The state space reconstruction is the major important quantitative index for describing non_linear chaotic time series. Based on the work of many scholars, such as: N.H.Packard, F.Takens, M. Casdagli, J.F.Gibson, CHEN...The state space reconstruction is the major important quantitative index for describing non_linear chaotic time series. Based on the work of many scholars, such as: N.H.Packard, F.Takens, M. Casdagli, J.F.Gibson, CHEN Yu_shu et al, the state space was reconstructed using the method of Legendre coordinate. Several different scaling regimes for lag time τ were identified. The influence for state space reconstruction of lag time τ was discussed. The result tells us that is a good practical method for state space reconstruction.展开更多
A new method of predicting chaotic time series is presented based on a local Lyapunov exponent, by quantitatively measuring the exponential rate of separation or attraction of two infinitely close trajectories in stat...A new method of predicting chaotic time series is presented based on a local Lyapunov exponent, by quantitatively measuring the exponential rate of separation or attraction of two infinitely close trajectories in state space. After reconstructing state space from one-dimensional chaotic time series, neighboring multiple-state vectors of the predicting point are selected to deduce the prediction formula by using the definition of the local Lyapunov exponent. Numerical simulations are carried out to test its effectiveness and verify its higher precision over two older methods. The effects of the number of referential state vectors and added noise on forecasting accuracy are also studied numerically.展开更多
The PPSV (Proportional Pulse in the System Variable) algorithm is a convenient method for the stabilization of the chaotic time series. It does not require any previous knowledge of the system. The PPSV method also ha...The PPSV (Proportional Pulse in the System Variable) algorithm is a convenient method for the stabilization of the chaotic time series. It does not require any previous knowledge of the system. The PPSV method also has a shortcoming, that is, the determination off. is a procedure by trial and error, since it lacks of optimization. In order to overcome the blindness, GA (Genetic Algorithm), a search algorithm based on the mechanics of natural selection and natural genetics, is used to optimize the λi The new method is named as GAPPSV algorithm. The simulation results show that GAPPSV algorithm is very efficient because the control process is short and the steady-state error is small.展开更多
This paper proposes a co-evolutionary recurrent neural network(CERNN) for the multi-step-prediction of chaotic time series,it estimates the proper parameters of phase space reconstruction and optimizes the structure o...This paper proposes a co-evolutionary recurrent neural network(CERNN) for the multi-step-prediction of chaotic time series,it estimates the proper parameters of phase space reconstruction and optimizes the structure of recurrent neural networks by co-evolutionary strategy.The searching space was separated into two subspaces and the individuals are trained in a parallel computational procedure.It can dynamically combine the embedding method with the capability of recurrent neural network to incorporate past experience due to internal recurrence.The effectiveness of CERNN is evaluated by using three benchmark chaotic time series data sets:the Lorenz series,Mackey-Glass series and real-world sun spot series.The simulation results show that CERNN improves the performances of multi-step-prediction of chaotic time series.展开更多
Nonlinear response of the driven Duffng oscillator to periodic or quasi-periodic signals has been well studied.In this paper,we investigate the nonlinear response of the driven Duffng oscillator to non-periodic,more s...Nonlinear response of the driven Duffng oscillator to periodic or quasi-periodic signals has been well studied.In this paper,we investigate the nonlinear response of the driven Duffng oscillator to non-periodic,more specifically,chaotic time series.Through numerical simulations,we find that the driven Duffng oscillator can also show regular nonlinear response to the chaotic time series with different degree of chaos as generated by the same chaotic series generating model,and there exists a relationship between the state of the driven Duffng oscillator and the chaoticity of the input signal of the driven Duffng oscillator.One real-world and two artificial chaotic time series are used to verify the new feature of Duffng oscillator.A potential application of the new feature of Duffng oscillator is also indicated.展开更多
A new class of support vector machine, nu-support vector machine, is discussed which can handle both classification and regression. We focus on nu-support vector machine regression and use it for phase space predictio...A new class of support vector machine, nu-support vector machine, is discussed which can handle both classification and regression. We focus on nu-support vector machine regression and use it for phase space prediction of compares nu-support vector machine with back propagation (BP) networks in order to better evaluate the performance of the proposed methods. The experimental results show that the nu-support vector machine regression obtains lower root mean squared error than the BP networks and provides an accurate chaotic time series prediction. These results can be attributable to the fact that nu-support vector machine implements the structural risk minimization principle and this leads to better generalization than the BP networks.展开更多
