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To accurately predict the remaining useful life (RUL) of cutting tool, a novel RUL prediction method is proposed. Firstly, the complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) is used to decompose original cutting tool vibration signals to get six intrinsic mode function (IMF) components from each sample. Secondly, high-frequency IMF components and low-frequency IMF components are obtained from IMF components and they are respectively fused into high-frequency data and low-frequency data using the improved fine-to-coarse reconstruction (IFTC), and high-frequency data and low-frequency data are reconstructed using phase space reconstruction (PSR). Thirdly, multiple prediction branches are adopted to construct an ensemble RUL prediction model for cutting tool, the high-frequency data and low-frequency data are input into bi-directional long short-term memory (BiLSTM) and convolutional neural network (CNN) to train a RUL prediction model respectively in each prediction branch. Finally, a series of experiments are conducted to verify the effectiveness of the proposed RUL prediction method, and the results show that the proposed method obtains a high score of RUL prediction for cutting tool.

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.

Early ventricular fibrillation (VF) prediction is critical for prevention of sudden cardiac death, and can improve patient survival. Generally, electrocardiogram (ECG) signal features are extracted to predict VF, a process which plays an important role in prediction accuracy. Therefore, this study first proposes a novel feature based on topological data analysis (TDA) to improve the accuracy of early ventricular fibrillation prediction. Firstly, the heart activity is regarded as a cardiac dynamical system, which is described by phase space reconstruction. Then the topological structure of the phase space is characterized with persistent homology, and its statistical features are further extracted and defined as TDA features. Finally, 60 subjects (30 VF, 30 healthy) from three public ECG databases are used to validate the prediction performance of the proposed method. Compared to heart rate variability features and box-counting features, TDA features achieve a superior accuracy of 91.7%. Additionally, the three types of features are combined as fusion features, achieving the optimal accuracy of 95.0%. The fusion features are then ranked, and the first seven components are all from the TDA features. It follows that the proposed features provide a significant effect in improving the predictive performance of early VF.

Stock market prediction is extremely important for investors because knowing the future trend of stock prices will reduce the risk of investing capital for profit. Therefore, seeking an accurate, fast, and effective approach to identify the stock market movement is of great practical significance. This study proposes a novel turning point prediction method for the time series analysis of stock price. Through the chaos theory analysis and application, we put forward a new modeling approach for the nonlinear dynamic system. The turning indicator of time series is computed firstly; then, by applying the RVFL-GMDH model, we perform the turning point prediction of the stock price, which is based on the fractal characteristic of a strange attractor with an infinite self-similar structure. The experimental findings confirm the efficacy of the proposed procedure and have become successful for the intelligent decision support of the stock trading strategy. Supplementary Information The online version contains supplementary material available at 10.1007/s10115-021-01602-3.

Nonlinear dynamics in Everglades STAs

The traditional circuit breaker fault diagnosis method suffers from insufficient feature information extraction and is easily affected by abnormal signal acquisition. To address this, this paper introduces the phase space reconstruction algorithm to reconstruct the current signal for fault diagnosis based on phase trajectory features. The proposed method uses a first-order forward differencing method and mutual information method to process abnormal data and select the parameters of the reconstruction, then extract overall and local inflection point features to construct a fault feature set. The support vector machine algorithm-based model is trained and tested using actual samples, and the results show that the proposed method can adaptively sample anomalous signals, exhibit strong robustness, and significantly improve the accuracy of fault classification.

Precipitation prediction is an important technical mean for flood and drought disaster early warning, rational utilization, and the development of water resources. Complementary ensemble empirical mode decomposition (CEEMD) can effectively reduce mode aliasing and white noise interference; extreme learning machines (ELM) can predict non-stationary data quickly and easily; and the fruit fly optimization algorithm (FFOA) has better local optimization ability. According to the multi-scale and non-stationary characteristics of precipitation time series, a new prediction approach based on the combination of complementary ensemble empirical mode decomposition (CEEMD), extreme learning machine (ELM), and the fruit fly optimization algorithm (FFOA) is proposed. The monthly precipitation data measured in Zhengzhou City from 1951 to 2020 was taken as an example to conduct a prediction experiment and compared with three prediction models: ELM, EMD-HHT, and CEEMD-ELM. The research results show that the sum of annual precipitation predicted by the CEEMD-ELM-FFOA model is 577.33 mm, which is higher than the measured value of 572.53 mm with an error of 4.80 mm. The average absolute error is 0.81 and the average relative error is 1.39%. The prediction value of the CEEMD-ELM-FFOA model can closely follow the changing trend of precipitation, which shows a better prediction effect than the other three models and can be used for regional precipitation prediction.

Cryptocurrencies are new kinds of electronic currencies based on communication technologies. These currencies have attracted the attention of investors. However, cryptocurrencies are very volatile and unpredictable. For investors, it is very difficult to make investment decisions in cryptocurrency market. Therefore, revealing changes in the dynamics of cryptocurrencies are valuable for investors. Bitcoin is the most popular and representative cryptocurrency in cryptocurrency market. In this study how dynamical properties of Bitcoin changed through time is analyzed with recurrence quantification analysis (RQA). RQA is a pattern recognition-based time series analysis method that reveals dynamics of the time series by calculating some metrics called RQA measures. This method has been successfully applied to nonlinear, nonstationary, short and chaotic time series and do not assume a statistical model. RQA can reveal important properties of time series data such as determinism, laminarity, stability, randomness, regularity and complexity. By using sliding window RQA we show that in 2021 RQA measures for Bitcoin prices collapse and Bitcoin becomes more unpredictable, more random, more unstable, more irregular and less complex. Therefore, dynamics and stability of the Bitcoin prices significantly changed in 2021.

