Chaotic Time Series Prediction Research Articles

Local Functional Coefficient Autoregressive Model for Multistep Prediction of Chaotic Time Series

A new methodology, which combines nonparametric method based on local functional coefficient autoregressive (LFAR) form with chaos theory and regional method, is proposed for multistep prediction of chaotic time series. The objective of this research study is to improve the performance of long-term forecasting of chaotic time series. To obtain the prediction values of chaotic time series, three steps are involved. Firstly, the original time series is reconstructed inm-dimensional phase space with a time delayτby using chaos theory. Secondly, select the nearest neighbor points by using local method in them-dimensional phase space. Thirdly, we use the nearest neighbor points to get a LFAR model. The proposed model’s parameters are selected by modified generalized cross validation (GCV) criterion. Both simulated data (Lorenz and Mackey-Glass systems) and real data (Sunspot time series) are used to illustrate the performance of the proposed methodology. By detailed investigation and comparing our results with published researches, we find that the LFAR model can effectively fit nonlinear characteristics of chaotic time series by using simple structure and has excellent performance for multistep forecasting.

Multivariate chaotic time series prediction using multiple kernel extreme learning machine

Multivariate chaotic time series is widely present in nature, such as in economy, society, industry and other fields. Modeling and predicting multivariate time series will help human to better manage, control, and make decision. A prediction method based on multiple kernel extreme learning machine is proposed in this paper to model the complex dynamics of multivariate chaotic time series. First, the multivariate chaotic time series is reconstructed in phase space, transforming the temporal correlation into spatial correlation. Then, a prediction model-multiple kernel extreme learning machine, which combines the multiple kernel learning and extreme learning machine with kernels, is proposed to approximate the nonlinear function of the input - output data in phase space. The proposed multiple kernel extreme learning machine could effectively combine the simple training of extreme learning machine with kernels and the data fusion capabilities of multiple kernel learning. Simulation results based on Lorenz chaotic time series prediction and San Francisco monthly runoff prediction demonstrate that, compared with other state-of-art chaotic time series prediction methods, the proposed multiple kernel extreme learning machine could get a better prediction accuracy.

Local Prediction of Chaotic Time Series Based on Polynomial Coefficient Autoregressive Model

We apply the polynomial function to approximate the functional coefficients of the state-dependent autoregressive model for chaotic time series prediction. We present a novel local nonlinear model called local polynomial coefficient autoregressive prediction (LPP) model based on the phase space reconstruction. The LPP model can effectively fit nonlinear characteristics of chaotic time series with simple structure and have excellent one-step forecasting performance. We have also proposed a kernel LPP (KLPP) model which applies the kernel technique for the LPP model to obtain better multistep forecasting performance. The proposed models are flexible to analyze complex and multivariate nonlinear structures. Both simulated and real data examples are used for illustration.

Using repetitive fuzzy method for chaotic time series prediction

In this paper, a new perspective is presented for prediction of chaotic time series by combining the phase space reconstruction and fuzzy approach. Before applying time series to the predictor system, the reconstruction parameters, including embedding dimension and time delay, are determined in an off-line manner by using nearest neighbor and mutual information methods. Then, the structure of the fuzzy system is specified and the input number of fuzzy system is set to the embedding dimensions. According to the embedding dimension and time delay, the phase space is reconstructed point-by-point at the entry of the fuzzy system. Fuzzy system is composed of two separated parts: predictor and tracker. The predictor part forecasts the next point for new entry with fine-tuned parameters using the last step, and the tracker part adjusts the parameters for the next step. This adjustment is done iteratively. The proposed method is compared with some references' results. Simplicity and appropriate speed with sufficient accuracy are the advantages of this method.

