Modeling Of Chaotic Time Series Research Articles

Dynamic adaptive graph convolutional transformer with broad learning system for multi-dimensional chaotic time series prediction

Chaotic time series data is extensively applied in financial stocks, climate monitoring, and sea clutter, in which data fusion from various sources and multi-sensor information make accurate predictions of chaotic time series challenging under complex nonlinear conditions. Previous works focus on designing different model frameworks to capture the temporal dependence and extract richer nonlinear features to improve the accuracy of univariate chaotic time series prediction, which ignores the spatial dependence of multivariable. However, in this paper, we argue that spatial correlation among multiple variables is essential to improve the prediction accuracy of chaotic time series. To fill the gap, we innovatively propose a Dynamic Adaptive Graph Convolutional Transformer with a Broad Learning System (DAGCT-BLS), a GCN and Transformer-based model utilizing multivariate spatial dependence for multi-dimensional chaotic time series forecasting. In DAGCT-BLS, the multivariate chaotic time series are reconstructed into the phase space, and the reconstructed data are rapidly feature-extracted using a cascade network BLS with frozen weights to maximize the retention of chaotic properties and nonlinear relationships. Then, the Dynamic Adaptive Graph Convolutional Network (DAGCN) is proposed to capture the spatial correlation among the multiple variables. Finally, improved multi-head attention of the Transformer Encoder is used to capture the temporal dependence of the phase point sequence. Experiments of our proposed model on three datasets (Lorenz, Rossler, and Sea clutter) show that DAGCT-BLS can achieve the best prediction performance and have strong interpretability, and multivariate-based joint modeling of chaotic time series helps to improve the prediction performance.

Multilayer Perceptron Network Optimization for Chaotic Time Series Modeling.

Chaotic time series are widely present in practice, but due to their characteristics-such as internal randomness, nonlinearity, and long-term unpredictability-it is difficult to achieve high-precision intermediate or long-term predictions. Multi-layer perceptron (MLP) networks are an effective tool for chaotic time series modeling. Focusing on chaotic time series modeling, this paper presents a generalized degree of freedom approximation method of MLP. We then obtain its Akachi information criterion, which is designed as the loss function for training, hence developing an overall framework for chaotic time series analysis, including phase space reconstruction, model training, and model selection. To verify the effectiveness of the proposed method, it is applied to two artificial chaotic time series and two real-world chaotic time series. The numerical results show that the proposed optimized method is effective to obtain the best model from a group of candidates. Moreover, the optimized models perform very well in multi-step prediction tasks.

Multistep-Ahead Chaotic Time Series Prediction Based on Hierarchical Echo State Network With Augmented Random Features

Multistep-ahead chaotic time series prediction is a kind of highly nonlinear problem, which puts forward higher requirements both for the dynamical memory and nonlinearity of the model. Echo state network (ESN) is frequently employed in the realm of chaotic time series modeling and prediction, but the basic ESN has been proved to have an antagonistic trade-off between nonlinear transformation and memory capacity. To overcome this trade-off, a new architecture named hierarchical echo state network with augmented random features (HESN-ARF) is proposed. On the basis of traditional linear random projection, the proposed HESN-ARF further leverages nonlinear kernel transformation to construct augmented random features, which can enable the linear and nonlinear properties to be fully represented. Moreover, the HESN-ARF utilizes low-rank kernel approximation to further reduce the computational cost, preserving the advantage of efficient modeling as much as possible while ensuring the capacities of nonlinear transformation and dynamical memory simultaneously. The proposed HESN-ARF can mine and learn the latent evolution patterns hidden in the dynamic system layer by layer through the hierarchical strategy, and achieves excellent performance in multi-step-ahead chaotic time series prediction, as demonstrated by experimental findings on two synthetic chaotic systems and a real-world meteorological dataset.

