Method For Chaotic Time Series Research Articles

Analysis of total electron content over the African low-latitude region during the maximum phase of solar cycle 24 (2012–2014)

This study examined the ionospheric variation over the African low-latitude region using Total Electron Content (TEC) and its wavelet power spectrum (WPS). TEC data were obtained from five Global Positioning System (GPS) stations located in the trough and crests of the equatorial ionization anomaly (EIA) in both hemispheres during quiet periods of the maximum phase of solar cycle 24 (2012–2014). Our results revealed a striking equinoctial asymmetry (EA) with stronger TEC and higher solar flux in September equinox than March equinox of 2012–2013, with the reverse being the case in 2014. As such, the difference in solar flux between both equinoxes might have played a dominant role in driving the EA. Furthermore, we found a hemispheric asymmetry in TEC with higher value towards the crest in the northern hemisphere (southern hemisphere) in September equinox (March equinox) of 2012–2013. In 2014, nevertheless, TEC was remarkably higher towards the crest located in the southern hemisphere in September equinox only. The chaotic time series methods used to examine the TEC variations showed positive Lyapunov exponents (LE) at all stations. This indicated the presence of chaoticity in the African EIA. Nevertheless, stations located towards the southern crest (Eldoret and Dodoma) had the maximum LE values while those at the magnetic equator (Adis Ababa) and northern crest (Nama and Al Wajh) had the minimum values. Consequently, the changes in the internal dynamics of the equatorial ionosphere have impacts on its chaotic behavior.

A high‐precision adaptive blind estimation method for chaotic time series

To improve the estimation accuracy of chaotic time series and reduce the computational complexity, a high-precision adaptive blind estimation method is proposed. By introducing the pre-treatment method, the adaptive identification model is established, and the linear estimation equation is derived. Through the organic combination of the model and the least square mechanism, the complexity of the model is limited. When changing the type and length of the sequence, the parameters can be changed adaptively, so as to construct the corresponding trajectory equation and realize the blind estimation of chaotic time series. The analysis and experimental results show that the estimation accuracy of this method can reach 10–16 order of magnitude.

Multimodality Prediction of Chaotic Time Series with Sparse Hard-Cut EM Learning of the Gaussian Process Mixture Model**Supported by the National Natural Science Foundation of China under Grant No 60972106, the China Postdoctoral Science Foundation under Grant No 2014M561053, the Humanity and Social Science Foundation of Ministry of Education of China under Grant No 15YJA630108, and the Hebei Province Natural Science

The contribution of this work is twofold: (1) a multimodality prediction method of chaotic time series with the Gaussian process mixture (GPM) model is proposed, which employs a divide and conquer strategy. It automatically divides the chaotic time series into multiple modalities with different extrinsic patterns and intrinsic characteristics, and thus can more precisely fit the chaotic time series. (2) An effective sparse hard-cut expectation maximization (SHC-EM) learning algorithm for the GPM model is proposed to improve the prediction performance. SHC-EM replaces a large learning sample set with fewer pseudo inputs, accelerating model learning based on these pseudo inputs. Experiments on Lorenz and Chua time series demonstrate that the proposed method yields not only accurate multimodality prediction, but also the prediction confidence interval. SHC-EM outperforms the traditional variational learning in terms of both prediction accuracy and speed. In addition, SHC-EM is more robust and insusceptible to noise than variational learning.

Nonparametric interval prediction of chaotic time series and its application to climatic system

This article is concerned with nonlinear time series analysis and it proposes an interval prediction method for chaotic time series. The selection algorithm for the number of neighbour points is introduced based on the local approximation technique of the classic chaotic time series model. The nonparametric statistics method is considered here, and we obtain an interval prediction for one or more steps under a certain confidence level assumption with the help of order statistics distribution. We also find a sufficient condition for the existence of such interval prediction, and a sufficient condition for the relationship between the number of neighbour points and the interval confidence level. In addition to the bootstrap multiple sample based on the selected neighbour points implemented on the computer, another interval prediction method is described too. We focus on the air pressure difference data from a climatic system and find that the series data has a positive Lyapunov exponent, which shows that it contains chaos. The application of the techniques on the data shows that both the interval predictions are reasonable in this data sample.

