MethodsThis paper introduced a novel approach for sleep stage classification using covariance feature matrix architecture with multivariate phase space reconstruction (MPSR). The goal was to capture the geometric properties and the hidden dynamic characteristics of multiple physiological signals. The covariance matrices constructed through the MPSR approach were considered as symmetric and positive definite (SPD) matrices, forming a Riemannian manifold space. These SPD matrices in the Riemannian manifold were mapped to the matrices in the tangent space, allowing them to be vectorized and treated as feature vectors in Euclidean space. Finally, an ensemble learning classifier was applied to perform various sleep stage tasks. ResultsOur proposed method was evaluated on three benchmark datasets to assess its effectiveness and robustness. For the tasks of both Five class and Six class sleep stages, the proposed approach in ten-fold cross validation achieved high accuracy of 93.57% and 92.56% for the Sleep EDF dataset, 86.36% and 84.18% for the DREAMS Subjects dataset, as well as 88.93% and 88.42% for the Sleep EDF Expanded dataset (95 subjects), respectively. In leave-one-subject-out cross validation, our proposed approach for the tasks of both Five class and Six class sleep stages yielded an accuracy of 84.46% and 80.73% for the Sleep EDF dataset, 82.50% and 79.51% for the DREAMS Subjects dataset, as well as 93.25% and 92.06% for the Sleep EDF Expanded-20 dataset (20 subjects), respectively. ConclusionCompared to the traditional sample covariance matrix (SCM), the covariance feature matrix using the MPSR method successfully captured the distinction of spatial information among various sleep stages. Moreover, our proposed method obtained good performance without requiring computationally large artifact suppression or a long signal decomposition process.

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.

Short-term Load Forecasting of Multi-Energy in Integrated Energy System Based on Multivariate Phase Space Reconstruction and Support Vector Regression Mode

In order to alleviate the energy crisis and improve the energy utilization rate, the integrated energy system (IES) has become an important way of energy utilization. IES integrates electricity, natural gas, heating and cooling energy supply. Accurate energy load forecasting is essential, which has a significant impact on the economic scheduling and optimal operation of the IES. Herein, a combined model prediction method of multivariate phase space reconstruction (MPSR) and support vector regression (SVR) is proposed in this paper. First, a quantitative analysis of the coupling relationship between different integrated energy subsystems is conducted, and Pearson correlation analysis theory is used to analyse the historical time series of electrical, cooling, heat, gas loads and environmental factors one by one, then the input variables of the combined forecasting model are obtained. After that, the multivariate phase space is reconstructed by the C-C method, and the SVR model is used to predict electricity, cooling, heating and gas loads. Final, the model is validated by the actual data of the IES in Arizona State University, the results of three cases show the efficiency and high accuracy of the proposed forecasting method that considers the coupling relationship between multi-energy loads of IES.

Short-term wind power forecast based on chaotic analysis and multivariate phase space reconstruction

The randomness and volatility of wind power time series, which are an external reflection of their internal chaotic dynamics, have always been important factors affecting the accuracy of wind power prediction. The chaotic characteristics of wind power data have not been studied deeply enough in existing research. Therefore, in this paper, a short-term wind power forecast method based on chaotic analysis is proposed, including chaotic time series estimation and multivariate phase space reconstruction (PSR). First, we calculate the largest Lyapunov exponent of wind power time series to measure the degree of chaos in wind power data. Then, the chaotic characteristics of several wind power time series from some neighboring wind farms are analyzed, based on which two multivariate multi-dimensional PSR models are established. Afterwards, the nearest neighbors (NNs) of the data to be predicted are selected in the phase space and delivered to the least-square support vector machine (LSSVM) model to experiment the forecasting. Wind power data collected from several adjacent wind sites are used to conduct the simulation studies, and the results have shown the effectiveness of the proposed method and its advantage over the classic LSSVM model in terms of accuracy and stability.

Adaptive symbolic transfer entropy and its applications in modeling for complex industrial systems.