Hermite orthogonal basis neural network based on improved teaching-learning-based optimization algorithm for chaotic time series prediction

Chaos phenomenon which exists widely in nature and society affects people's production and life. It has great important significance to find out the regularity of chaotic time series from a chaotic system. Since chaotic system has extremely complex dynamic characteristics and unpredictability, and chaotic time series prediction through traditional methods has low prediction precision, slow convergence speed and complex model structure, a prediction model about Hermite orthogonal basis neural network based on improved teaching-learning-based optimization algorithm is proposed. Firstly, according to the chaotic time series, autocorrelation method and Cao method are used to determine the best delay time and the minimum embedding dimension respectively, then a phase space is reconstructed to obtain the refactoring delay time vector. Secondly, on the basis of phase space reconstruction and best square approximation theory, combined with the neural network topology, a prediction model about Hermite orthogonal basis neural network with excitation functions based on the Hermite orthogonal basis functions is put forward. Thirdly, in order to optimize the parameters of the prediction model, an improved teaching-learning-based optimization algorithm is proposed, where a feedback stage is introduced at the end of the learning stage based on the teaching-learning-based optimization algorithm. Finally, the parameter optimization problem is transformed into a function optimization problem in the multidimensional space, then the improved teaching-learning-based optimization algorithm is used for parameter optimization of the prediction model so as to establish it and analyze it. Lorenz and Liu chaotic systems are taken as models respectively, then the chaotic time series which will be used as simulation object is produced by the fourth order Runge-Kutta method. The comparison experiments with other prediction models are conducted on single-step and multi-step prediction for the chaotic time series. The simulation results and numerical analysis show that compared with radial basis function neural network, echo state network, least square support vector machine prediction model and Hermite orthogonal basis neural network based on teaching-learning-based optimization algorithm, the proposed prediction model has the mean absolute error and root mean square error reduced significantly, has a decision coefficient close to 1, meanwhile, has a mean modeling time shortened greatly. So the proposed prediction model can improve the prediction precision, accelerate the convergence speed and simplify the model structure, thus the prediction model is effective and feasible, which makes it promoted and applied easily.

Impact of Noise on a Dynamical System: Prediction and Uncertainties from a Swarm-Optimized Neural Network.

An artificial neural network (ANN) based on particle swarm optimization (PSO) was developed for the time series prediction. The hybrid ANN+PSO algorithm was applied on Mackey-Glass chaotic time series in the short-term x(t + 6). The performance prediction was evaluated and compared with other studies available in the literature. Also, we presented properties of the dynamical system via the study of chaotic behaviour obtained from the predicted time series. Next, the hybrid ANN+PSO algorithm was complemented with a Gaussian stochastic procedure (called stochastic hybrid ANN+PSO) in order to obtain a new estimator of the predictions, which also allowed us to compute the uncertainties of predictions for noisy Mackey-Glass chaotic time series. Thus, we studied the impact of noise for several cases with a white noise level (σ N) from 0.01 to 0.1.

Integrated photonic reservoir computing based on hierarchical time-multiplexing structure.

An integrated photonic reservoir computing (RC) based on hierarchical time-multiplexing structure is proposed by numerical simulations. A micro-ring array (MRA) is employed as a typical time delay implementation of RC. At the output port of the MRA, a secondary time-multiplexing is achieved by multi-mode interference (MMI) splitter and delay line array. This hierarchical time-multiplexing structure can ensure a large reservoir size with fast processing speed. Simulation results indicate that the proposed RC system yields better performance than previously reported ones. The achieved normalized mean square error between the system output and target sequence are 0.5% and 2.7% for signal classification and chaotic time series prediction, respectively, while the sample rate is as high as 1.3 Gbps.

Study on Prediction of Chaotic Time Series Using Least Squares Support Vector Machines

The predictions of chaotic time series by applying the least squares support vector machine (LS-SVM), with comparison with the traditional-SVM and-SVM, were specified. The results show that, compared with the traditional SVM, the prediction accuracy of LS-SVM is better than the traditional SVM and more suitable for time series online prediction.

A simple Mathematical Fuzzy Model of Brain Emotional Learning to Predict Kp Geomagnetic Index

In this paper, we propose fuzzy mathematical model of brain limbic system (LS) which is responsible for emotional stimuli. Here the proposed model is utilized to predict the chaotic activity of the earth’s magnetosphere. Numerical results show that the correlation of the results obtained from the proposed fuzzy model is higher than non-fuzzy models. Hence, the proposed model can be applied in real time chaotic time series prediction.