Local Piecewise-Linearity Prediction Method of Chaotic Time Series

A Local Piecewise-Linearity Prediction method is presented, Based on the advantages and limitations of local prediction of chaotic time series. Taking time series of rainfall as example for prediction the rainfall of one city in Liaoning province, which includes the application of the largest Lyapunov exponent, Local-region method and Local Piecewise-Linearity method. The method proposed is proved practical in comparison with the observed data.

The largest Lyapunov exponent and the turbulent fluctuation of the time series from air turbulent jets

The velocity time series of round air jets are acquired with the hot-wire anemometer. We compute the largest Lyapunov exponent and the turbulent incoherent perturbation of the velocity signals at 939 ≤ Re ≤ 3758 using an analytical method of chaotic time series which is based on the invariant of the largest Lyapunov exponent. The results show that the largest Lyapunov exponent increases with exit Reynolds number, and decreases with the distance away from nozzle exit. The reciprocal of the largest Lyapunov exponent is positively correlated with correlation time. The incoherent perturbation of turbulence increases either with exit Reynolds number or with the distance away from nozzle exit. The incoherent perturbation of turbulence is negatively correlated with Kolmogorov scale.

Study on Apparent Kinetic Prediction Model of the Smelting Reduction Based on the Time‐Series

A series of direct smelting reduction experiment has been carried out with high phosphorous iron ore of the different bases by thermogravimetric analyzer. The derivative thermogravimetric (DTG) data have been obtained from the experiments. One‐step forward local weighted linear (LWL) method , one of the most suitable ways of predicting chaotic time‐series methods which focus on the errors, is used to predict DTG. In the meanwhile, empirical mode decomposition‐autoregressive (EMD‐AR), a data mining technique in signal processing, is also used to predict DTG. The results show that (1) EMD‐AR(4) is the most appropriate and its error is smaller than the former; (2) root mean square error (RMSE) has decreased about two‐thirds; (3) standardized root mean square error (NMSE) has decreased in an order of magnitude. Finally in this paper, EMD‐AR method has been improved by golden section weighting; its error would be smaller than before. Therefore, the improved EMD‐AR model is a promising alternative for apparent reaction rate (DTG). The analytical results have been an important reference in the field of industrial control.

Research on the Phase Space Reconstruction Method of Chaotic Time Series

The puzzle over the recognition of the quality of the Chaotic Dynamics based on single variable time series brings forward the new method of phase space reconstruction—identify the embedding dimension with the FNN after delay time is fixed through the method of autocorrelation function. By means of the numerical verification of a few typical examples of chaotic dynamic system, the result shows that this method can be able to efficiently reconstruct the phase space of the original system out of the time series and relatively completely reduce the dynamic characteristics of the original system and thus the validity of the method is testified on chaotic signal recognition.

Study on Grey Forecast Method of Chaotic Time Series

In this paper, the research evolving on chaotic time series forecast is firstly analyzed, and then the grey forecast model of chaotic time series is set up and is applied to chaotic time series forecast to solve the grey forecast problem of chaotic time series.

Predictability of Iteration Method for Chaotic Time Series

The approximation expression about error accumulation of a long-term prediction is derived. By analyzing this formula, we find that the factors that can affect the long-term predictability include the model parameters, prediction errors and the derivates of the used basis functions. To enlarge the maximum attempting time, we present that more suitable basis functions should be those with smaller derivative functions and a fast attenuation where out of the time series range. We compare the long-term predictability of a non-polynomial based algorithm and a polynomial one to prove the success of our method.

ARRBFNs with SVR for prediction of chaotic time series with outliers

In this article, annealing robust radial basis function networks (ARRBFNs), which consist of a radial basis function network and a support vector regression (SVR), and an annealing robust learning algorithm (ARLA) are proposed for the prediction of chaotic time series with outliers. In order to overcome the initial structural problems of the proposed neural networks, the SVR is utilized to determine the number of hidden nodes, the initial parameters of the kernel, and the initial weights for the proposed ARRBFNs. Then the ARLA that can conquer the outliers is applied to tune the parameters of the kernel and the weights in the proposed ARRBFNs under the initial structure with SVR. The simulation results of Mackey-Glass time series show that the proposed approach with different SVRs can cope with outliers and give a fast learning speed. The results of the simulation are also given to demonstrate the validity of proposed method for chaotic time series with outliers.