Directed coupling between variables is the foundation of studying the dynamical behavior of complex systems. We propose an adaptive symbolic transfer entropy (ASTE) method based on the principle of equal probability division. First, the adaptive kernel density method is used to obtain an accurate probability density function for an observation series. Second, the complete phase space of the system can be obtained by using the multivariable phase space reconstruction method. This provides common parameters for symbolizing a time series, including delay time and embedding dimension. Third, an optimization strategy is used to select the appropriate symbolic parameters of a time series, such as the symbol set and partition intervals, which can be used to convert the time series to a symbol sequence. Then the transfer entropy between the symbolic sequences can be carried out. Finally, the proposed method is analyzed and validated using the chaotic Lorenz system and typical complex industrial systems. The results show that the ASTE method is superior to the existing transfer entropy and symbolic transfer entropy methods in terms of measurement accuracy and noise resistance, and it can be applied to the network modeling and performance safety analysis of complex industrial systems.

Ultra-Short-Term Wind Power Prediction Based on Multivariate Phase Space Reconstruction and Multivariate Linear Regression

In order to improve the accuracy of wind power prediction (WPP), we propose a WPP based on multivariate phase space reconstruction (MPSR) and multivariate linear regression (MLR). Firstly, the multivariate time series (TS) are constructed through reasonable selection of wind power and weather factors, which are closely associated with wind power. Secondly, the phase space of the multivariate time series is reconstructed based on the chaos theory and C-C method. Thirdly, an auto regression model for multivariate phase space is created by regarding phase variables as state variables, and the very-short-term wind power is predicted by using a multi-linear regression algorithm. Finally, a parallel algorithm based on map/reduce is presented to improve computing speed. A cloud computing platform, Hadoop consisting of five nodes, is established as a matter of convenience, followed by the prediction of wind power of a wind farm in the Hunan province of China. The experimental results show that the model based on MPSR and MLR is more accurate than both the continuous method and the simple approximation method, and the parallel algorithm based on map/reduce effectively accelerates the computing speed.

Dynamic fault prognosis for multivariate degradation process

Abstract Stochastic characteristics of degradation process provides a great challenge for multi-step fault prognosis. Prediction accuracy is a key feature for multivariable time series in the degradation process. In order to achieve good prediction accuracy, it is necessary to take time varying feature of time series into consideration. This paper develops a dynamic multi-step fault prognosis approach for the multivariable time series in degradation process. Firstly, a time varying dynamic model is established via multivariable phase space reconstruction (PSR) method. Then a hybrid algorithm combining iterative dynamic least square-support vector regression (LS-SVR) method and moving window mechanism is proposed for multi-step degradation evolution prediction before fault happening. Finally, a comparative case study on fault prognosis is provided to validate the effectiveness and the feasibility of the proposed algorithms.

Short-term traffic flow prediction model using particle swarm optimization–based combined kernel function-least squares support vector machine combined with chaos theory

Short-term traffic flow prediction is an important part of intelligent transportation systems research and applications. For further improving the accuracy of short-time traffic flow prediction, a novel hybrid prediction model (multivariate phase space reconstruction–combined kernel function-least squares support vector machine) based on multivariate phase space reconstruction and combined kernel function-least squares support vector machine is proposed. The C-C method is used to determine the optimal time delay and the optimal embedding dimension of traffic variables’ (flow, speed, and occupancy) time series for phase space reconstruction. The G-P method is selected to calculate the correlation dimension of attractor which is an important index for judging chaotic characteristics of the traffic variables’ series. The optimal input form of combined kernel function-least squares support vector machine model is determined by multivariate phase space reconstruction, and the model’s parameters are optimized by particle swarm optimization algorithm. Finally, case validation is carried out using the measured data of an expressway in Xiamen, China. The experimental results suggest that the new proposed model yields better predictions compared with similar models (combined kernel function-least squares support vector machine, multivariate phase space reconstruction–generalized kernel function-least squares support vector machine, and phase space reconstruction–combined kernel function-least squares support vector machine), which indicates that the new proposed model exhibits stronger prediction ability and robustness.