Activation Function Modulation in Generative Triangular Recurrent Neural Networks

Autonomous generation of time series is challenging because the network must capture short-term features while tracking long-term time dependencies. This paper introduces the modulation of the activation function slopes of the upper-lower triangular recurrent neural networks (ULTRNNs) for dynamic variation of memory through a secondary recurrent network with its own independent states. A zigzag propagation algorithm for weight updates is proposed that accounts for the dynamic interaction of the states between the ULTRNN and the secondary network. A novel training method is proposed that distributes the eigenvalues of the closed-loop system around the unit circle in the complex z-plane to ensure that the network behaves as a nonlinear oscillator with an output that neither collapses nor saturates but continues to emulate the target. Examples encompassing the Lorenz series, Santa Fe laser data, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">kolam</i> patterns, electrocardiogram (ECG) signals, stock pricing data, and smart grid data are presented to demonstrate that the proposed approach is highly effective in the generative modeling of complex periodic, chaotic, and nonstationary time series. The qualitative and quantitative performance of the ULTRNN obtained with the proposed activation-function modulation technique is comparable to that of state-of-the-art techniques including feedforward networks and generative adversarial networks, but with far fewer trainable parameters and shorter computation times.

Maximum Information Exploitation Using Broad Learning System for Large-Scale Chaotic Time-Series Prediction.

How to make full use of the evolution information of chaotic systems for time-series prediction is a difficult issue in dynamical system modeling. In this article, we propose a maximum information exploitation broad learning system (MIE-BLS) for extreme information utilization of large-scale chaotic time-series modeling. An improved leaky integrator dynamical reservoir is introduced in order to capture the linear information of chaotic systems effectively. It can not only capture the information of the current state but also achieve the compromise with historical states in the dynamical system. Furthermore, the feature is mapped to the enhancement layer by nonlinear random mapping to exploit nonlinear information. The cascading mechanism promotes the information propagation and achieves feature reactivation in dynamical modeling. Discussions about maximum information exploration and the comparisons with ResNet, DenseNet, and HighwayNet are presented in this article. Simulation results on four large-scale data sets illustrate that MIE-BLS could achieve better performance of information exploration in large-scale dynamical system modeling.

Selection of Embedding Dimension and Delay Time in Phase Space Reconstruction via Symbolic Dynamics.

The modeling and prediction of chaotic time series require proper reconstruction of the state space from the available data in order to successfully estimate invariant properties of the embedded attractor. Thus, one must choose appropriate time delay and embedding dimension p for phase space reconstruction. The value of can be estimated from the Mutual Information, but this method is rather cumbersome computationally. Additionally, some researchers have recommended that should be chosen to be dependent on the embedding dimension p by means of an appropriate value for the time delay , which is the optimal time delay for independence of the time series. The C-C method, based on Correlation Integral, is a method simpler than Mutual Information and has been proposed to select optimally and . In this paper, we suggest a simple method for estimating and based on symbolic analysis and symbolic entropy. As in the C-C method, is estimated as the first local optimal time delay and as the time delay for independence of the time series. The method is applied to several chaotic time series that are the base of comparison for several techniques. The numerical simulations for these systems verify that the proposed symbolic-based method is useful for practitioners and, according to the studied models, has a better performance than the C-C method for the choice of the time delay and embedding dimension. In addition, the method is applied to EEG data in order to study and compare some dynamic characteristics of brain activity under epileptic episodes

Air Combat Maneuver Trajectory Prediction Model of Target Based on Chaotic Theory and IGA-VNN

Target maneuver trajectory prediction is an important prerequisite for air combat situation awareness and threat assessment. Aiming at the problem of low prediction accuracy in traditional trajectory prediction methods, combined with the chaotic characteristics of the target maneuver trajectory time series, a target maneuver trajectory prediction model based on chaotic theory and improved genetic algorithm-Volterra neural network (IGA-VNN) model is proposed, mathematically deducing and analyzing the consistency between Volterra functional model and back propagation (BP) neural network in structure. Firstly, the C-C method is used to reconstruct the phase space of the target trajectory time series, and the maximum Lyapunov exponent of the time series of the target maneuver trajectory is calculated. It is proved that the time series of the target maneuver trajectory has chaotic characteristics, so the chaotic method can be used to predict the target trajectory time series. Then, the practicable Volterra functional model and BP neural network are combined together, learning the advantages of both and overcoming the difficulty in obtaining the high-order kernel function of the Volterra functional model. At the same time, an adaptive crossover mutation operator and a combination mutation operator based on the difference degree of gene segments are proposed to improve the traditional genetic algorithm; the improved genetic algorithm is used to optimize BP neural network, and the optimal initial weights and thresholds are obtained. Finally, the IGA-VNN model of chaotic time series is applied to the prediction of target maneuver trajectory time series, and the experimental results show that its estimated performance is obviously superior to other prediction algorithms.