Time series prediction using ensembles of ANFIS models with genetic optimization of interval type-2 and type-1 fuzzy integrators

This paper describes an optimization of interval type-2 and type-1 fuzzy integrators in ensembles of ANFIS models with genetic algorithms (GAs), this with emphasis on its application to the prediction of chaotic time series, where the goal is to minimize the prediction error. The Mackey-Glass time series was considered to validate the proposed ensemble approach. The methods used for the integration of the ensembles of ANFIS are: type-1 and interval type-2 Mamdani fuzzy inference systems (FIS). Genetic Algorithms are used for optimization of the membership function parameters of the FIS in each integrator. In the experiments we changed the type of the membership functions for each type-1 and interval type-2 FIS, thereby increasing the complexity of the training, The output (Forecast) generated by each integrator is calculated with the RMSE (root mean square error) to minimize the prediction error, therefore we compared the performance obtained by each FIS.

IDENTIFICATION AND FORECASTING OF SHARP CHANGES IN ECONOMICAL SYSTEMS BY TRACKING OF LOCAL LYAPUNOV EXPONENTS

An approach to detect sharp changes in economical systems was developed in this paper. We will give brief introduction in current methods of prediction of nonlinear and chaotic time-series and give definition of local Lyapunov exponents (LLE). Then author’s approach will be described. Also some numerical results and discussions will be given.

A Novel Fuzzy Based Method for Heart Rate Variability Prediction

Abstract In this paper, a novel technique based on fuzzy method is presented for chaotic nonlinear time series prediction. Fuzzy approach with the gradient learning algorithm and methods constitutes the main components of this method. This learning process in this method is similar to conventional gradient descent learning process, except that the input patterns and parameters are stored in memory as a look-up table after upgrade. In the testing phase according to input patterns, the nearest neighbors and the weights corresponding to the test pattern, similar patterns are extracted from memory. Eventually by extracted weights and input pattern, prediction is performed. In order to validate the proposed method for predicting, the Mackey-Glass, Lorenz and biological Heart Rate Variability (HRV) time series is used. Finally, results of proposed method with the conventional methods of time-series prediction are also compared. The results demonstrate the capability of proposed method in chaotic time series prediction.

Explicit Recursive and Adaptive Filtering in Reproducing Kernel Hilbert Spaces

This brief presents a methodology to develop recursive filters in reproducing kernel Hilbert spaces. Unlike previous approaches that exploit the kernel trick on filtered and then mapped samples, we explicitly define the model recursivity in the Hilbert space. For that, we exploit some properties of functional analysis and recursive computation of dot products without the need of preimaging or a training dataset. We illustrate the feasibility of the methodology in the particular case of the γ-filter, which is an infinite impulse response filter with controlled stability and memory depth. Different algorithmic formulations emerge from the signal model. Experiments in chaotic and electroencephalographic time series prediction, complex nonlinear system identification, and adaptive antenna array processing demonstrate the potential of the approach for scenarios where recursivity and nonlinearity have to be readily combined.

A hybridization of teaching–learning-based optimization and differential evolution for chaotic time series prediction

Chaotic time series prediction problems have some very interesting properties and their prediction has received increasing interest in the recent years. Prediction of chaotic time series based on the phase space reconstruction theory has been applied in many research fields. It is well known that prediction of a chaotic system is a nonlinear, multivariable and multimodal optimization problem for which global optimization techniques are required in order to avoid local optima. In this paper, a new hybrid algorithm named teaching---learning-based optimization (TLBO)---differential evolution (DE), which integrates TLBO and DE, is proposed to solve chaotic time series prediction. DE is incorporated into update the previous best positions of individuals to force TLBO jump out of stagnation, because of its strong searching ability. The proposed hybrid algorithm speeds up the convergence and improves the algorithm's performance. To demonstrate the effectiveness of our approaches, ten benchmark functions and three typical chaotic nonlinear time series prediction problems are used for simulating. Conducted experiments indicate that the TLBO---DE performs significantly better than, or at least comparable to, TLBO and some other algorithms.

Fast photonic information processing using semiconductor lasers with delayed optical feedback: Role of phase dynamics

Semiconductor lasers subject to delayed optical feedback have recently shown great potential in solving computationally hard tasks. By optically implementing a neuro-inspired computational scheme, called reservoir computing, based on the transient response to optical data injection, high processing speeds have been demonstrated. While previous efforts have focused on signal bandwidths limited by the semiconductor laser's relaxation oscillation frequency, we demonstrate numerically that the much faster phase response makes significantly higher processing speeds attainable. Moreover, this also leads to shorter external cavity lengths facilitating future on-chip implementations. We numerically benchmark our system on a chaotic time-series prediction task considering two different feedback configurations. The results show that a prediction error below 4% can be obtained when the data is processed at 0.25 GSamples/s. In addition, our insight into the phase dynamics of optical injection in a semiconductor laser also provides a clear understanding of the system performance at different pump current levels, even below solitary laser threshold. Considering spontaneous emission noise and noise in the readout layer, we obtain good prediction performance at fast processing speeds for realistic values of the noise strength.

