Bayesian system identification and chaotic prediction from data for stochastic Mathieu-van der Pol-Duffing energy harvester

In this paper,the approximate Bayesian computation combines the particle swarm optimization and se-quential Monte Carlo methods,which identify the parameters of the Mathieu-van der Pol-Duffing chaotic energy harvester system.Then the proposed method is applied to estimate the coefficients of the chaotic model and the response output paths of the identified coefficients compared with the observed,which verifies the effectiveness of the proposed method.Finally,a partial response sample of the regular and chaotic responses,determined by the maximum Lyapunov exponent,is applied to detect whether chaotic motion occurs in them by a 0-1 test.This paper can provide a reference for data-based parameter iden-tification and chaotic prediction of chaotic vibration energy harvester systems.

New prediction of chaotic time series based on local Lyapunov exponent

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 recon- structing 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 locaI Lyapunov exponent. Numerical simulations are carded 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.

Chaotic time series multi-step direct prediction with partial least squares regression

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.

Chaotic time series prediction using fuzzy sigmoid kernel-based support vector machines

Support vector machines （SVM） have been widely used in chaotic time series predictions in recent years. In order to enhance the prediction efficiency of this method and implement it in hardware, the sigmoid kernel in SVM is drawn in a more natural way by using the fuzzy logic method proposed in this paper. This method provides easy hardware implementation and straightforward interpretability. Experiments on two typical chaotic time series predictions have been carried out and the obtained results show that the average CPU time can be reduced significantly at the cost of a small decrease in prediction accuracy, which is favourable for the hardware implementation for chaotic time series prediction.

Multimodality Prediction of Chaotic Time Series with Sparse Hard-Cut EM Learning of the Gaussian Process Mixture Model

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 expec- tation maximization （SHC-EM） learning algorithm for the GPM model is proposed to improve the prediction performance. SHO-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 1earning in terms of both prediction accuracy and speed. In addition, SHC-EM is more robust and insusceptible to noise than variational learning.

GEKF,GUKF and GGPF based prediction of chaotic time-series with additive and multiplicative noises

On the assumption that random interruptions in the observation process are modelled by a sequence of independent Bernoulli random variables, this paper generalize the extended Kalman filtering （EKF）, the unscented Kalman filtering （UKF） and the Gaussian particle filtering （GPF） to the case in which there is a positive probability that the observation in each time consists of noise alone and does not contain the chaotic signal （These generalized novel algorithms are referred to as GEKF, GUKF and GGPF correspondingly in this paper）. Using weights and network output of neural networks to constitute state equation and observation equation for chaotic time-series prediction to obtain the linear system state transition equation with continuous update scheme in an online fashion, and the prediction results of chaotic time series represented by the predicted observation value, these proposed novel algorithms are applied to the prediction of Mackey-Glass time-series with additive and multiplicative noises. Simulation results prove that the GGPF provides a relatively better prediction performance in comparison with GEKF and GUKF.

On the Chaotic Behavior and Predictability of the Real Atmosphere

In this paper the concept of Chaos and its applications to the study of predictability theory is introduced. The author's attempt is to give a general overview of ideas and methods involved in this problem to scientists,who are interested in the problem of predictability but not familiar with the theory of chaos. The problem is discussed in 4 sections. In the first section, the concept of chaos and the study methods are outlined briefly; in the second section, the methods of quantitatively measuring the main characteristics of chaos which are the basis for the predictability theory are introduced; the third section discusses the time series analysis for directly studying chaotic phenomena in practical problems; and the last section presents some research results on the chaotic characteristics and the predictability of the real atmosphere.