Combination prediction method of chaotic time series

In the present paper, we propose an approach of combination prediction of chaotic time series. The method is based on the adding-weight one-rank local-region method of chaotic time series. The method allows us to define an interval containing a future value with a given probability, which is obtained by studying the prediction error distribution. Its effectiveness is shown with data generated by Logistic map.

A novel multi-step adaptive prediction method for chaotic time series

Based on the short-term predictability of chaotic time series and the adaptive tracking chaotic trajectory of adaptive algorithm, a novel multi-step adaptive prediction method is proposed in this paper to resolve the problem of adjusting the filter parameters of the local adaptive prediction method during multi-step prediction. Simulation results show that this multi-step adaptive prediction method can be successfully used to make multi-step predictions, and its performance is better than that of the local adaptive prediction algorithm.

Modelling of chaotic time series using a variable-length windowing approach

Abstract A multidimensional feature extraction method for chaotic time series is presented. The method uses a variable-length windowing approach. Mackey–Glass delay-difference equation is used to create chaotic sample signals. Among many possible alternatives, the length of the data segment having the smallest variance fractal dimension (VFD) value is found and used as the feature value. Multidimensional feature vectors are formed to model each sample signal. A probabilistic neural network (PNN) is trained and tested with these vectors. It is shown that the application of the new feature extraction method improves the classification performance of the PNN as compared to a VFD based feature extraction method using a fix-length windowing approach.

A Noise Cleaning Method for Chaotic Time Series and ItsApplication in Communication

A new algorithm for filtering highly noisy contaminated chaotic timeseries is proposed and realized. It is indicated by computer simulationthat this algorithm can effectively reduce noise on chaotic signal nomater how parameter of chaos generator varies with time. In comparisonwith the extended-Kalman-filter-based method, this algorithm has abetter filtering performance in the case of low signal-to-noise (SNR)ratios, and has the similar performance in the case of high SNR. Inaddition, a chaotic modulation communication system is also used toevaluate the performance of the algorithm, the result shows theeffectiveness of the method.

Recovering strange attractors from noisy interspike intervals of neuronal firings

Both deterministic chaos and stochastic noise may play important roles in information transmission for neurons; however, their effects will be mixed and make it difficult to analyze such as using chaotic time series methods. In this letter, we show for the first time that fine structures of strange attractors can be recovered from noisy interspike interval (ISI) time series of both a theoretical model and real neurons by using the nonlinear filtering method. Thus improved analyzing results can be expected based on such a simple nonlinear filter. Different properties of noise in numerical and experimental ISI data are also compared.

A noise reduction method for chaotic time series

AbstractIntensive studies have recently been made on the noise reduction method for the chaotic time series. In those approaches, the trajectory of the attractor is reconstructed in the multidimensional space form the single‐variable time‐series data, and then the deviation of the trajectory is corrected. In the conventional method, the noise is assumed to be Gaussian distribution. Consequently, when the outliers are contained in the real data, they are difficult to delete.This paper intends to solve this problem and proposes a noise reduction method for the chaotic timeseries data, considering not only the noise with the Gaussian distribution, but also the outliers. A method is devised that eliminates the outliers efficiently using Biweight estimation and similar methods, and a program is constructed based on the idea. the experimental data are constructed by adding the pseudorandom variable with the Cauchy or Gaussian distribution to the chaotic timeseries data generated from the Hénon map and the Lorenz model. Using these data, the noise reduction experiment is conducted; the relative error, the phase plot and correlation dimensions are examined. It is verified as a result that the proposed method can reduce drastically the effect of the outliers compared to the conventional methods, and has nearly the same noise reduction performance as in the conventional methods for the noise with the Gaussian distribution.