Non-specific sympathetic arousal responses to different stressful elicitations can be easily recognized from the analysis of physiological signals. However, neural patterns of sympathetic arousal during physical and mental fatigue are clearly not unitary. In the context of physiological monitoring through wearable and non-invasive devices, electrodermal activity (EDA) is the most effective and widely used marker of sympathetic activation. This study presents ComEDA, a novel approach for the characterization of complex dynamics of EDA. ComEDA overcomes the methodological limitations related to the application of nonlinear analysis to EDA dynamics, is not parameter-sensitive and is suitable for the analysis of ultra-short time series. We validated the proposed algorithm using synthetic series of white noise and 1/f noise, varying the number of samples from 50 to 5000. By applying our approach, we were able to discriminate a statistically significant increase of complexity in the 1/f noise with respect to white noise, obtaining p-values in the range [4.35 × 10

We present a novel method for interpolating univariate time series data. The proposed method combines multi-point fractional Brownian bridges, a genetic algorithm, and Takens’ theorem for reconstructing a phase space from univariate time series data. The basic idea is to first generate a population of different stochastically-interpolated time series data, and secondly, to use a genetic algorithm to find the pieces in the population which generate the smoothest reconstructed phase space trajectory. A smooth trajectory curve is hereby found to have a low variance of second derivatives along the curve. For simplicity, we refer to the developed method as PhaSpaSto-interpolation, which is an abbreviation for phase-space-trajectory-smoothing stochastic interpolation. The proposed approach is tested and validated with a univariate time series of the Lorenz system, five non-model data sets and compared to a cubic spline interpolation and a linear interpolation. We find that the criterion for smoothness guarantees low errors on known model and non-model data. Finally, we interpolate the discussed non-model data sets, and show the corresponding improved phase space portraits. The proposed method is useful for interpolating low-sampled time series data sets for, e.g., machine learning, regression analysis, or time series prediction approaches. Further, the results suggest that the variance of second derivatives along a given phase space trajectory is a valuable tool for phase space analysis of non-model time series data, and we expect it to be useful for future research.

Flight Maneuver Recognition (FMR) refers to the automatic recognition of a series of aircraft flight patterns and is a key technology in many fields. The chaotic nature of its input data and the professional complexity of the identification process make it difficult and expensive to identify, and none of the existing models have general generalization capabilities. A general framework is proposed in this paper, which can be used for all kinds of flight tasks, independent of the aircraft type. We first preprocessed the raw data with unsupervised clustering method, segmented it into maneuver sequences, then reconstructed the sequences in phase space, calculated their approximate entropy, quantitatively characterized the sequence complexity, and distinguished the flight maneuvers. Experiments on a real flight training dataset have shown that the framework can quickly and correctly identify various flight maneuvers for multiple aircraft types with minimal human intervention.

The change and development of leakage current (LC) during the flashover process of insulators are closely related to the discharge process. Characteristic quantities of LC play an important role in the identification and prediction of ice-covered flashover development stages. In this paper, the feature extraction method for LC of ice-covered insulator is discussed based on the shape characteristics of chaotic attractor. Firstly, the power spectrum and the largest Lyapunov exponent are used to analyze the LC of ice-covered insulator qualitatively and quantitatively, which confirmed positive chaotic features during the flashover process. On this basis, the LC is reconstructed in phase space. To solve the problem that the attractor in the high-dimensional space cannot be observed in the phase space reconstruction, the principal vector analysis method is used to project the high-dimensional reconstruction vector into the three-dimensional space to reduce the dimension. Finally, the evolution law of chaotic attractor in each stage of LC under different pollution degree is analyzed and discussed. The results show that the LC of ice-covered insulator has obvious characteristics of attractor shape at different stages of pollution, and is sensitive to the degree of pollution and each stage of development. So the shape of the attractor of LC has important application value for the identification of pollution flashover state and prediction of pollution flashover process of ice-covered insulators.

In order to better characterize the performance degradation trend of rolling bearings, a new performance degradation evaluation method based on principal component analysis (PCA), phase space reconstruction (PSR) and kernel extreme learning machine (KELM), namely PAPRKM is proposed to evaluate the performance degradation of rolling bearings in this paper. In the PAPRKM method, the time-domain and frequency-domain features of the vibration signal are extracted to construct the high-dimension feature matrix. Then the PCA is used to reduce the dimension of the feature matrix in order to represent the running state and the declining trend of rolling bearings, so as to eliminate the redundancy and information conflict among these features. Nextly, the PSR is adopted to obtain more relevant information from the time series. By determining the delay time and embedding dimension, the time series are reconstructed to obtain a new performance degradation index, which is regarded as the input data to input into KELM, and the degradation trend prediction model is established to realize the performance degradation trend prediction. Finally, the actual vibration signals of rolling bearings are applied to prove the effectiveness of the PAPRKM. The obtained experimental results show that the PAPRKM method can effectively predict the performance degradation trend of rolling bearings. The predicted results are more accurate than the other compared methods.

In view of the difficulty of fault prediction for aviation hydraulic pumps and the poor real-time performance of state monitoring in practical applications, a hydraulic pump pressure signal prediction method is proposed to accomplish the monitoring and prediction of the health status of hydraulic pumps in advance. First, based on the on-line real-time acquisition of time series flight parameters and pressure signal data, the chaotic characteristics of the system are analyzed using chaos theory, so that the time series pressure signal is predictable. Second, phase space reconstruction (PSR) of the one-dimensional time series data is conducted. The embedding dimension $m$ and time delay $\tau $ are obtained by the C-C method. The reconstructed matrix is used as the training set and test set of the support vector regression (SVR) algorithm model according to a certain proportion, and the genetic algorithm (GA) is then used to optimize the parameters of the SVR model. Finally, the SVR model optimized by the genetic algorithm based on phase space reconstruction (PSR-GA-SVR) is used to test the test set data. The results show that the prediction accuracy of the proposed method is higher than that of the BP neural network based on phase space reconstruction (PSR-BPNN) and the SVR model based on phase space reconstruction (PSR-SVR). Relative to PSR-BPNN and PSR-SVR, PSR-GA-SVR produces a minimum mean square error (MSE) reduced by 73.40% and 68.0%, respectively, and a mean absolute error (MAE) decreased by 90.41% and 90.87%, respectively. The confidence level for PSR-GA-SVR was increased, and the coefficient of determination was greater than 0.98.

