Chaotic Time Series Data Research Articles

Empirical analysis of Deep Learning Models in forecasting Chaotic Time series data

Empirical analysis of Deep Learning Models in forecasting Chaotic Time series data

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.

Parallel multivariate deep learning models for time-series prediction: A comparative analysis in Asian stock markets

<span lang="EN-US">This study investigates deep learning models for financial data prediction and examines whether the architecture of a deep learning model and time-series data properties affect prediction accuracy. Comparing the performance of convolutional neural network (CNN), long short-term memory (LSTM), Stacked-LSTM, CNN-LSTM, and convolutional LSTM (ConvLSTM) when used as a prediction approach to a collection of financial time-series data is the main methodology of this study. In this instance, only those deep learning architectures that can predict multivariate time-series data sets in parallel are considered. This research uses the daily movements of 4 (four) Asian stock market indices from 1 January 2020 to 31 December 2020. Using data from the early phase of the spread of the Covid-19 pandemic that has created worldwide economic turmoil is intended to validate the performance of the analyzed deep learning models. Experiment results and analytical findings indicate that there is no superior deep learning model that consistently makes the most accurate predictions for all states' financial data. In addition, a single deep learning model tends to provide more accurate predictions for more stable time-series data, but the hybrid model is preferred for more chaotic time-series data.</span>

A method for quantifying the complexity of graph structures obtained from chaotic time series data

A method for quantifying the complexity of graph structures obtained from chaotic time series data

LP-Conjugation: Unraveling Time Series Dynamics through Chaotic Map Convolution

LP-Conjugation, introduced by Padua and Libao, is a versatile method used in solving the Inverse Frobenius-Perron problem (IFPP) for chaotic systems based on time series data. While their initial work focused on the logistic map and prime gap data, this article highlights the broader significance of LP-Conjugation as a powerful tool for the IFPP. Any chaotic map can be employed with established chaotic time series data to reconstruct unknown maps. LP-Conjugation utilizes a known chaotic map ?(?) and invariant distribution F(?) to construct an unknown chaotic map ?(?) with known invariant distribution H(?). The article presents a detailed proof, illustrating the method's robustness and adaptability. Further illustrations on how to construct a chaotic map from this logistic-based chaotic map sample is also provided. Real-world examples demonstrate LP-Conjugation's efficacy in diverse applications, solidifying its role as a valuable approach for solving the IFPP and understanding chaotic dynamics.

Urban Mobility and Knowledge Extraction from Chaotic Time Series Data: A Comparative Analysis for Uncovering COVID-19 Effects

The COVID-19 pandemic exerted devastating effects on the global economy, public health, and urban life. The geographical consequences lie in spatial time series dimensions, but the chaotic characteristics hinder intuitive understanding. Aiming to discover the variance of urban mobility in the pandemic context, this article presents a hybrid clustering technique called whole time sequence mixture (WTSM). The combination of whole time and subsequence techniques ensures robustness to data volume, dimensionality, sampling, distortion tasks, and a prior constraint. For yellow taxi trips in New York City, the case study spanned the pre-, mid-, and postpandemic periods and compared the performance of the established methods (symbolic aggregate approximation and dynamic time warping) and WTSM techniques. Findings revealed COVID-19 trends and social restriction-induced mobility variations and determined that vaccine supply did not lead to immediate mobility restoration. Meanwhile, concepts of validation indexes and computational complexity corroborate the superiority of WTSM. The transient cluster arising from temporal dissimilarity is a unique finding of WTSM, which led to high cluster cohesion and separation. WTSM can obtain new knowledge and rationale for urgent government intervention and epidemic management, not sacrificing computational efficiency and clustering quality. With the improved storage capability and desire for multidimensional pattern extraction, the increased accessibility to the chaotic time series data sets could promote further studies and help administrative schemes.

An entropy-based measure of correlation for time series

In this paper, an information-based measure of association between time series, called information-based correlation coefficient (ICC), is introduced to potentially overcome some of the problems related to Pearson's correlation coefficient and other commonly used coefficients. The ICC, besides its simple definition, seems to have other interesting advantages over traditional correlation coefficients, such as a more natural interpretation in terms of the reciprocal informativeness between two series, higher reliability for independent white noise, Student's t, and AR(1) processes, higher reliability in the presence of spurious correlation and outliers, and a well-developed mathematical theory based on entropy. The ICC can also detect a number of nonlinear relationships, although it may not be equitable and general. Moreover, the ICC can be computed also for ordinal data, and it offers higher reliability for independent ordinal-valued time series. Monte Carlo simulations, applications to real data, properties of the estimator and asymptotics under independence are also discussed in this paper. In particular, the paper includes applications to signal processing, chaotic time series, macroeconomic and weather data.