In order to make more exact predictions of gas emissions, information fusion and chaos time series are com- bined to predict the amount of gas emission in pits. First, a multi-sensor information fusion frame is establ...In order to make more exact predictions of gas emissions, information fusion and chaos time series are com- bined to predict the amount of gas emission in pits. First, a multi-sensor information fusion frame is established. The frame includes a data level, a character level and a decision level. Functions at every level are interpreted in detail in this paper. Then, the process of information fusion for gas emission is introduced. On the basis of those data processed at the data and character levels, the chaos time series and neural network are combined to predict the amount of gas emission at the decision level. The weights of the neural network are gained by training not by manual setting, in order to avoid subjectivity introduced by human intervention. Finally, the experimental results were analyzed in Matlab 6.0 and prove that the method is more accurate in the prediction of the amount of gas emission than the traditional method.展开更多
Considering chaotic time series multi-step prediction,multi-step direct prediction model based on partial least squares (PLS) is proposed in this article,where PLS,the method for predicting a set of dependent variable...Considering chaotic time series multi-step prediction,multi-step direct prediction model based on partial least squares (PLS) is proposed in this article,where PLS,the method for predicting a set of dependent variables forming a large set of predictors,is used to model the dynamic evolution between the space points and the corresponding future points.The model can eliminate error accumulation with the common single-step local model algorithm,and refrain from the high multi-collinearity problem in the reconstructed state space with the increase of embedding dimension.Simulation predictions are done on the Mackey-Glass chaotic time series with the model. The satisfying prediction accuracy is obtained and the model efficiency verified.In the experiments,the number of extracted components in PLS is set with cross-validation procedure.展开更多
This paper proposes a Genetic Programming-Based Modeling (GPM) algorithm on chaotic time series. GP is used here to search for appropriate model structures in function space, and the Particle Swarm Optimization (PSO) ...This paper proposes a Genetic Programming-Based Modeling (GPM) algorithm on chaotic time series. GP is used here to search for appropriate model structures in function space, and the Particle Swarm Optimization (PSO) algorithm is used for Nonlinear Parameter Estimation (NPE) of dynamic model structures. In addition, GPM integrates the results of Nonlinear Time Series Analysis (NTSA) to adjust the parameters and takes them as the criteria of established models. Experiments showed the effectiveness of such improvements on chaotic time series modeling.展开更多
Artificial neural network (NN) is such a model as to imitate the structure and intelligence feature of human brain. It has strong nonlinear mapping function. To introduce NN into the study of earthquake prediction is ...Artificial neural network (NN) is such a model as to imitate the structure and intelligence feature of human brain. It has strong nonlinear mapping function. To introduce NN into the study of earthquake prediction is not only an extension of the application of artificial neural network model but also a new try for precursor observation to serve the earthquake prediction. In this paper, we analyzed the predictability of time series and gave a method of application of artificial neural network in forecasting earthquake precursor chaotic time series. Besides, taking the ground tilt observation of Jiangning and Xuzhou Station, the bulk strain observation of Liyang station as examples, we analyzed and forecasted their time series respectively. It is indicated that the precision of this method can meet the needs of practical task and therefore of great value in the application to the future practical earthquake analysis and prediction.展开更多
To predict the trend of chaotic time series in time series analysis and time series data mining fields,a novel predicting algorithm of chaotic time series trend is presented,and an on-line segmenting algorithm is prop...To predict the trend of chaotic time series in time series analysis and time series data mining fields,a novel predicting algorithm of chaotic time series trend is presented,and an on-line segmenting algorithm is proposed to convert a time series into a binary string according to ascending or descending trend of each subsequence.The on-line segmenting algorithm is independent of the prior knowledge about time series.The naive Bayesian algorithm is then employed to predict the trend of chaotic time series according to the binary string.The experimental results of three chaotic time series demonstrate that the proposed method predicts the ascending or descending trend of chaotic time series with few error.展开更多