Electrocardiogram (ECG) biometric provides an authentication to identify an individual on the basis of specific cardiac potential measured from a living body. Convolutional neural networks (CNN) outperform traditional ECG biometrics because convolutions can produce discernible features from ECG through machine learning. Phase space reconstruction (PSR), using a time delay technique, is one of the transformations from ECG to a feature map, without the need of exact R-peak alignment. However, the effects of time delay and grid partition on identification performance have not been investigated. In this study, we developed a PSR-based CNN for ECG biometric authentication and examined the aforementioned effects. Based on a population of 115 subjects selected from the PTB Diagnostic ECG Database, a higher identification accuracy was achieved when the time delay was set from 20 to 28 ms, since it produced a well phase-space expansion of P, QRS, and T waves. A higher accuracy was also achieved when a high-density grid partition was used, since it produced a fine-detail phase-space trajectory. The use of a scaled-down network for PSR over a low-density grid with 32 × 32 partitions achieved a comparable accuracy with using a large-scale network for PSR over 256 × 256 partitions, but it had the benefit of reductions in network size and training time by 10 and 5 folds, respectively.

Accurate wind speed forecasting exerts a critical role in energy conversion and management of wind power. In term of this purpose, a hybrid model based on multi-stage principal component extraction, kernel extreme learning machine (KELM) and gated recurrent unit (GRU) network is developed in this paper, where the multi-stage principal component extraction combines complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN), singular spectrum analysis (SSA) and phase space reconstruction (PSR). Firstly, CEEMDAN is employed to decompose the raw wind speed data into a sequence of intrinsic mode functions (IMFs) and a residual component. Then the principal components and residual components of all IMFs are captured by SSA. Meanwhile, all residual components obtained by CEEMDAN decomposition and SSA processing are added to form a new component. Subsequently, PSR is utilized to construct each forecasting component obtained by CEEMDAN-SSA into the input and output of training set and testing set for the prediction model. Later, KELM and GRU neural network are conducted to predict the high-frequency and low-frequency components, respectively. Eventually, the prediction values of each component are accumulated to acquire the final prediction result. To evaluate the performance of the proposed model, four datasets from Sotavento Galicia wind farm are adopted to conduct experimental research. The experimental results manifest that the proposed model achieves higher accuracy of multi-step prediction than other comparative models.

Since traffic is a time-dependent and complicated non-linear system. Chaos Theory has been specifically designed to identify chaotic behaviour and properties for such systems. Previous researches have been restricted to single traffic parameter, which fails to get actual behaviour of the traffic systems. In this work, we have devised the multi-parameters chaos prediction method to describe the tendency of traffic from different aspects for prediction of single parameter chaotic time series. The fusion method is considered the relationship of different traffic parameters according to the phase space reconstruction. Taking the exit ramp of Xishanping expressway in Beibei district, Chongqing Municipality as the example, the feasibility and reliability of traffic state prediction were tested. The trail results indicates that more features of realistic traffic conditions are reflected by fusing multiple traffic parameters, and gain real accuracy improvements in traffic prediction. Compared with three single-parameter time series prediction methods, the mean absolute relative error of the multi-parameter prediction method decreased by 2.42, 2.39 and 0.8 respectively, the mean absolute relative error decreased by 2.33, 3.25 and 1.27, and the equal coefficient reached 0.9528 with a slight increase. Proposed method will definitely generates the opportunity for next generation of traffic control that are better able to detect the dynamic states of traffic, and therefore more effectively prevent the traffic congestion and pollution in the urban areas worldwide.

In order to address the problem that high energy consumption, high memory usage and low clustering effect in traditional data set high-dimensional clustering algorithms, we propose the high-dimensional clustering algorithm of incomplete mixed data set based on artificial intelligence. First, we construct the phase space reconstruction to ensure the invariance of features of incomplete mixed data set by analyzing the incomplete mixed data set and introduce the correlation dimension to obtain the feature correlation value. Second, we introduce the standard deviation and realize the extraction of features of incomplete mixed data set through calculating the sparsity of sample features. Third, we conduct repeated clustering for the mixed data set in the subspace according to the degree of correlation between incomplete mixed data sets in the multidimensional subspace. Last, we realize the design of high-dimensional clustering method for incomplete mixed data set in accordance with the stronger relevance in the mixed data sets. Experimental results show that the proposed algorithm has good correlation dimension processing effects, lower memory usage, time-consuming, lower and concentrated ensemble energy consumption (within 300J), good clustering effects, as high as 92%, which has some advantages and practical application value.

In order to further improve the prediction accuracy of the chaotic time series and overcome the defects of the single model, a multi-model hybrid model of chaotic time series is proposed. First, the Discrete Wavelet Transform (DWT) is used to decompose the data and obtain the approximate coefficients (low-frequency sequence) and detailed coefficients (high-frequency sequence) of the sequence. Secondly, phase space reconstruction is performed on the decomposed data. Thirdly, the chaotic characteristics of each sequence are judged by correlation integral and Kolmogorov entropy. Fourthly, in order to explore the deeper features of the time series and improve the prediction accuracy, a sequence of Volterra adaptive prediction models is established for the components with chaotic characteristics according to the different characteristics of each component. For the components without chaotic characteristics, a JGPC prediction model without chaotic feature sequences is established. Finally, the multi-model fusion prediction of the above multiple sequences is carried out by the LSTM algorithm, and the final prediction result is obtained through calculation, which further improves the prediction accuracy. Experiments show that the multi-model hybrid method of Volterra-JGPC-LSTM is more accurate than other comparable models in predicting chaotic time series.

Power quality disturbances (PQD) degrades the quality of power. Detection of these PQDs in real time using smart systems connected to the power grid is a challenge due to the integration of energy generation units and electronic devices. Deep learning methods have shown advantages for PQD classification accurately. PQD events are non-stationary and occur at discrete events. Pre-processing of power signal using dual tree complex wavelet transform in localizing the disturbances according to time-frequency-phase information improves classification accuracy.Phase space reconstruction of complex wavelet sub bands to 2D data and use of fully connected feed forward neural network improves classification accuracy. In this work, a combination of DTCWT-PSR and FC-FFNN is used to classify different complex PSDs accurately.The proposed algorithm is evaluated for its performance considering different network configurations and the most optimum structure is developed. The classification accuracy is demonstrated to be 99.71% for complex PQDs and is suitable for real time activity with reduced complexity.