Chaotic time series prediction using wavelet transform and multi-model hybrid method

In order to further improve the prediction accuracy of the chaotic time series and overcome the defects of the single model, a multi-model hybrid model of chaotic time series is proposed. First, the Discrete Wavelet Transform (DWT) is used to decompose the data and obtain the approximate coefficients (low-frequency sequence) and detailed coefficients (high-frequency sequence) of the sequence. Secondly, phase space reconstruction is performed on the decomposed data. Thirdly, the chaotic characteristics of each sequence are judged by correlation integral and Kolmogorov entropy. Fourthly, in order to explore the deeper features of the time series and improve the prediction accuracy, a sequence of Volterra adaptive prediction models is established for the components with chaotic characteristics according to the different characteristics of each component. For the components without chaotic characteristics, a JGPC prediction model without chaotic feature sequences is established. Finally, the multi-model fusion prediction of the above multiple sequences is carried out by the LSTM algorithm, and the final prediction result is obtained through calculation, which further improves the prediction accuracy. Experiments show that the multi-model hybrid method of Volterra-JGPC-LSTM is more accurate than other comparable models in predicting chaotic time series.

Selective ensemble of multiple local model learning for nonlinear and nonstationary systems

This paper proposes a selective ensemble of multiple local model learning for modeling and identification of nonlinear and nonstationary systems, in which the set of local linear models are self adapted to capture the newly emerging process characteristics and the prediction of the process output is also self adapted based on an optimally selected ensemble of subset linear local models. Specifically, our selective ensemble of multiple local model learning approach performs the model adaptation at two levels. At the level of local model adaptation, a newly emerging process state in the incoming data is automatically identified and a new local linear model is fitted to this newly emerged process state. At the level of online prediction, a subset of candidate local linear models are optimally selected and the prediction of the process output is computed as an optimal linear combiner of the selected subset local linear models. Two case studies involving chaotic time series prediction and modeling of a real-world industrial microwave heating process are used to demonstrate the effectiveness of our proposed approach, in comparison with other existing methods for modeling and identification of nonlinear and time-varying systems.

Soft Measurement Modeling Based on Chaos Theory for Biochemical Oxygen Demand (BOD)

The precision of soft measurement for biochemical oxygen demand (BOD) is always restricted due to various factors in the wastewater treatment plant (WWTP). To solve this problem, a new soft measurement modeling method based on chaos theory is proposed and is applied to BOD measurement in this paper. Phase space reconstruction (PSR) based on Takens embedding theorem is used to extract more information from the limited datasets of the chaotic system. The WWTP is first testified as a chaotic system by the correlation dimension (D), the largest Lyapunov exponents (λ1), the Kolmogorov entropy (K) of the BOD and other water quality parameters time series. Multivariate chaotic time series modeling method with principal component analysis (PCA) and artificial neural network (ANN) is then adopted to estimate the value of the effluent BOD. Simulation results show that the proposed approach has higher accuracy and better prediction ability than the corresponding modeling approaches not based on chaos theory.

NARX prediction of some rare chaotic flows: Recurrent fuzzy functions approach

The nonlinear and dynamic accommodating capability of time domain models makes them a useful representation of chaotic time series for analysis, modeling and prediction. This paper is devoted to the modeling and prediction of chaotic time series with hidden attractors using a nonlinear autoregressive model with exogenous inputs (NARX) based on a novel recurrent fuzzy functions (RFFs) approach. Case studies of recently introduced chaotic systems with hidden attractors plus classical chaotic systems demonstrate that the proposed modeling methodology exhibits better prediction performance from different viewpoints (short term and long term) compared to some other existing methods.

