Intergroup Cascade Broad Learning System With Optimized Parameters for Chaotic Time Series Prediction

Long-term prediction of chaotic time series is very difficult, for the chaos restricts predictability. In this paper a new method is studied to model and predict chaotic time series based on minimax probability machine regression (MPMR). Since the positive global Lyapunov exponents lead the errors to increase exponentially in modelling the chaotic time series, a weighted term is introduced to compensate a cost function. Using mean square error (MSE) and absolute error (AE) as a criterion, simulation results show that the proposed method is more effective and accurate for multistep prediction. It can identify the system characteristics quite well and provide a new way to make long-term predictions of the chaotic time series.

One of the features of deterministic chaos is sensitive to initial conditions. This feature limits the prediction horizons of many chaotic systems. In this paper, we propose a new prediction technique for chaotic time series. In our method, some neighbouring points of the predicted point, for which the corresponding local Lyapunov exponent is particularly large, would be discarded during estimating the local dynamics, and thus the error accumulated by the prediction algorithm is reduced. The model is tested for the convection amplitude of Lorenz systems. The simulation results indicate that the prediction technique can improve the prediction of chaotic time series.

The prediction accuracy and generalization ability of neural/neurofuzzy models for chaotic time series prediction highly depends on employed network model as well as learning algorithm. In this study, several neural and neurofuzzy models with different learning algorithms are examined for prediction of several benchmark chaotic systems and time series. The prediction performance of locally linear neurofuzzy models with recently developed Locally Linear Model Tree (LoLiMoT) learning algorithm is compared with that of Radial Basis Function (RBF) neural network with Orthogonal Least Squares (OLS) learning algorithm, MultiLayer Perceptron neural network with error back-propagation learning algorithm, and Adaptive Network based Fuzzy Inference System. Particularly, cross validation techniques based on the evaluation of error indices on multiple validation sets is utilized to optimize the number of neurons and to prevent over fitting in the incremental learning algorithms. To make a fair comparison between neural and neurofuzzy models, they are compared at their best structure based on their prediction accuracy, generalization, and computational complexity. The experiments are basically designed to analyze the generalization capability and accuracy of the learning techniques when dealing with limited number of training samples from deterministic chaotic time series, but the effect of noise on the performance of the techniques is also considered. Various chaotic systems and time series including Lorenz system, Mackey-Glass chaotic equation, Henon map, AE geomagnetic activity index, and sunspot numbers are examined as case studies. The obtained results indicate the superior performance of incremental learning algorithms and their respective networks, such as, OLS for RBF network and LoLiMoT for locally linear neurofuzzy model.

Prediction and analysis of chaotic time series on the basis of support vector

A nonlinear behavior may exist intrinsically with a deterministic dynamic system which shows high sensitivity with initial condition. Such behavior is characterized as chaotic behavior. Strange attractor confines the dynamic behavior of chaotic system in a finite state space region, and the available state variables show the stochastic behavior with time. In this research work, time delay neural network has applied to predict the various chaotic time series. Adaptive social behavior optimization has applied to evolve the optimal set of weights of the time delay neural network. Comparison of learning performance has given with popularly gradient descent-based learning. Performance evaluation has defined in terms of coefficient of determination along with root-mean-square error in prediction under learning and test phase of chaotic time series. The three benchmarks of chaotic time series (logistic differential equation, Mackey–Glass, and Lorenz system) have taken for predicting purpose. We have shown with experimental results that the proposed new method of neural network learning is very efficient and has delivered the better prediction for various complex chaotic time series.

Configuring radial basis function network using fractal scaling process with application to chaotic time series prediction

Chaotic systems are complex dynamical systems that play a very important role in the study of the atmosphere, aerospace engineering, finance, etc. To improve the accuracy of chaotic time series prediction, this study proposes a hybrid model CEEMDAN-LSTM which combines Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN) and long short-term memory (LSTM). In the model, the original time series is decomposed into several intrinsic mode functions (IMFs) and a residual component. To reduce the difficulty of predicting chaotic time series and provide a high level of predictive accuracy, the LSTM prediction model is built for all each characteristic series from CEEMDAN deposition. Finally, the final prediction results are obtained by combining all the prediction sequences. To test the effectiveness of this model we proposed, we examined the CEEMDAN-LSTM model using the Lorenz-63 system. Further compared to Autoregressive Integrated Moving Average (ARIMA ), Support Vector Regression (SVR), multilayer perceptron (MLP), and the single LSTM model, the results of the experiment show that the proposed model performs better in the prediction of chaotic time series. Besides, the hybrid model proposed in this paper has better results than the LSTM model alone. Therefore, hybrid models based on deep learning methods and signal decomposition methods have great potential in the field of chaotic time series prediction.

