Phase Space Of Time Series Research Articles

Chaos-generalized regression neural network prediction model of mine water inflow

Artificial neural network (ANN) provides a new way for mine water inflow prediction. However, the effectiveness of prediction using ANN model would not be guaranteed if the influencing factors of water inflow are difficult to quantify or there are only a few observation data. Chaos theory can recover the rich dynamic information hidden in time series. By reconstructing water inflow time series in phase space, the multi-dimensional matrix could be obtained, with each column representing an influencing factor of water inflow and its value representing the change of the influencing factor with time. Therefore, a new prediction model of mine water inflow can be established by combining ANN with chaos theory when lacking data on the influencing factors of water inflow. In the present study, the No. 12 coal mine of Pingdingshan China was selected as the study site. The Chaos-GRNN model and Chaos- BPNN model of mine, water inflow were established by using the water inflow data from February 1976 to December 2013. The model was verified by using the water inflow values in the 24 months from 2014 to 2015. The number embedded dimension (M) of influencing factors of water inflow determined by phase space reconstruction was 7, meaning that there were 7 influencing factors of water inflow and 7 neurons in GRNN input layer, and the time delay was 13 months. The value of GRNN input layer neurons was determined accordingly. The maximum Lyapunov index was 0.0530, and the prediction time of GRNN was 19 months. The two models were evaluated by using four evaluation indices (R, RMSE, MAPE, NSE) and violin plot. It was found that both models can realize the long-term prediction of water inflow, and the prediction effectiveness of Chaos-GRNN model is better than that of Chaos-BPNN model.

Optimal Data Selection Rule Mining for Transformer Condition Assessment

The condition of the transformer can be accurately assessed based on dissolved gas online monitoring and analysis technology. With the widespread use of Dissolved Gas Analysis (DGA) technology, a large amount of dissolved gas time series data is obtained and forms DGA big data, which brings challenges to the data selection process of traditional condition assessment methods. Inaccurate results might be generated by selecting excessive or insufficient data from a long-span dissolved gas time series. The shorter dissolved gas time series fails to characterize the operation law of transformers fully, and the longer dissolved gas time series contains redundant information that results in inaccurate analysis models and excessive calculation consumption. This paper attempts to mine the optimal data selection rule of long-span dissolved gas time series for more accurate and efficient analysis. In this paper, C-C method was employed to calculate the phase space parameters of the time series. By analyzing the convergence of the correlation integral, the boundary conditions that maintain the time series phase space stability were revealed, and so was the relationship between the optimal length and the embedding dimension. The real cases show that the optimal data set provides the same accuracy, and significantly improves the computation speed by 113.54 times. The optimal data selection rule eliminates redundant information in the long-span dissolved gas time series and retains the “optimal data”, which not only preserves the characteristics of the original data, but also improves the computation speed of transformer condition assessment.

Temporal gap statistic: A new internal index to validate time series clustering

Abstract Unsupervised Machine Learning techniques have been developed to find out structures in datasets without considering any prior information. In such a context, the main challenge is to confirm whether the obtained structure indeed contains relevant data patterns. Aiming at solving this issue, there are several validation indexes proposed under different categories (e.g. internal, external, and relative) that allow to, for example, compare clustering algorithms or define the best parameter configurations. However, most of those indices are applied to data characterized for being collected in an independent and identically distributed manner. Thus, after performing a Systematic Literature Review, we noticed there are few researches investigating validation indexes specifically designed to deal with time-dependent data. The absence of researches for such context has motivated this work that was devoted to developing a new internal index based on Gap Statistic. Our index supports the estimation of the optimal number of clusters in a dataset only composed of time series. To reach this goal, we performed three important modifications in Gap Statistic: i) the use of a measure to calculate the distance between time series; ii) the adoption of a clustering method based on medoid; and iii) the modeling of time series in phase space using Dynamical System tools. Our results emphasize the importance of the proposed index, by accurately clustering sets of chaotic time series.

Symbolic diffusion entropy rate of chaotic time series as a surrogate measure for the largest Lyapunov exponent.

Existing methods for estimating the largest Lyapunov exponent from a time series rely on the rate of separation of initially nearby trajectories reconstructed from the time series in phase space. According to Ueda, chaotic dynamical behavior is viewed as a manifestation of random transitions between unstable periodic orbits in a chaotic attractor, which are triggered by perturbations due to experimental observation or the roundoff error characteristic of the computing machine, and consequently consists of a sequence of piecewise deterministic processes instead of an entirely deterministic process. Chaotic trajectories might have no physical reality. Here, we propose a mathematical method for estimating a surrogate measure for the largest Lyapunov exponent on the basis of the random diffusion of the symbols generated from a time series in a chaotic attractor, without resorting to initially nearby trajectories. We apply the proposed method to numerical time series generated by chaotic flow models and verify its validity.