A chaotic time series prediction model for speech signal encoding based on genetic programming

In this paper, a novel solving method for speech signal chaotic time series prediction model was proposed. A phase space was reconstructed based on speech signal's chaotic characteristics and the genetic programming (GP) algorithm was introduced for solving the speech chaotic time series prediction models on the phase space with the embedding dimension m and time delay τ. And then, the speech signal's chaotic time series models were built. By standardized processing of these models and optimizing parameters, a speech signal's coding model of chaotic time series with certain generalization ability was obtained. At last, the experimental results showed that the proposed method can get the speech signal chaotic time series prediction models much more effectively, and had a better coding accuracy than linear predictive coding (LPC) algorithms and neural network model.

Solution Modeling Using Postfix Genetic Programming

This article introduces Postfix Genetic Programming (GP), a postfix notation-based GP, approach to symbolic regression for solving empirical modeling problems. The main features of Postfix-GP are presented. These features include (1) postfix-based, variable-length individual-representation and its stack-based evaluation and (2) subtree crossover operator and semantic-aware subtree crossover operator. The article also presents two different constant creation approaches to evolve useful numeric constants for symbolic regression problems. The first approach uses an explicit list of constants and the second allows the Postfix-GP algorithm to evolve constants. Use of Postfix-GP as a solution modeling tool is demonstrated by solving two function identification problems and two deterministic chaotic time series modeling problems. Effectiveness of the evolved models is tested by statistically analyzing their performance on training and out-of-sample datasets of the problems. Experimental results on test problems suggest that Postfix-GP offers a new possibility for solving empirical modeling problems. We also compared the performance of Postfix-GP with the lil-GP system for all four problems. The results suggest that the quality of solutions found by Postfix-GP was better than that found by lil-GP, for the tested problems.

The Combinatorial Prediction about Chaotic Times Series of Natural Circulation Flow under Rolling Motion

The paper has established a combinatorial prediction model of chaotic time series based on history data and coupling data. Through the study of the flow characteristic about natural circulation under rolling motion, the single variable reconstruction and coupling multivariate reconstruction are discussed for chaotic time series based on phase space reconstruction technique, and the combinatorial prediction model has been built which bases on developing trend of history data and coupling relationship of correlative data. The paper also studied an example of coolant volume flow prediction with a relative precision of 0.9804 with the established model. The result indicated that the model with high precision and robustness could apply for natural circulation coolant volume flow prediction under rolling motion.

A variable projection approach for efficient estimation of RBF-ARX model.

The radial basis function network-based autoregressive with exogenous inputs (RBF-ARX) models have much more linear parameters than nonlinear parameters. Taking advantage of this special structure, a variable projection algorithm is proposed to estimate the model parameters more efficiently by eliminating the linear parameters through the orthogonal projection. The proposed method not only substantially reduces the dimension of parameter space of RBF-ARX model but also results in a better-conditioned problem. In this paper, both the full Jacobian matrix of Golub and Pereyra and the Kaufman's simplification are used to test the performance of the algorithm. An example of chaotic time series modeling is presented for the numerical comparison. It clearly demonstrates that the proposed approach is computationally more efficient than the previous structured nonlinear parameter optimization method and the conventional Levenberg-Marquardt algorithm without the parameters separated. Finally, the proposed method is also applied to a simulated nonlinear single-input single-output process, a time-varying nonlinear process and a real multiinput multioutput nonlinear industrial process to illustrate its usefulness.

A Chaotic Time Series Forecasting Model Based on Parameters Simultaneous Optimization Algorithm

Phase space reconstruction and parameters optimization were two key steps in the chaotic time series forecasting model, and tradition models could not get the optimal forecasting results without considering the relationship between the two steps. In order to improve forecasting accuracy of the chaotic time series, a chaotic time series forecasting model based on parameters simultaneous optimization was proposed. Firstly, the parameters were analyzed and designed based on uniform design, and then parameters were optimized by self-calling least square support vector machine, finally, the optimal forecasting model of chaotic time series is established, and the simulation experiment was carried out on by two chaotic time series. The simulation results show that the proposed model had improved the forecasting accuracy and computational complexity is much lower compared with the traditional models. Parameters simultaneous optimization algorithm has provided a new way to solve the parameters optimization problem of chaotic time series forecasting model.