Based on the measured displacements,the change laws of the effect of distance in phase space on the deformation of mine lane were analyzed and the chaotic time series model to predict the surrounding rocks deformation...Based on the measured displacements,the change laws of the effect of distance in phase space on the deformation of mine lane were analyzed and the chaotic time series model to predict the surrounding rocks deformation of deep mine lane in soft rock by nonlinear theory and methods was established.The chaotic attractor dimension(D) and the largest Lyapunov index(Emax) were put forward to determine whether the deformation process of mine lane is chaotic and the degree of chaos.The analysis of examples indicates that when D>2 and Emax>0,the surrounding rock's deformation of deep mine lane in soft rock is the chaotic process and the laws of the deformation can still be well demonstrated by the method of the reconstructive state space.Comparing with the prediction of linear time series and grey prediction,the chaotic time series prediction has higher accuracy and the prediction results can provide theoretical basis for reasonable support of mine lane in soft rock.The time of the second support in Maluping Mine of Guizhou,China,is determined to arrange at about 40 d after the initial support according to the prediction results.展开更多
Natural and chaotic time series are predicted using an artificial neural network(ANN)based on particle swarm optimization(PSO).Firstly,the hybrid ANN+PSO algorithm is applied on Mackey–Glass series in the short-term ...Natural and chaotic time series are predicted using an artificial neural network(ANN)based on particle swarm optimization(PSO).Firstly,the hybrid ANN+PSO algorithm is applied on Mackey–Glass series in the short-term prediction𝑥(𝑡+6),using the current value𝑥(𝑡)and the past values:𝑥(𝑡−6),𝑥(𝑡−12),𝑥(𝑡−18).Then,this method is applied on solar radiation data using the values of the past years:𝑥(𝑡−1),...,𝑥(𝑡−4).The results show that the ANN+PSO method is a very powerful tool for making predictions of natural and chaotic time series.展开更多
This paper presents an adaptive step-size modified fractional least mean square(AMFLMS) algorithm to deal with a nonlinear time series prediction. Here we incorporate adaptive gain parameters in the weight adaptation ...This paper presents an adaptive step-size modified fractional least mean square(AMFLMS) algorithm to deal with a nonlinear time series prediction. Here we incorporate adaptive gain parameters in the weight adaptation equation of the original MFLMS algorithm and also introduce a mechanism to adjust the order of the fractional derivative adaptively through a gradient-based approach. This approach permits an interesting achievement towards the performance of the filter in terms of handling nonlinear problems and it achieves less computational burden by avoiding the manual selection of adjustable parameters. We call this new algorithm the AMFLMS algorithm. The predictive performance for the nonlinear chaotic Mackey Glass and Lorenz time series was observed and evaluated using the classical LMS, Kernel LMS, MFLMS,and the AMFLMS filters. The simulation results for the Mackey glass time series, both without and with noise, confirm an improvement in terms of mean square error for the proposed algorithm. Its performance is also validated through the prediction of complex Lorenz series.展开更多
Due to the error in the measured value of the initial state and the sensitive dependence on initial conditions of chaotic dynamical systems,the error of chaotic time series prediction increases with the prediction ste...Due to the error in the measured value of the initial state and the sensitive dependence on initial conditions of chaotic dynamical systems,the error of chaotic time series prediction increases with the prediction step.This paper provides a method to improve the prediction precision by adjusting the predicted value in the course of iteration according to the evolution information of small intervals on the left and right sides of the predicted value.The adjusted predicted result is a non-trajectory which can provide a better prediction performance than the usual result based on the trajectory.Numerical simulations of two typical chaotic maps demonstrate its effectiveness.When the prediction step gets relatively larger,the effect is more pronounced.展开更多
Based on discussion on the theories of support vector machines (SVM),an one-step prediction model for time series prediction is presented,wherein the chaos theory is incorporated.Chaotic character of the time series i...Based on discussion on the theories of support vector machines (SVM),an one-step prediction model for time series prediction is presented,wherein the chaos theory is incorporated.Chaotic character of the time series is taken into account in the prediction procedure; parameters of reconstruction-delay and embedding-dimension for phase-space reconstruction are calculated in light of mutual-information and false-nearest-neighbor method,respectively.Precision and functionality have been demonstrated by the experimental results on the basis of the prediction of Lorenz chaotic time series.展开更多