Magnetotelluric (MT) sounding data can easily be damaged by various types of noise, especially in industrial areas, where the quality of measured data is poor. Most traditional de-noising methods are ineffective to the low signal-to-noise ratio of data. To solve the above problem, we propose the use of a de-noising method for the detection of noise in MT data based on discrete wavelet transform and singular value decomposition (SVD), with multiscale dispersion entropy and phase space reconstruction carried out for pretreatment. No “over processing” takes place in the proposed method. Compared with wavelet transform and SVD decomposition in synthetic tests, the proposed method removes the profile of noise more completely, including large-scale noise and impulse noise. For high levels or low levels of noise, the proposed method can increase the signal-to-noise ratio of data more obviously. Moreover, application to the field MT data can prove the performance of the proposed method. The proposed method is a feasible method for the elimination of various noise types and can improve MT data with high noise levels, obtaining a recovery in the response. It can improve abrupt points and distortion in MT response curves more effectively than the robust method can.

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.

The evaporation duct is a particular type of atmospheric structure that always appears on the open ocean. Predicting the evaporation duct height (EDH) accurately and in a timely manner is of great significance for the practical application of marine wireless communication equipment. Understanding the characteristics of EDH time series is an essential prerequisite for establishing an appropriate prediction model. Moreover, the sampling timescales of EDH data may influence the dynamic characteristics of the EDH time series as well. In this study, EDH time series datasets at three timescales, hourly, daily, and monthly, were constructed as the case study. Statistical methods, namely the augmented Dickey–Fuller test and Ljung–Box test, were adopted to verify the stationary and white noise characteristics of the EDH time series. Then, rescaled range analysis was applied to calculate the Hurst exponent to study the fractal characteristics of the EDH time series. An extensive analysis and discussion of the chaotic dynamics of the EDH time series are provided. From the perspective of nonlinear dynamics, the phase space was constructed from the time delay τ and embedding dimension m, which were calculated from the mutual information method and the Grassberger–Procaccia algorithm, respectively. The maximum Lyapunov exponent was also calculated by the small data volume method to explore the existence of chaos in the EDH time series. According to our analysis, the EDH time series are stationary and have a non-white noise characteristic. The Hurst exponents for all three timescales were greater than 0.5, indicating the predictability of the EDH time series. The phase space diagrams exhibited strange attractors in a well-defined region for all the timescales, suggesting that the evolution of the EDH time series can possibly be explained by deterministic chaos. All of the maximum Lyapunov exponents were positive, confirming the chaos in the EDH time series. Further, stronger chaotic characteristics were found for the finer-resolution time series than the coarser-resolution time series. This study provides a new perspective for scholars to understand the fluctuation principles of the evaporation duct at different timescales. The findings from this study also lay a theoretical and scientific foundation for the future application of chaotic prediction methods in the research on the evaporation duct.

This paper describes a comparison of three types of feature sets. The feature sets were intended to classify 13 faults in a centrifugal pump (CP) and 17 valve faults in a reciprocating compressor (RC). The first set comprised 14 non-linear entropy-based features, the second comprised 15 information-based entropy features, and the third comprised 12 statistical features. The classification was performed using random forest (RF) models and support vector machines (SVM). The experimental work showed that the combination of information-based features with non-linear entropy-based features provides a statistically significant accuracy higher than the accuracy provided by the Statistical Features set. Results for classifying the 13 conditions in the CP using non-linear entropy features showed accuracies of up to 99.50%. The same feature set provided a classification accuracy of 97.50% for the classification of the 17 conditions in the RC.

The transient pulses caused by local faults of rolling bearings are an important measurement information for fault diagnosis. However, extracting transient pulses from complex nonstationary vibration signals with a large amount of background noise is challenging, especially in the early stage. To improve the anti-noise ability and detect incipient faults, a novel signal de-noising method based on enhanced time-frequency manifold (ETFM) and kurtosis-wavelet dictionary is proposed. First, to mine the high-dimensional features, the C-C method and Cao’s method are combined to determine the embedding dimension and delay time of phase space reconstruction. Second, the input parameters of the liner local tangent space arrangement (LLTSA) algorithm are determined by the grid search method based on Renyi entropy, and the dimension is reduced by manifold learning to obtain the ETFM with the highest time-frequency aggregation. Finally, a kurtosis-wavelet dictionary is constructed for selecting the best atom and eliminating the noise and reconstruct the defective signal. Actual simulations showed that the proposed method is more effective in noise suppression than traditional algorithms and that it can accurately reproduce the amplitude and phase information of the raw signal.

Snow cover over the Tibetan Plateau plays a vital role in the regional and global climate system because it affects not only the climate but also the hydrological cycle and ecosystem. However, high-quality snow data are hindered due to the sparsity of observation networks and complex terrain in the region. In this study, a nonlinear time series analysis method called phase space reconstruction was used to obtain the Tibetan Plateau snow depth by combining the FY-3C satellite data andin situdata for the period 2014–2017. This method features making a time delay reconstruction of a phase space to view the dynamics. Both of the grids and their nearbyin situsnow depth time series were reconstructed with two appropriate parameters called time delay and embedding dimension. The values of the snow depth for grids were averaged over thein situobservations and retrieval of the satellite if their two parameters were the same. That implies that the two trajectories of the time series had the same evolution trend. Otherwise, the snow depth values for grids were averaged over thein situobservation. If there were noin situsites within the grids, the retrieval of the satellite remained. The results show that the integrated Tibetan Plateau snow depth (ITPSD) had an average bias of –1.35 cm and 1.14 cm, standard deviation of the bias of 3.96 cm and 5.67 cm, and root mean square error of 4.18 cm and 5.79 cm compared with thein situdata and FY-3C satellite data, respectively. ITPSD expressed the issue that snow depth is usually overestimated in mountain regions by satellites. This is due to the introduction of more station observations using a dynamical statistical method to correct the biases in the satellite data.