Application research on PM2.5 concentration prediction of multivariate chaotic time series

PM2.5 is affected by complex factors such as meteorological elements in the air system and other pollutants in the air. So PM2.5 has chaotic property, which makes the prediction of PM2.5 concentration extremely difficult. In order to improve the prediction accuracy of PM2.5 concentration, this paper introduces the chaotic time series prediction method to establish multivariate time for PM2.5 concentration. The sequence prediction model achieves short-term predictions based on the hour concentration of PM2.5 in Beijing. Firstly, the chaotic time series phase space of the relevant unit is expanded into the multi-time sequence phase space, and the multi-time sequence phase space matrix of PM2.5 concentration is constructed. Then the RBF neural network is used to predict the state point in the multi-phase space system. The phase space points of the PM2.5 concentration sequence is separated for prediction. Finally, the comparison between the prediction model and the traditional prediction model is carried out. The results show that the root mean square error of the predicted PM2.5 concentration in the multivariate chaotic time series prediction model based on the phase space reconstruction is 4.92% in the next 5 hours. The average absolute error is 2.40%, which is more effective than the commonly used statistical prediction method.

Short term load forecasting based on phase space reconstruction algorithm and bi-square kernel regression model

Short term load forecasting (STLF) is an important issue for an electricity power system, to enhance its management efficiency and reduce its operational costs. However, STLF is affected by lots of exogenous factors, it demonstrates complicate characteristics, particularly, the multi-dimensional nonlinearity. Therefore, it is desired to extract some valuable features embedded in the time series, to demonstrate the relationships of the nonlinearity, eventually, to improve the forecasting accuracy. Due to the superiorities of phase space reconstruction (PSR) algorithm in reconstructing the phase space of time series, and of bi-square kernel (BSK) regression model in simultaneously considering original spectral signature and spatial information, this paper proposes a novel electricity load forecasting model by hybridizing PSR algorithm with BSK regression model, namely PSR-BSK model. The electricity load data can be sufficiently reconstructed by PSR algorithm to extract the evolutionary trends of the electricity power system and the embedded valuable features information to improve the reliability of the forecasting performances. The BSK model reasonably illustrates the spatial structures among regression points and their neighbor points to receive the rules of rotation rules and disturbance in each dimension. Eventually, the proposed PSR-BSK model including multi-dimensional regression is successfully established. The short term load data from the New South Wales (NSW, Australia) market and the New York Independent System Operator (NYISO, USA) are employed to illustrate the forecasting performances with different alternative forecasting models. The results demonstrate that, in these two employed numerical examples, the proposed PSR-BSK models all significantly receive the smallest forecasting errors in terms of MAPE (less than 2.20%), RMSE (less than 30.0), and MAE (less than 2.30), and the shortest running time (less than 400 s) than other forecasting models.

Semi-supervised time series classification on positive and unlabeled problems using cross-recurrence quantification analysis

When dealing with semi-supervised scenarios, the Positive and Unlabeled (PU) problem is a special case in which few labeled examples from a single class of interest are received to proceed with the classification of unseen instances, according to their similarities with the known class. In the scope of time series, most of the current studies propose to address this subject using a self-training approach based on the 1-Nearest Neighbor algorithm. In order to compute the most similar instance, they compare features along the time domain using the Euclidean Distance and the Dynamic Time Warping-Delta. Despite time-domain measurements permit the analysis of local series shapes, they disconsider temporal recurrences commonly found in natural phenomena (e.g. population growth, climate studies) and are more sensitive to local noise and fluctuations, leading to poor classification performances as confirmed in this paper. This drawback motivated us to propose the use of the Maximum Diagonal Line of the Cross-Recurrence Quantification Analysis (MDL-CRQA), applied on the time series phase space, as similarity measurement. The phase space is obtained after applying Takens embedding theorem on the series, unfolding temporal relationships and dependencies among data observations. As consequence, by comparing phase spaces rather than the series themselves, we can assess how their trajectories evolve along time, including their periodicities and temporal cycles, as well as decreasing noise influences. Experimental results confirm MDL-CRQA improves classification results for PU time series when compared against the mostly used time-domain similarity measurements.