The paper not only studies the noise reduction methods of chaotic time series with noise and its reconstruction techniques,but also discusses prediction techniques of chaotic time series and its applications based on ...The paper not only studies the noise reduction methods of chaotic time series with noise and its reconstruction techniques,but also discusses prediction techniques of chaotic time series and its applications based on chaotic data noise reduction.In the paper,we first decompose the phase space of chaotic time series to range space and null noise space.Secondly we restructure original chaotic time series in range space.Lastly on the basis of the above,we establish order of the nonlinear model and make use of the nonlinear model to predict some research.The result indicates that the nonlinear model has very strong ability of approximation function,and Chaos predict method has certain tutorial significance to the practical problems.展开更多
基金the National Natural Science Foundation of China(19990510)
文摘The state space reconstruction is the major important quantitative index for describing non_linear chaotic time series. Based on the work of many scholars, such as: N.H.Packard, F.Takens, M. Casdagli, J.F.Gibson, CHEN Yu_shu et al, the state space was reconstructed using the method of Legendre coordinate. Several different scaling regimes for lag time τ were identified. The influence for state space reconstruction of lag time τ was discussed. The result tells us that is a good practical method for state space reconstruction.
基金Project supported by the National Natural Science Foundation of China (Grant No. 61201452)
文摘A new method of predicting chaotic time series is presented based on a local Lyapunov exponent, by quantitatively measuring the exponential rate of separation or attraction of two infinitely close trajectories in state space. After reconstructing state space from one-dimensional chaotic time series, neighboring multiple-state vectors of the predicting point are selected to deduce the prediction formula by using the definition of the local Lyapunov exponent. Numerical simulations are carried out to test its effectiveness and verify its higher precision over two older methods. The effects of the number of referential state vectors and added noise on forecasting accuracy are also studied numerically.
文摘The PPSV (Proportional Pulse in the System Variable) algorithm is a convenient method for the stabilization of the chaotic time series. It does not require any previous knowledge of the system. The PPSV method also has a shortcoming, that is, the determination off. is a procedure by trial and error, since it lacks of optimization. In order to overcome the blindness, GA (Genetic Algorithm), a search algorithm based on the mechanics of natural selection and natural genetics, is used to optimize the λi The new method is named as GAPPSV algorithm. The simulation results show that GAPPSV algorithm is very efficient because the control process is short and the steady-state error is small.
基金Project supported by the State Key Program of National Natural Science of China (Grant No 30230350)the Natural Science Foundation of Guangdong Province,China (Grant No 07006474)
文摘This paper proposes a co-evolutionary recurrent neural network(CERNN) for the multi-step-prediction of chaotic time series,it estimates the proper parameters of phase space reconstruction and optimizes the structure of recurrent neural networks by co-evolutionary strategy.The searching space was separated into two subspaces and the individuals are trained in a parallel computational procedure.It can dynamically combine the embedding method with the capability of recurrent neural network to incorporate past experience due to internal recurrence.The effectiveness of CERNN is evaluated by using three benchmark chaotic time series data sets:the Lorenz series,Mackey-Glass series and real-world sun spot series.The simulation results show that CERNN improves the performances of multi-step-prediction of chaotic time series.
基金supported by the National Natural Science Foundation of China (Grant Nos 40574051 and 40774054)
文摘Nonlinear response of the driven Duffng oscillator to periodic or quasi-periodic signals has been well studied.In this paper,we investigate the nonlinear response of the driven Duffng oscillator to non-periodic,more specifically,chaotic time series.Through numerical simulations,we find that the driven Duffng oscillator can also show regular nonlinear response to the chaotic time series with different degree of chaos as generated by the same chaotic series generating model,and there exists a relationship between the state of the driven Duffng oscillator and the chaoticity of the input signal of the driven Duffng oscillator.One real-world and two artificial chaotic time series are used to verify the new feature of Duffng oscillator.A potential application of the new feature of Duffng oscillator is also indicated.