Effective carbon pricing policies have become an effective tool for many countries to encourage emission reduction. An accurate carbon price prediction model is helpful for the implementation of energy conservation and emission reduction policies and the decision-making of governments and investors. However, it is difficult for a single prediction model to achieve high prediction accuracy because of the high complexity of the carbon price series. Many studies have proved the nonlinear characteristics of carbon trading prices, but there are very few studies on the chaotic nature of carbon price series. As a consequence, this paper proposes an innovative hybrid model for carbon price prediction. A decomposition-reconstruction-prediction-integration scheme is designed to predict carbon prices. Firstly, several intrinsic mode functions (IMFs) and one residue were obtained from the raw data decomposed by ICEEMDAN. Next, the decomposed subsection is reconstructed into a new sequence according to the calculation results by the Lempel-Ziv complexity algorithm. Then, considering the chaotic characteristics of sequence, the input variables of the models are determined through the phase space reconstruction (PSR) algorithm combined with the partial autocorrelation function (PACF). Finally, the Sparrow search algorithm (SSA) is introduced to optimize the extreme learning machine (ELM) model, which is applied in the carbon price prediction for the purpose of verifying the validity of the proposed combination model, which is applied to the pilots of Hubei, Beijing, and Guangdong. The empirical results show that the combination model outperformed the 13 other models in predicting accuracy, speed, and stability. The decomposition-reconstruction-prediction-integration strategy is a method for predicting the carbon price efficiently.

As a frequent natural disaster, red tide has attracted more and more attentions. In fact, red tide results from the joint actions of multiple complex marine environmental factors. Unfortunately, there is no work on the interaction analysis between these factors. To inaugurate a systematic research of this area, a novel machine learning based framework is developed for marine environmental series analysis. It combines cross recurrence plot (CRP), cross recurrence quantification analysis (CRQA) and statistical analysis. This framework provides a general way to transform two marine series into a high-dimensional space. CRP is used to visualize internal dynamics, while the influence of factors is quantitatively analyzed through CRQA. Finally, the representative factors in each field are statistically determined by boxplot. This is the first analysis framework attempting to reveal the similarity in intrinsic dynamics of marine factors. Experimental results show that the framework is competent to perform the visualization of marine time series. Besides, the results also demonstrate the degree of similarity between different marine factors through quantitative analysis.

The Conceptor network is a newly proposed reservoir computing (RC) model, which outperforms traditional classifiers, which can fail to model new classes of data for a supervised learning task. However, the reservoir structure design for the Conceptor is single, involving just a traditional random network, which has strong coupling between nodes and limits computing ability. This study focused on the reservoir topology design problem, and we propose a complex network Conceptor-based phase space reconstruction of time series. Several dynamical systems were chosen to build complex networks using a phase space reconstruction algorithm. The experiment results obtained using a mix of two irrational-period sines showed that the proposed phase space reconstruction reservoir topologies with the appropriate values of threshold provide Conceptors with extra reconstruction precision. Among them, the phase space reconstruction reservoir-based Lorenz system shows the best performance. Further experiments also identified the appropriate values of threshold of the phase space reconstruction method required to obtain optimal performance. The precision showed a non-linear decline with increase in memory load, and the proposed Lorenz phase space reconstruction reservoir maintained its advantages under different memory loads.

Considering the characteristics of network traffic on the data link layer, such as massive high-speed data flow, information camouflaged easily, and the phenomenon that abnormal traffic is much smaller than the normal one, an intrusion detection system (IDS) based on the quantitative model of interaction mode between ports is proposed. The model gives the quantitative expression of Port Interaction Mode in Data Link Layer (PIMDL), focusing on improving the accuracy and efficiency of the intrusion detection by taking the arrival time distribution of traffic. The feasibility of the model proposed is proved by the phase space reconstruction and visualization method. According to the characteristics of long and short sessions, a neural network based on CNN and LSTM is designed to mine the differences between normal and abnormal models. On this basis, an improved Intrusion Detection algorithm based on a multi-model scoring mechanism is designed to classify sessions in model space. And the experiments show that the quantitative model and the improved algorithm proposed can not only effectively avoid camouflage identity information, but also improve computational efficiency, as well as increase the accuracy of small sample anomaly detection.

Este projeto descreve uma abordagem padronizada de análise de dados coletados de redes reais e de dados gerados sinteticamente, que apresentam características lineares e não-lineares, dentro do espectro da Teoria do Caos e seus procedimentos canônicos, avaliando o desempenho de uma gama de algoritmos. O objetivo principal deste trabalho consiste em propor um método de predição e alocação dinâmica de taxa de transmissão para servidores de rede com base no expoente de Lyapunov e no parâmetro de Hurst, tendo em vista a análise de uma variedade de amostras de tráfego e dados sintéticos, de forma a quantificá-los com o uso de métodos matemáticos que revelem suas particularidades intrínsecas. São descritos processos característicos tais como autossimilaridade, dependência de longa duração entre as amostras e comportamento multi-escala. Para isso, reconstrói-se o espaço de fase e o atrator com o atraso ideal $\tau$, descrevendo métodos de cálculo do parâmetro; determina-se a dimensão de incorporação $m$ e a dimensão de correlação $D_2$; calcula-se o expoente de Lyapunov $\lambda$ e o parâmetro de Hurst $H$; e analizam-se estatisticamente os componentes principais; para então prever o tráfego dentro do limite de máxima duração da predição delimitada pelo inverso do expoente de Lyapunov, e alocar dinamicamente a taxa de transmissão correspondente para os servidores de rede. As simulações realizadas comprovaram a eficiência da abordagem proposta, quanto à sequência seguida neste trabalho e a classificação da natureza dos dados, especialmente nos casos de comportamento misto entre aleatoriedade e determinismo. This project describes a standard approach to analyze and measure experimental and synthetic data, which display linear and non-linear characteristics, within the spectrum of Chaos Theory and its canonical procedures, evaluating the performance of several algorithms. The main goal of this work is to propose a method to predict traffic and perform a dynamic traffic rate allocation for the network servers based on the Lyapunov exponent and the Hurst parameter, considering the analysis of a range of traffic samples and synthetic data, in order to quantify them with the use of mathematical methods that reveal their intrinsic features. Some of the characteristic processes described are self-similarity, long-range dependency among samples and multiscale behavior. Thus, it is necessary to: reconstruct phase space and the attractor with the ideal delay $\tau$, while describing a couple of methods to compute it; to determine embedding and correlation dimensions ($m$ and $D_2$, respectively); to calculate the Lyapunov exponent $\lambda$ and Hurst parameter $H$; to perform principal component analysis ($PCA$); to predict the traffic within the longest predictable duration constrained by the inverse of the Lyapunov exponent, and finally perform dynamic transmission rate allocation for the network servers. The simulations performed confirmed the efficiency of the proposed approach, regarding the sequence adopted in this work and the classification of data, especially in cases of mixed behavior between randomness and determinism. Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES

An effective water resources management requires accurate predictions of possible risks. Drought is one of the most devastating phenomena that has a certain risk of occurrence. Understanding the variability of the drought indices is of great importance in determining the spatiotemporal behavior of the drought phenomenon. Moreover, determination of the variability and short-term prediction of the drought indices enables us to take necessary steps in hydrological and agricultural issues. In this study, drought indices have been predicted via Support Vector Regression, SVR. This method originated from a linear regression method in a high dimensional feature space. SVR necessitates a special input matrix. In this study, this matrix has been constructed on the basis of Chaotic Approach, CA. Commonly used drought indices are used in the prediction stage. These indices consist of monthly Palmer Drought Severity Index, PDSI, Palmer Hydrological Drought Index, PHDI, Palmer Z-Index, ZNDX, Modified Palmer Drought Severity Index, PMDI, and Standard Precipitation Index, SPI. One-step ahead prediction has been realized for a 36-month period. Most results show that predictions of the drought indices using SVR are quite promising., Italian Journal of Agrometeorology, No. 2 (2020)

Background Atrial fibrillation (AF) represents the most common arrhythmia worldwide, related to increased risk of ischemic stroke or systemic embolism. It is critical to screen and diagnose AF for the benefits of better cardiovascular health in lifetime. The ECG-based AF detection, the gold standard in clinical care, has been restricted by the need to attach electrodes on the body surface. Recently, ballistocardiogram (BCG) has been investigated for AF diagnosis, which is an unobstructive and convenient technique to monitor heart activity in daily life. However, here is a lack of high-dimension representation and deep learning analysis of BCG. Method Therefore, this paper proposes an attention-based multi-scale features fusion method by using BCG signal. The 1-D morphology feature extracted from Bi-LSTM network and 2-D rhythm feature extracted from reconstructed phase space are integrated by means of CNN network to improve the robustness of AF detection. To the best of our knowledge, this is the first study where the phase space trajectory of BCG is conducted. Results 2000 segments (AF and NAF) of BCG signals were collected from 59 volunteers suffering from paroxysmal AF in this survey. Compared to the classical time and frequency features and the state-of-the-art energy features with the popular machine learning classifiers, AF detection performance of the proposed method is superior, which has 0.947 accuracy, 0.935 specificity, 0.959 sensitivity, and 0.937 precision, for the same BCG dataset. The experimental results show that combined feature could excavate more potential characteristics, and the attention mechanism could enhance the pertinence for AF recognition. Conclusions The proposed method can provide an innovative solution to capture the diverse scale descriptions of BCG and explore ways to involve the deep learning method to accurately screen AF in routine life.

The non-invasiveness of photoplethysmographic (PPG) acquisition systems, together with their cost-effectiveness and easiness of connection with IoT technologies, is opening up to the possibility of their widespread use. For this reason, the study of the reliability of PPG and pulse rate variability (PRV) signal quality has become of great scientific, technological, and commercial interest. In this field, sensor location has been demonstrated to play a crucial role. The goal of this study was to investigate PPG and PRV signal quality acquired from two body locations: finger and wrist. We simultaneously acquired the PPG and electrocardiographic (ECG) signals from sixteen healthy subjects (aged 28.5 &plusmn, 3.5, seven females) who followed an experimental protocol of affective stimulation through visual stimuli. Statistical tests demonstrated that PPG signals acquired from the wrist and the finger presented different signal quality indexes (kurtosis and Shannon entropy), with higher values for the wrist-PPG. Then we propose to apply the cross-mapping (CM) approach as a new method to quantify the PRV signal quality. We found that the performance achieved using the two sites was significantly different in all the experimental sessions (p &lt, 0.01), and the PRV dynamics acquired from the finger were the most similar to heart rate variability (HRV) dynamics.

Transformer state forecasting and fault forecasting are important for the stable operation of power equipment and the normal operation of power systems. Forecasting of the dissolved gas content in oil is widely conducted for transformer faults, but its accuracy is affected by data scale and data characteristics. Based on phase space reconstruction (PSR) and weighted least squares support vector machine (WLSSVM), a forecasting model of time series of dissolved gas content in transformer oil is proposed in this paper. The phase spaces of time series of the dissolved gas content sequence are reconstructed by chaos theory, and the delay time and dimension are obtained by the C-C method. The WLSSVM model is used to forecast time series of dissolved gas content, the chemical reaction optimization (CRO) algorithm is used to optimize training parameters, the bootstrap method is used to build forecasting intervals. Finally, the accuracy and generalization ability of the forecasting model are verified by the analysis of actual case and the comparison of different models.

This paper takes the time series of short-term traffic flow as research object. The delay time and embedding dimension are calculated by C-C algorithm, and the chaotic characteristics of the time series are verified by small data sets method.Then based on the neural network prediction model and the chaotic phase space reconstruction theory, the network topology is determined, and the prediction is conducted by the wavelet neural network and RBF neural network using Lan-Hai expressway experimental data. The results show that the prediction effect of RBF neural network is better. Due to the poor stability of the network caused by the initial parameters randomness, the genetic algorithm is used to optimize the initial parameters. The results show that the prediction error of the optimized wavelet neural network or RBF neural network is reduced by more than 10%, and prediction accuracy of the latter is better.