Chaotic time series prediction of E-nose sensor drift in embedded phase space

Chemical sensor drift shows a chaotic behavior and unpredictability in long-term observation which makes it difficult to construct an appropriate sensor drift treatment. The main purpose of this paper is to study a new methodology for chaotic time series modeling of chemical sensor observations in embedded phase space. This method realizes a long-term prediction of sensor baseline and drift based on phase space reconstruction (PSR) and radial basis function (RBF) neural network. PSR can memory all of the properties of a chaotic attractor and clearly show the motion trace of a time series, thus PSR makes the long-term drift prediction using RBF neural network possible. Experimental observation data of three metal oxide semiconductor sensors in a year demonstrate the obvious chaotic behavior through the Lyapunov exponents. Results demonstrate that the proposed model can make long-term and accurate prediction of chemical sensor baseline and drift time series.

Developing a local least-squares support vector machines-based neuro-fuzzy model for nonlinear and chaotic time series prediction.

Local modeling approaches, owing to their ability to model different operating regimes of nonlinear systems and processes by independent local models, seem appealing for modeling, identification, and prediction applications. In this paper, we propose a local neuro-fuzzy (LNF) approach based on the least-squares support vector machines (LSSVMs). The proposed LNF approach employs LSSVMs, which are powerful in modeling and predicting time series, as local models and uses hierarchical binary tree (HBT) learning algorithm for fast and efficient estimation of its parameters. The HBT algorithm heuristically partitions the input space into smaller subdomains by axis-orthogonal splits. In each partitioning, the validity functions automatically form a unity partition and therefore normalization side effects, e.g., reactivation, are prevented. Integration of LSSVMs into the LNF network as local models, along with the HBT learning algorithm, yield a high-performance approach for modeling and prediction of complex nonlinear time series. The proposed approach is applied to modeling and predictions of different nonlinear and chaotic real-world and hand-designed systems and time series. Analysis of the prediction results and comparisons with recent and old studies demonstrate the promising performance of the proposed LNF approach with the HBT learning algorithm for modeling and prediction of nonlinear and chaotic systems and time series.

Adaptive Chaotic Prediction Algorithm of RBF Neural Network Filtering Model Based on Phase Space Reconstruction

With the analysis of the technology of phase space reconstruction, a modeling and forecasting technique based on the Radial Basis Function (RBF) neural network for chaotic time series is presented in this paper. The predictive model of chaotic time series is established with the adaptive RBF neural networks and the steps of the chaotic learning algorithm with adaptive RBF neural networks are expressed. The network system can enhance the stabilization and associative memory of chaotic dynamics and generalization ability of predictive model even by imperfect and variation inputs during the learning and prediction process by selecting the suitable nonlinear feedback term. The dynamics of network become chaotic one in the weight space. Simulation experiments of chaotic time series produced by Lorenz equation are proceeded by a RBF neural network .The experimental and simulating results indicated that the forecast method of the adaptive RBF neutral network compared with the forecast method of back propagation (BP) neutral network based on the chaotic learning algorithm has faster learning capacity and higher accuracy of forecast．The method provides a new way for the chaotic time series prediction.

A hybrid prediction model of multivariate chaotic time series based on error correction

Considering the problem that simply modifying the reservoir algorithm cannot significantly improve the prediction accuracy of chaotic multivariate time series, in this paper we propose a hybrid prediction model based on error correction. The observed data includes both linear and nonlinear features. First, we use autoregressive and moving average model to capture the linear features, then build a regularized echo state network to portray the dynamic nonlinear features. Finally, we add the predicted nonlinear value to the predicted linear value, in order to improve forecasting accuracy achieved by either of the models used separately. The experimental results of Lorenz and Sunspot-Runoff in the Yellow River time series demonstrate the effectiveness and characteristics of the proposed model herein.