文摘A new class of support vector machine, nu-support vector machine, is discussed which can handle both classification and regression. We focus on nu-support vector machine regression and use it for phase space prediction of compares nu-support vector machine with back propagation (BP) networks in order to better evaluate the performance of the proposed methods. The experimental results show that the nu-support vector machine regression obtains lower root mean squared error than the BP networks and provides an accurate chaotic time series prediction. These results can be attributable to the fact that nu-support vector machine implements the structural risk minimization principle and this leads to better generalization than the BP networks.
基金Project BK2001073 supported by Natural Science Foundation of Jiangsu
文摘In order to make more exact predictions of gas emissions, information fusion and chaos time series are com- bined to predict the amount of gas emission in pits. First, a multi-sensor information fusion frame is established. The frame includes a data level, a character level and a decision level. Functions at every level are interpreted in detail in this paper. Then, the process of information fusion for gas emission is introduced. On the basis of those data processed at the data and character levels, the chaos time series and neural network are combined to predict the amount of gas emission at the decision level. The weights of the neural network are gained by training not by manual setting, in order to avoid subjectivity introduced by human intervention. Finally, the experimental results were analyzed in Matlab 6.0 and prove that the method is more accurate in the prediction of the amount of gas emission than the traditional method.
文摘Considering chaotic time series multi-step prediction,multi-step direct prediction model based on partial least squares (PLS) is proposed in this article,where PLS,the method for predicting a set of dependent variables forming a large set of predictors,is used to model the dynamic evolution between the space points and the corresponding future points.The model can eliminate error accumulation with the common single-step local model algorithm,and refrain from the high multi-collinearity problem in the reconstructed state space with the increase of embedding dimension.Simulation predictions are done on the Mackey-Glass chaotic time series with the model. The satisfying prediction accuracy is obtained and the model efficiency verified.In the experiments,the number of extracted components in PLS is set with cross-validation procedure.
基金Project (Nos. 60174009 and 70071017) supported by the National
Natural Science Foundation of China
文摘This paper proposes a Genetic Programming-Based Modeling (GPM) algorithm on chaotic time series. GP is used here to search for appropriate model structures in function space, and the Particle Swarm Optimization (PSO) algorithm is used for Nonlinear Parameter Estimation (NPE) of dynamic model structures. In addition, GPM integrates the results of Nonlinear Time Series Analysis (NTSA) to adjust the parameters and takes them as the criteria of established models. Experiments showed the effectiveness of such improvements on chaotic time series modeling.
基金Project supported by the National Natural Science Foundation of China （Grant No 60602034） and the Natural Science Foundation of Jiangxi Province, China （Grant No 0611031）.
文摘Artificial neural network (NN) is such a model as to imitate the structure and intelligence feature of human brain. It has strong nonlinear mapping function. To introduce NN into the study of earthquake prediction is not only an extension of the application of artificial neural network model but also a new try for precursor observation to serve the earthquake prediction. In this paper, we analyzed the predictability of time series and gave a method of application of artificial neural network in forecasting earthquake precursor chaotic time series. Besides, taking the ground tilt observation of Jiangning and Xuzhou Station, the bulk strain observation of Liyang station as examples, we analyzed and forecasted their time series respectively. It is indicated that the precision of this method can meet the needs of practical task and therefore of great value in the application to the future practical earthquake analysis and prediction.
文摘To predict the trend of chaotic time series in time series analysis and time series data mining fields,a novel predicting algorithm of chaotic time series trend is presented,and an on-line segmenting algorithm is proposed to convert a time series into a binary string according to ascending or descending trend of each subsequence.The on-line segmenting algorithm is independent of the prior knowledge about time series.The naive Bayesian algorithm is then employed to predict the trend of chaotic time series according to the binary string.The experimental results of three chaotic time series demonstrate that the proposed method predicts the ascending or descending trend of chaotic time series with few error.