Analysis and Prediction for Time Series on Torque Friction of Rolling Bearings

Abstract Based on the Cao, mutual information, and small-data methods, the embedding dimension, delay time, and maximum Lyapunov exponent are calculated, respectively, to analyze the chaos characteristics of rolling bearings. The curve attractor of x(t) − x(t + (m − 1)τ)x(t) is constructed to parse the dynamics features on friction torque in the phase space of time series. According to the five prediction methods utilized—one-rank local-region, adding-weight one-rank local-region, improved adding-weight one-rank local-region, dial basis function (RBF) neural network, and Volterra series—the time series of bearings A, B, and C are forecasted with the first 400 experiment data as training values and the latter 57 data as test values to verify the prediction models’ feasibility. Finally, the bootstrap-maximum-entropy method is proposed to effectively fuse the results of these five prediction methods, and obtain the estimation interval and true value of friction torque. Experimental investigation shows that the friction torque phase trajectory has a linear increasing trend. These five forecasting models are effective for friction torque time series prediction with small error and high precision. The range of the fused estimation interval is relatively small, and the maximum error between the estimated true value and the experiment value is only 5.183 %, so the fluctuation information and the change trends of friction torque are accurately described. Moreover, the proposed models do not consider the probability distribution and trend information of the research system, breaking from the features of traditional statistical models.

Secure communication scheme analysis via complex networks

SummaryRecently, some existing works have introduced novel way to construct complex networks from embedded time series, which provides new sights into the organizational properties of the time series in phase space. In this paper, we attempt to answer the fundamental question of “how much information regarding the dynamic property of the original time series can be extracted from these networks.” To this end, we propose a new method for reconstructing time series from the networks. We compare the reconstructed time series from these networks and that from the recurrence plot. We find that these networks contain topological information of the embedded time series to a certain degree. In general, they are more powerful than the recurrence plot method in the reconstruction of embedded time series. In addition, we study a new generalized projective synchronization (GPS) of coupled complex dynamical networks with different sizes via feedback control and impulsive control. Based on the stability analysis of impulsive system, a network synchronization criterion is established. These works may find potential application for secure communication via networks.

Application of Nonlinear Dynamical Methods for Arc Welding Quality Monitoring

Owing to its diverse, the stability of arc signals in high-powered submerged arc welding is not very salient, and weld defects are difficult to detect automatically. Aimed at this problem, this paper proposes a noise robustness algorithm for calibrating the singularity points and denoting the kinetics and stability of arc. Firstly, reconstruct a vector, which is the calculation of the approximate entropy in phase space, denotes the distortion of arc. Then, a algorithm for calculation is given based on reconstruction of chaotic time series in phase space. Finally, we apply the calculation of approximate entropy algorithm in phase space to flaw detection for arc signals, which is efficient proved by experimental results.

Spatial variation of deterministic chaos in mean daily temperature and rainfall over Nigeria

Daily rainfall and temperature data from 47 locations across Nigeria for the 36-year period 1979–2014 were treated to time series analysis technique to investigate some nonlinear trends in rainfall and temperature data. Some quantifiers such as Lyapunov exponents, correlation dimension, and entropy were obtained for the various locations. Positive Lyapunov exponents were obtained for the time series of mean daily rainfall for all locations in the southern part of Nigeria while negative Lyapunov exponents were obtained for all locations in the Northern part of Nigeria. The mean daily temperature had positive Lyapunov exponent values (0.35–1.6) for all the locations. Attempts were made in reconstructing the phase space of time series of rainfall and temperature.

Applying the Hybrid Model of EMD, PSR, and ELM to Exchange Rates Forecasting

Financial time series forecasting has been a challenge for time series analysts and researchers because it is noisy, nonstationary and chaotic. To overcome this limitation, this study uses empirical mode decomposition (EMD) and phase space reconstruction (PSR) to assist in the task of financial time series forecasting. In addition, we propose an approach that combines these two data preprocessing methods with extreme learning machine (ELM). The approach contains four steps as follows. (1) EMD is used to decompose the dynamics of the exchange rate time series into several components of intrinsic mode function (IMF) and one residual component. (2) The IMF and residual time series phase space is reconstructed to reveal its unseen dynamics according to the optimum time delay $$\tau $$ź and embedding dimension m. (3) The reconstructed time series datasets are divided into two datasets: training and testing, in which the training datasets are used to build ELM models. (4) A regression forecast model is set up for each IMF as well as the residual component by using ELM. The final prediction results are obtained by compositing the prediction values. To verify the effectiveness of the proposed approach, four exchange rates are chosen as the forecasting targets. Compared with some existing state-of-the-art models, the proposed approach yields superior results. Academically, we demonstrated the validity and superiority of the proposed approach that integrates EMD, PSR, and ELM. Corporations or individuals can apply the results of this study to acquire accurate exchange rate information and reduce exchange rate expenses.