Adaptive step-size modified fractional least mean square algorithm for chaotic time series prediction

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.

Chaotic Time Series Prediction Algorithm for Lori Mapping Based on Optimized BP Neural Network

Based on the particle swarm optimization (PSO) and BP neural network (BPNN), an algorithm for BP neural network optimized particle swarm optimization (PSOBPNN) is proposed. In the algorithm, PSO is used to obtain better network initial threshold and weight to compensate the defect of connection weight and thresholds of BPNN, thus it can make BPNN have faster convergence and greater learning ability. The efficiency of the proposed prediction method is tested by the simulation of the chaotic time series for Lori mapping. The simulations results show that the proposed method has higher forecasting accuracy compared with the BPNN, so it is proved that the algorithm is feasible and effective in the chaotic time series prediction.

Self-organization and lateral interaction in echo state network reservoirs

Echo state networks (ESNs) are recurrent structures that give rise to an interesting trade-off between achievable performance and tractability. This is a consequence of the fact that the key element of these networks – the recurrent intermediate layer known as dynamical reservoir – is not, as a rule, subject to supervised training, which is restricted to the linear output layer, also termed as readout. This trade-off, aside from being of theoretical significance, establishes ESNs as most attractive tools for both online and offline information processing. There are two key aspects to be taken into account in the ESN design: (i) the unsupervised definition of the synaptic weights of the reservoir and (ii) the definition of the structure and of the training strategy associated with the readout. This work is concerned with the first of these aspects: it proposes novel strategies for ESN reservoir design based on the theoretical framework built by Kohonen׳s classical works on self-organization – which includes the notions of short-range positive feedback and lateral inhibition – and also on the related and more recent notion of neural gas. It is shown, with the aid of a representative set of simulation results, that the proposed methodologies are capable of leading to significant performance improvements in the context of relevant information processing tasks – channel equalization and chaotic time series prediction – particularly when the input data suits well a cluster-based profile.

Annealed cooperative–competitive learning of Mahalanobis-NRBF neural modules for nonlinear and chaotic differential function approximation

This work explores annealed cooperative–competitive learning of multiple modules of Mahalanobis normalized radial basis functions (NRBF) with applications to nonlinear function approximation and chaotic differential function approximation. A multilayer neural network is extended to be composed of multiple Mahalanobis-NRBF modules. Each module activates normalized outputs of radial basis functions, determining Mahalanobis radial distances based on its own adaptable weight matrix. An essential cooperative scheme well decomposes learning a multi-module network to sub-tasks of learning individual modules. Adaptable network interconnections are asynchronously updated module-by-module based on annealed cooperative–competitive learning for function approximation under a physical-like mean-field annealing process. Numerical simulations show outstanding performance of annealed cooperative–competitive learning of a multi-module Mahalanobis-NRBF network for nonlinear function approximation and long term look-ahead prediction of chaotic time series.

A modified fractional least mean square algorithm for chaotic and nonstationary time series prediction

A method of modifying the architecture of fractional least mean square (FLMS) algorithm is presented to work with nonlinear time series prediction. Here we incorporate an adjustable gain parameter in the weight adaptation equation of the original FLMS algorithm and absorb the gamma function in the fractional step size parameter. This approach provides an interesting achievement in the performance of the filter in terms of handling the nonlinear problems with less computational burden by avoiding the evaluation of complex gamma function. We call this new algorithm as the modified fractional least mean square (MFLMS) algorithm. The predictive performance for the nonlinear Mackey glass chaotic time series is observed and evaluated using the classical LMS, FLMS, kernel LMS, and proposed MFLMS adaptive filters. The simulation results for the time series with and without noise confirm the superiority and improvement in the prediction capability of the proposed MFLMS predictor over its counterparts.