Short-term Traffic Flow Prediction Based on Multivariable Phase Space Reconstruction and LSSVM

Real-time and accurate short-term traffic flow prediction is the premise and key of intelligent traffic control and guidance system. According to this problem, this paper put forward a prediction model based on multivariable phase space reconstruction and least squares support vector machine (LSSVM). First, the model confirms embedding dimension and delay time of the traffic flow, occupancy and average speed time series by analyzing their chaotic characteristics, and reconstructs multivariable state space. Second, the phase points obtained after reconstruction are as input, and the last traffic flow parameters came from the following phase points are as output. Finally, the LSSVM which be trained is adapted to realize short-term traffic flow prediction. This research compares this model with a model based on univariate phase space reconstruction and LSSVM, and the results show that the model proposed in this paper predicts better.

Short-term Traffic Flow Prediction Based on Multivariable Phase Space Reconstruction and LSSVM

Short-term Traffic Flow Prediction Based on Multivariable Phase Space Reconstruction and LSSVM

Blast Furnace System Modeling by Multivariate Phase Space Reconstruction and Neural Networks

AbstractTime series data collected from a medium‐size blast furnace (BF) is analyzed using the phase space reconstruction. To achieve better reconstruction, multivariate correlation analysis is first applied to screen out correlated variables, which shows that three important variables, i.e., silicon content in hot metal ([Si]), permeability index (FF), and coal injection (PM), are most appropriate for multivariate reconstruction. The time delay and embedding dimension are determined via the autocorrelation function and false nearest neighbor method. With the reconstructed time series, the neural networks model is applied to construct the predictive model for silicon content in hot metal. The simulation shows that the models based on multivariate reconstruction give better predictions than those obtained by univariate reconstruction. Moreover, it reveals that multivariate reconstruction can greatly mitigate the drawbacks caused by insufficiency of data.

Applying One Fuzzy Neural Network Model Based on Multivariate Phase Space Reconstruction to Predict Deformation of Deep Foundation Pit

Applying One Fuzzy Neural Network Model Based on Multivariate Phase Space Reconstruction to Predict Deformation of Deep Foundation Pit

Multivariate chaotic time series phase space reconstruction based on extending dimension by conditional entropy

For multivariate chaotic time series, a method of conditional entropy extending dimension(CEED) in the reconstructed phase space is proposed. First, the delay time of any variable time series is selected by mutual information method, and then the embedding dimension of phase space is extended by the conditional entropy. This method can ensure the independence of reconstructed coordinates from low space to high space and eliminate the redundancy of phase space, because the largest condition entropy is choosen. The effective input vector for the prediction of multivariate time series is given. Simulations of the Lorenz system and Henon system show that the neural network predictions of multivariate time series are much better than the prediction of univariate and existing multivariate. Therefore, CEED is effective for multivariate chaotic systems.

An approach to phase space reconstruction from multivariate data based on data fusion

To cope with the problem of phase space reconstruction from univariate time series and multivariate time series, the novel approach to phase space reconstruction from multivariate data based on data fusion is presented in this paper. According to Bayes estimation theory, the phase points in the same phase space reconstructed from multivariate data are fused, then an optimal fusion phase space could be determined. The approach is applied to multivariate phase space reconstructions of Lorenz system and Rssler system, respectively. Compared with the figures reconstructed from univariate datas, the information reconstructed from multivariate data includes the main characters of all univariate data and represents the comprehensive information of system attractor, which makes the phase space reconstructed more abundant. At last, the approach is applied to multivariate phase space reconstruction of oil film whirling in the rotor system. The information reconstructed includes all the characters of the system, which improves the veracity for fault diagnosis. So all the analysis further shows that the approach presented here is effective.

Multivariate phase space reconstruction by nearest neighbor embedding with different time delays

A recently proposed nearest neighbor based selection of time delays for phase space reconstruction is extended to multivariate time series, with an iterative selection of variables and time delays. A case study of numerically generated solutions of the x- and z coordinates of the Lorenz system, and an application to heart rate and respiration data, are used for illustration.