Determining Reconstruction Variables and Parameters on Improved Multi-Objective Immune Algorithm

Aimed at multi-dimension of evaluating reconstruction effect and the correlation between reconstruction variables and reconstruction parameters, improved multi-objective immune algorithm (IMOIA) was proposed to simultaneously determine reconstruction variables and parameters in multivariate time series phase space. In IMOIA, antibody coding was binary strings cascade of multi-parameter, antibody evaluating was three functions, and a new population evaluating S - is presented. Immune operators were designed: overall clone operator to improve speed, mutation operator with multiple changing probability on gene loci and S enhanced local and global search capability to ensure algorithm convergence, Clonal Selection operator on Rank and Crowding-distance to ensure solutions uniformity, repetitive antibodies determination and deletion operator to ensure solutions diversity. Use our method to two benchmarking chaos system, simulation results showed that our method can simultaneously solved multiple combinations of optimal reconstruction variables and parameters and had obvious superiority to other intelligent optimization algorithms and determining reconstruction parameter 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.

Chaotic Characteristics of Workface Gas Emission Time-Series Data

The chaotic time series phase space reconstruction theory based in this paper. First, the appropriate embedding dimension and delay time are selected by minimum entropy rate. Followed the chaotic behavior are analyzed by the use of the Poincare section map and Power spectrum of time series from the qualitative point of view. Based on NLSR LLE the quantitative study of the chaotic time series characteristics indicators is proposed. Finally, the gas emission workface of Hebi 10th Mine Coal is studied. The several analytical results of the above methods show that: the gas emission time-series data of this workface has chaotic characteristics.

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.

Network Security Situation Prediction Using Artificial Immune System and Phase Space Reconstruction

To solve network security situation prediction problem, a novel Prediction approach for network security situation based on artificial Immune system and Phase Space Reconstruction (PIPSR) is proposed. In PIPSR, we use phase space reconstruction to analyze the time series of network security situation and to reconstruct proper time series phase space; then, we use immune evolution mechanism to construct corresponding prediction model for network security situation; and lastly we use this prediction model to forecast network security situation. The simulation results show that PIPSR can more exactly forecast the future network security situation than GAP and BPNNP.

Superfamily classification of nonstationary time series based on DFA scaling exponents

The superfamily phenomenon of time series with different dynamics can be characterized by the motif-rank patterns observed in the nearest-neighbor networks of the time series in phase space. However, the determinants of superfamily classification are unclear. We attack this problem by studying the influence of linear temporal correlations and multifractality using fractional Brownian motions and multifractal random walks. Numerical investigations unveil that the classification of a superfamily phenomenon is uniquely determined by the detrended fluctuation analysis (DFA) scaling exponent α of the time series. Only four motif patterns are observed in the simulated data, which are delimited by three DFA scaling exponents α ≃ 0.25, α ≃ 0.35 and α ≃ 0.45. The validity of the result is confirmed by stock market indices and turbulence velocity signals.

Chaotic dynamic characteristics of Z component in geomagnetic variation field

对地球变化磁场进行了混沌动力学特性分析，主要进行了以下两个方面的工作：1)运用多种方法对多个地磁台站同一时段变化磁场分量数据（分为磁扰较小和磁扰较大时段）进行了混沌特性分析，主要是为了以更充分的判据来验证地球变化磁场是否具有混沌特性；2)同一地磁台站不同时段变化磁场分量数据的混沌特性分析，主要是验证地球变化磁场是否具有变参数混沌特性.分析结果表明：1)变化磁场时间序列确实具有混沌特性；2)从混沌时间序列相空间重构的角度可初步认为模拟地球变化磁场动力学系统所必须的独立变量应该为6左右；3)变化磁场时间序列不

High-dimensional delay selection for regression models with mutual information and distance-to-diagonal criteria

Delay selection for time series phase space reconstruction may be performed using a mutual information (MI) criterion. However, the delay selection is in that case limited to the estimation of a single delay using MI between two variables only. A high-dimensional estimator of the MI may be used to select more than one delay between more than two variables but this approach is rather time consuming. In this paper, an alternative fast criterion is proposed to optimize all delays for a high-dimensional phase space reconstruction: the distance-to-diagonal (DD) criterion, based on a geometrical heuristic. The use of the distance to diagonal criterion is illustrated and compared to MI on artificial and benchmark time series.