In this paper, we present some improvements to the convergent cross mapping (CCM) algorithm for detecting causality in uni-directionally connected chaotic systems. The basic concept of the CCM algorithm is that the causal influence of system X on system Y appears as mapping of the neighbouring states in the reconstructed d-dimensional manifold, My, to the neighbouring states in the reconstructed d-dimensional manifold, Mx, and this effect is evaluated using the correlation coefficient between the estimated and observed values of Mx. We proposed a composite indicator of causality as the ratio between the correlation coefficient and the Shannon entropy of the distribution of the residuals between the estimated and observed values of Mx. Application of the proposed approach to four master-slave Rössler and Lorenz systems and real-world data showed that the new algorithm allowed a slight increase in capability to reveal the presence and direction of couplings.

In the analysis of biological time series, the state space is comprised of a framework for the study of systems with presumably deterministic and stationary properties. However, a physiological experiment typically captures an observable that characterizes the temporal response of the physiological system under study, the dynamic variables that make up the state of the system at any time are not available. Only from the acquired observations should state vectors be reconstructed to emulate the different states of the underlying system. This is what is known as the reconstruction of the state space, called the phase space in real-world signals, in many cases satisfactorily resolved using the method of delays. Each state vector consists of m components, extracted from successive observations delayed a time &tau, The morphology of the geometric structure described by the state vectors, as well as their properties depends on the chosen parameters &tau, and m. The real dynamics of the system under study is subject to the correct determination of the parameters &tau, and m. Only in this way can be deduced features have true physical meaning, revealing aspects that reliably identify the dynamic complexity of the physiological system. The biological signal presented in this work, as a case study, is the photoplethysmographic (PPG) signal. We find that m is five for all the subjects analyzed and that &tau, depends on the time interval in which it is evaluated. The H&eacute, non map and the Lorenz flow are used to facilitate a more intuitive understanding of the applied techniques.

With population explosion and globalization, the spread of infectious diseases has been a major concern. In 2019, a newly identified type of Coronavirus caused an outbreak of respiratory illness, popularly known as COVID-19, and became a pandemic. Although enormous efforts have been made to understand the spread of COVID-19, our knowledge of the COVID-19 dynamics still remains limited. The present study employs the concepts of chaos theory to examine the temporal dynamic complexity of COVID-19 around the world. The false nearest neighbor (FNN) method is applied to determine the dimensionality and, hence, the complexity of the COVID-19 dynamics. The methodology involves: (1) reconstruction of a single-variable COVID-19 time series in a multi-dimensional phase space to represent the underlying dynamics; and (2) identification of “false” neighbors in the reconstructed phase space and estimation of the dimension of the COVID-19 series. For implementation, COVID-19 data from 40 countries/regions around the world are studied. Two types of COVID-19 data are analyzed: (1) daily COVID-19 cases; and (2) daily COVID-19 deaths. The results for the 40 countries/regions indicate that: (1) the dynamics of COVID-19 cases exhibit low- to medium-level complexity, with dimensionality in the range 3 to 7; and (2) the dynamics of COVID-19 deaths exhibit complexity anywhere from low to high, with dimensionality ranging from 3 to 13. The results also suggest that the complexity of the dynamics of COVID-19 deaths is greater than or at least equal to that of the dynamics of COVID-19 cases for most (three-fourths) of the countries/regions. These results have important implications for modeling and predicting the spread of COVID-19 (and other infectious diseases), especially in the identification of the appropriate complexity of models.

In this paper, carbon emissions and the related problems are studied based on carbon emission time series data and the chaos theory, in order to make clear the relationship among the data, and we reconstruct the time series by phase space reconstruction. Finally, the predicting model of the carbon emission is established with BP neural network. The simulation results show that the hybrid of chaos theory and BP neural network can be used to fit and predict the carbon emissions time series without considering other factors, which is easier and more accurate than other predicting method.

Subcutaneous Implantable Cardioverter-Defibrillators (S-ICDs) are used for prevention of sudden cardiac death triggered by ventricular arrhythmias. T Wave Over Sensing (TWOS) is an inherent risk with S-ICDs which can lead to inappropriate shocks. A major predictor of TWOS is a high T:R ratio (the ratio between the amplitudes of the T and R waves). Currently patients' Electrocardiograms (ECGs) are screened over 10 seconds to measure the T:R ratio, determining the patients' eligibility for S-ICD implantation. Due to temporal variations in the T:R ratio, 10 seconds is not long enough to reliably determine the normal values of a patient's T:R ratio. In this paper, we develop a convolutional neural network (CNN) based model utilising phase space reconstruction matrices to predict T:R ratios from 10-second ECG segments without explicitly locating the R or T waves, thus avoiding the issue of TWOS. This tool can be used to automatically screen patients over a much longer period and provide an in-depth description of the behaviour of the T:R ratio over that period. The tool can also enable much more reliable and descriptive screenings to better assess patients' eligibility for S-ICD implantation.

Wind power has already brought lots of benefits for energy market and environment. To promote the utilization of wind energy further and achieve the ambitious goals of applying sustainable energy, the improvement of the performance of wind speed forecasting will play a significant role. The whole system will benefit from an accurate wind forecasting system with high resolution. This paper focused on short-term wind speed forecasting and proposed a method providing the wind speed forecasting with short-term horizon and high resolution. In this paper, a novel method based on the wavelet decomposition, the phase space reconstruction and the self-organizing map (SOM) artificial neural network was proposed. Study case was provided to demonstrate the performance of the proposed forecasting method. Comparative results from several classic forecasting methods were provided as well. Based on the outcomes of study cases and comparison results, it can be concluded that the proposed method performed well in short term forecasting horizon.

The incipient damages of mechanical equipment excite weak impulse vibration, which is hidden, almost unobservable, in the collected signal, making fault detection and failure prevention at the inchoate stage rather challenging. Traditional feature extraction techniques, such as bandpass filtering and time-frequency analysis, are suitable for matrix processing but challenged by the higher-order data. To tackle these problems, a novel method of impulse feature extraction for vibration signals, based on sparse non-negative tensor factorization is presented in this paper. Primarily, the phase space reconstruction and the short time Fourier transform are successively employed to convert the original signal into time-frequency distributions, which are further arranged into a three-way tensor to obtain a time-frequency multi-aspect array. The tensor is decomposed by sparse non-negative tensor factorization via hierarchical alternating least squares algorithm, after which the latent components are reconstructed from the factors by the inverse short time Fourier transform and eventually help extract the impulse feature through envelope analysis. For performance verification, the experimental analysis on the bearing datasets and the swashplate piston pump has confirmed the effectiveness of the proposed method. Comparisons to the traditional methods, including maximum correlated kurtosis deconvolution, singular value decomposition, and maximum spectrum kurtosis, also suggest its better performance of feature extraction.