High rock slope engineering is typical nonlinear system, its evolution process is chaotic, dissipated and uncertain. Chaotic system can’t be forecasted for long-term and needs discuss about maximum time scale of predictability. Nonlinear theory is proposed to research maximum time scale of predictability of safety monitoring chaotic time series and construct the model APSO-RBFNN to predict the chaotic time series in maximum time scale of predictability. The largest Lyapunov exponent and maximum time scale is calculate with small data sets method. In the maximum time scale of predictability, the essay applies APSO-RBFNN to chaotic time series for risk assessment. The engineering cases studies reveal that the forecasting values are in good agreement with the measured values and this model has high accuracy and a good prospect for risk assessment of nonlinear chaotic time series of geotechnical engineering.

The prediction of chaotic time series is an important research issue. To improve the prediction accuracy, a hybrid approach called WNN-PSO is proposed, which based on the self-learning ability of wavelet neural network, whose parameters are optimized by particle swarm optimization. The WNN-PSO method has higher prediction accuracy, fast convergence, and heightens the ability of jumping the local optimums. The experiment results of the prediction for chaotic time series show the feasibility and effectiveness of the proposed method. Compared with wavelet neural network and BP neural network, the proposed method are superior to them. Finally, the WNN-PSO is applied to predict the life energy consumption of china in our lives.

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.

The main purpose of this paper is to study a new method to model and predict a chaotic time series using a fuzzy model. First, the GK fuzzy clustering method is used to confirm the input space of the fuzzy model. The goal is to divide the training patterns into representative groups so that patterns within one cluster are more similar than those belonging to other clusters. Then, the Kalman filtering algorithm with singular value decomposition is applied to estimate the consequent parameters of the fuzzy model in order to avoid error delivery and error accumulation. The effectiveness of the proposed method is evaluated through simulated examples, including Mackey-Glass time series and Lorenz chaotic systems. The results show that the proposed method provides effective and accurate prediction. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

This chapter predicts the chaotic ferroresonance time series model by the dynamic fuzzy neural network (D-FNN). Through modeling chaotic system and analysis of the results of the chaotic model, the purpose of understanding the chaotic system and control has been successfully achieved. The method does not rely on the deterministic equations and initial conditions of chaotic time series. On the basis of given training data, we use reconstruction methods for data processing and initialization. Next, we make prediction with combination of the data and the dynamic fuzzy neural network. Experiments show that this method has preferable predictive ability and accuracy. The prediction obtains satisfactory results.

The broad learning system (BLS) is designed based on the technology of compressed sensing and pseudo-inverse theory, and consists of feature nodes and enhancement nodes, has been proposed recently. Compared with the popular deep learning structures, such as deep neural networks, BLS has the ability of rapid incremental learning and can remodel the system without the usual tedious retraining process. However, given that BLS is still in its infancy, it still needs analysis, improvements, and verification. In this article, we first analyze the principle of fast incremental learning ability of BLS in depth. Second, in order to provide an in-depth analysis of the BLS structure, according to the novel structure design concept of deep neural networks, we present four brand-new BLS variant networks and their incremental realizations. Third, based on our analysis of the effect of feature nodes and enhancement nodes, a new BLS structure with a semantic feature extraction layer has been proposed, which is called SFEBLS. The experimental results show that SFEBLS and its variants can increase the accuracy rate on the NORB dataset 6.18%, Fashion-MNIST dataset by 3.15%, ORL data by 5.00%, street view house number dataset by 12.88%, and CIFAR-10 dataset by 18.42%, respectively, and the four brand-new BLS variant networks also obviously outperform the original BLS.

In this paper, a novel model, named double-reservoir echo state networks (DR-ESN), is proposed. DR-ESN is constructed by two reservoirs which are connected in series, thus the performance of abstracting the characteristics from the prediction task is improved. A sufficient condition is provided to ensure the stability of DR-ESN. The batch gradient method and ridge regression method are utilized to optimize the six parameters of DR-ESN and train the readouts, respectively. DR-ESN is verified by two different experiments, chaotic time series prediction and real-valued function time series prediction. The simulation results demonstrates that DR-ESN has a more precise result than leaky-ESN in predicting the time series.

In this paper, an integrated model based on efficient extreme learning machine (EELM) and differential evolution (DE) is proposed to predict chaotic time series. In the proposed model, a novel learning algorithm called EELM is presented and used to model the chaotic time series. The EELM inherits the basic idea of extreme learning machine (ELM) in training single hidden layer feedforward networks, but replaces the commonly used singular value decomposition with a reduced complete orthogonal decomposition to calculate the output weights, which can achieve a much faster learning speed than ELM. Moreover, in order to obtain a more accurate and more stable prediction performance for chaotic time series prediction, this model abandons the traditional two-stage modeling approach and adopts an integrated parameter selection strategy which employs a modified DE algorithm to optimize the phase space reconstruction parameters of chaotic time series and the model parameter of EELM simultaneously based on a hybrid validation criterion. Experimental results show that the proposed integrated prediction model can not only provide stable prediction performances with high efficiency but also achieve much more accurate prediction results than its counterparts for chaotic time series prediction.