基金Project(50490274) supported by the National Natural Science Foundation of China
文摘Based on the measured displacements,the change laws of the effect of distance in phase space on the deformation of mine lane were analyzed and the chaotic time series model to predict the surrounding rocks deformation of deep mine lane in soft rock by nonlinear theory and methods was established.The chaotic attractor dimension(D) and the largest Lyapunov index(Emax) were put forward to determine whether the deformation process of mine lane is chaotic and the degree of chaos.The analysis of examples indicates that when D>2 and Emax>0,the surrounding rock's deformation of deep mine lane in soft rock is the chaotic process and the laws of the deformation can still be well demonstrated by the method of the reconstructive state space.Comparing with the prediction of linear time series and grey prediction,the chaotic time series prediction has higher accuracy and the prediction results can provide theoretical basis for reasonable support of mine lane in soft rock.The time of the second support in Maluping Mine of Guizhou,China,is determined to arrange at about 40 d after the initial support according to the prediction results.
基金by the Direction of Research of the University of La Serena,and the Department of Physics of the University of La Serena.
文摘Natural and chaotic time series are predicted using an artificial neural network(ANN)based on particle swarm optimization(PSO).Firstly,the hybrid ANN+PSO algorithm is applied on Mackey–Glass series in the short-term prediction𝑥(𝑡+6),using the current value𝑥(𝑡)and the past values:𝑥(𝑡−6),𝑥(𝑡−12),𝑥(𝑡−18).Then,this method is applied on solar radiation data using the values of the past years:𝑥(𝑡−1),...,𝑥(𝑡−4).The results show that the ANN+PSO method is a very powerful tool for making predictions of natural and chaotic time series.
基金Project supported by the Higher Education Commission of Pakistan
文摘This paper presents an adaptive step-size modified fractional least mean square(AMFLMS) algorithm to deal with a nonlinear time series prediction. Here we incorporate adaptive gain parameters in the weight adaptation equation of the original MFLMS algorithm and also introduce a mechanism to adjust the order of the fractional derivative adaptively through a gradient-based approach. This approach permits an interesting achievement towards the performance of the filter in terms of handling nonlinear problems and it achieves less computational burden by avoiding the manual selection of adjustable parameters. We call this new algorithm the AMFLMS algorithm. The predictive performance for the nonlinear chaotic Mackey Glass and Lorenz time series was observed and evaluated using the classical LMS, Kernel LMS, MFLMS,and the AMFLMS filters. The simulation results for the Mackey glass time series, both without and with noise, confirm an improvement in terms of mean square error for the proposed algorithm. Its performance is also validated through the prediction of complex Lorenz series.
文摘Due to the error in the measured value of the initial state and the sensitive dependence on initial conditions of chaotic dynamical systems,the error of chaotic time series prediction increases with the prediction step.This paper provides a method to improve the prediction precision by adjusting the predicted value in the course of iteration according to the evolution information of small intervals on the left and right sides of the predicted value.The adjusted predicted result is a non-trajectory which can provide a better prediction performance than the usual result based on the trajectory.Numerical simulations of two typical chaotic maps demonstrate its effectiveness.When the prediction step gets relatively larger,the effect is more pronounced.
文摘Based on discussion on the theories of support vector machines (SVM),an one-step prediction model for time series prediction is presented,wherein the chaos theory is incorporated.Chaotic character of the time series is taken into account in the prediction procedure; parameters of reconstruction-delay and embedding-dimension for phase-space reconstruction are calculated in light of mutual-information and false-nearest-neighbor method,respectively.Precision and functionality have been demonstrated by the experimental results on the basis of the prediction of Lorenz chaotic time series.
基金Project supported by the National Natural Science Foundation of China(Nos.70271071,19990510,D0200201)
文摘The paper not only studies the noise reduction methods of chaotic time series with noise and its reconstruction techniques,but also discusses prediction techniques of chaotic time series and its applications based on chaotic data noise reduction.In the paper,we first decompose the phase space of chaotic time series to range space and null noise space.Secondly we restructure original chaotic time series in range space.Lastly on the basis of the above,we establish order of the nonlinear model and make use of the nonlinear model to predict some research.The result indicates that the nonlinear model has very strong ability of approximation function,and Chaos predict method has certain tutorial significance to the practical problems.