Recently, a large number of distributed photovoltaic (PV) power generations have been connected to the power grid, which resulted in an increased fluctuation of the net load. Therefore, load forecasting has become more difficult. Considering the characteristics of the net load, an ultrashort-term forecasting model based on phase space reconstruction and deep neural network (DNN) is proposed, which can be divided into two steps. First, the phase space reconstruction of the net load time series data is performed using the C-C method. Second, the reconstructed data is fitted by the DNN to obtain the predicted value of the net load. The performance of this model is verified using real data. The accuracy is high in forecasting the net load under high PV penetration rate and different weather conditions.

A built-in sensor in a smart device, such as the accelerometer and the gyroscope, will produce an obvious nonlinear output when it receives voice signal. In this paper, based on the chaotic theory, the nonlinearity of smartphone built-in accelerometer is revealed by phase space reconstructing after we calculate several nonlinearity characteristics, such as best delay time, embedding dimension, and the attractor of accelerometer system, under the condition of voice commands inputting. The results of theoretical calculation and experiments show that this specific nonlinearity could lay a foundation for further signal extraction and analysis.

Adequate understanding of streamflow dynamics is important for a wide range of applications in hydrology and water resources. A basic idea in streamflow studies is to establish connections that generally exist between the different components of the catchment system. Despite the development and applications of numerous approaches, our understanding of the connections in streamflow remains inadequate. The concepts of complex networks offer new avenues for studying connections in complex and dynamically-evolving systems, such as catchments. Applications of the complex networks concepts to streamflow are an emerging area of research. This thesis aims to study the spatial, temporal, and spatio-temporal connections in streamflow using complex networks concepts. Since the construction of the network (i.e., nodes and links) using data is a key step in complex networks-based methods, an improvement to network construction is proposed. Specifically, the phase space reconstruction concept, which allows reconstruction of a single time series in a multi-dimensional phase space to represent the system, is used for constructing the temporal and spatio-temporal streamflow networks. The spatial, temporal, and spatio-temporal connections in streamflow are primarily examined using the clustering coefficient, which quantifies the network’s tendency to cluster. The degree distribution, a measure of the spread in the node degree, is also used to examine the spatial connections. For implementation, monthly streamflow data observed over a period of 53 years from a large network of 639 monitoring stations in the contiguous United States are studied. The influence of the correlation threshold in the spatial analysis and distance threshold in the temporal and spatio-temporal analysis is investigated. Based on the clustering coefficients, the type of network is identified. The clustering coefficients are also interpreted in terms of catchment and flow characteristics. The clustering coefficients for the spatial, temporal, and spatio-temporal connections reveal that the streamflow networks are not purely random networks, but may be small-world or scale-free or some other type. The clustering coefficients also offer reliable interpretations in terms of catchment and flow characteristics, especially about the complexity of network and strength of connections. These outcomes clearly indicate the usefulness of the complex networks-based methods for streamflow analysis.

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.

One of the primary responsibilities of any meteorological and hydrological services is to provide information on weather warnings and forecasts to the general public for necessary precaution and prevention. This necessitated a better understanding of the underlying dynamics in weather parameters for modelling, prediction and control. This research examines chaos in the trend of the meteorological parameters such as air temperature, relative humidity and wind speed in eight meteorological stations in Nigeria using nonlinear time series analysis approach. The meteorological data were obtained and subjected to various components of nonlinear analyses which includes, time series, phase-space reconstruction, actual mutual information (AMI), false nearest neighbours (FNNs), and Lyapunov exponent. The delay time (τ) and embedded dimension (m) are estimated to be 10 and m ≥ 5 ≥ 9 produced from actual mutual information (AMI) and false nearest neighbours (FNNs) respectively and are used to obtain the ultimate possible attractor reconstruction. A significant chaotic quantifier, the Lyapunov exponent was also estimated for all the locations. Our result shows positive largest Lyapunov exponent was evident with the parameters across the eight stations. The largest Lyapunov exponent was found to be within the range 0.0034 - 0.1200 for all the parameters; which indicates strong presence of chaos. The results therefore provide a good source of information for constructing weather prediction model.

In this paper, a novel algorithm based on the local projection noise reduction approach is applied to smooth noise for strongly contaminated chaotic oscillators. Specifically, one-dimensional time series are embedded into a high dimensional phase space and the noise level is defined through orthogonal projections of the data points within the neighbourhood of the reference point onto linear subspaces. The current vector of the phase space is denoised by performing two-dimensional discrete stationary wavelet transform (SWT)-based filtering in the neighbourhood of the phase point. Numerical results show that our algorithm effectively recovers continuous-time chaotic signals in heavy-noise environments and outperforms the classical local projection noise reduction approach for simulated data from the Rössler system and Duffing oscillator at signal-to-noise ratios (SNRs) from 15 to 0 dB, either for the real world data – human breath time series.

Aiming at the problems existing in previous chaos time series prediction methods, a novel chaos times series prediction method, which applies modified GM(1, 1) model with optimizing parameters to study evolution laws of phase point L1 norm in reconstructed phase space, is proposed in this paper. Phase space reconstruction theory is used to reconstruct the unobserved phase space for chaotic time series by C-C method, and L1 norm series of phase points can be obtained in the reconstructed phase space. The modified GM(1, 1) model, which is improved by optimizing background value and optimizing original condition, is used to study the change law of phase point L1 norm for forecasting. The measured data from stabilized platform experiment and three traditional chaos time series are applied to evaluate the performance of the proposed model. To test the prediction method, three accuracy evaluation standards are employed here. The empirical results of stabilized platform are encouraging and indicate that the newly proposed method is excellent in prediction of chaos time series of chaos systems.