Noise Removal in Chaotic Time-series Data Considering the Motion Characteristics of Small Fishing Boats

Noise Removal in Chaotic Time-series Data Considering the Motion Characteristics of Small Fishing Boats

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

The prediction of chaotic time series can well describe the physical properties and influencing factors of a sequence of numerical data points in successive order and has played an increasingly crucial role in various scientific and engineering communities. These time series data require the prediction model to have strong dynamic feature representation ability, which often leads to excessive calculation burden. As an emerging and effective learning method, a broad learning system (BLS) can be trained quickly by incremental learning, and the system can be reconstructed without requiring a lengthy retraining process, which has been shown to facilitate superior time-saving performance. However, the historical information of time series data is often ignored, resulting in the lack of dynamic feature representation ability of the original BLS algorithm. In this article, a novel intergroup cascade BLS (ICBLS) method and its variant with optimized parameters, termed MOICBLS, are proposed, which can combine the previous knowledge with current information to determine the output results by constructing a novel intergroup cascade structure for the enhancement nodes. The new algorithms try to retain the time-saving advantage of BLS and greatly enhance the extraction ability of dynamic features of chaotic time series data. In addition, it cannot be ignored that the network structure parameters have an unquestionable impact on the performance of a BLS. In response, two conflicting goals of prediction accuracy and diversity are formulated as multiobjective optimization functions in the training phase to generate the optimized network structure parameters needed by multiple candidate models. The experimental results on two benchmark chaotic time series dataset, the Lorenz system and Rossler system, and a geomagnetic disturbance storm time chaotic time series dataset demonstrate that our proposed algorithms are effective for prediction of chaotic time series, and the prediction accuracy is better than some existing approaches.

Prediction Modelling Through Chaotic Approach on Ozone (O3) Pollutant Time Series in Shah Alam City

Studies on Ozone (O3) particles and their effect have become a significant concern. Exposure to high concentrations of O3 can cause an adverse reaction in the human respiratory system. Therefore, this study aims to predict the monthly O3 time series in the highly populated area of Shah Alam through a chaotic approach. The phase space plot and Cao methods successfully detected the chaotic behaviour O3 time series data. The three parameters are determined before the prediction process: time delay t, embedded dimension m, and nearest neighbour k. This study used t=1 and the value of m is calculated through the Cao method. The last parameter that needs to be determined is by graph plotting k versus correlation coefficient(cc). The combination of parameters t and m will be used for prediction, and the performance measure will be recorded. The prediction process is done using three non-linear methods, namely LMAM, LLAM and ILLAM. LMAM gives the best prediction value by using the combination of parameter t and the value of the performance measure is 0.8486. Trial-and-error method, m = 6, LMAM gives cc = 0.8977 with k = 17, LLAM gives cc = 0.8033 with k =1, and ILLAM gives cc = 0.8265 with k =24. Mean-top-bottom method, m = 9, LMAM gives cc = 0.8863 with k =10, LLAM gives cc = 0.8841 with k = 6 and ILLAM gives cc = 0.9370 with k = 22. This finding indicates that the O3 time series can be predicted using a chaotic approach and the improved method in determining the value can compute better prediction performance.

Associative reasoning-based interpretable continuous decision making in industrial production process

Adaptive decision making in continuous production process is challenging for the uncertain characteristics of dynamics system state, decision-making time, as well as the correlation between the previous decision and the subsequent decision. In this paper, a data-driven continuous decision-making model with self-adaption and interpretability is proposed. For the uncertainty of industrial data, a new form of information granule with variable parameters is designed to describe the knowledge extracted from data. For the continuity of the decision-making process, a data-driven interpretable associative reasoning model is established by considering the similarity of knowledge and the attention mechanism. Aiming at the dynamic feature in the decision-making process, a data-driven associative learning method is proposed to obtain the adaptivity of the proposed model, which generates and updates routes according to the co-activation and simultaneous activities of the information granules. To reason decision results, the proposed model searches and records the reasoning routes according to the reasoning results and the objective at the decision points. Noisy chaotic time series data and a number of classification datasets are employed for experimental analyses, and the results indicate that the proposed method exhibits high reasoning accuracy, anti-noise ability, and interpretability. In addition, the experimental results based on industrial data indicate that the decision-making model is capable of giving a reasonable, effective and interpretable decision scheme and suitable for establishing a credible decision-making system.

Retracted] Research on the Spread Path and Evolution Causes of Oral Language in the Digital Era

Visual orientation seems to indicate the decline of oral communication, but oral communication has its own living space under the new media ecology. Research has found that in the digital media era, voice communication is manifested as a single‐level feature that simulates current interaction and information communication. Although voice communication is a lie constructed by individuals, the interaction between the subject’s discourse and the actual field of interaction separate the emotional distance, but the situation is harmonious and inclusive. The following voice communication and new media technologies are still trustworthy. Aiming at multifactor evolutionary algorithm (MFEA), the most classical multifactor evolutionary algorithm in multitask computation, we theoretically analyze the inherent defects of MFEA in dealing with multitask optimization problems with different subfunction dimensions and propose an improved version of the multifactor evolutionary algorithm, called HD‐MFEA. In HD‐MFEA, we proposed heterodimensional selection crossover and adaptive elite replacement strategies, enabling HD‐MFEA to better carry out gene migration in the heterodimensional multitask environment. At the same time, we propose a benchmark test problem of multitask optimization with different dimensions, and HD‐MFEA is superior to MFEA and other improved algorithms in the test problem. Secondly, we extend the application scope of multitask evolutionary computation, and for the first time, the training problem of neural networks with different structures is equivalent to the multitask optimization problem with different dimensions. At the same time, according to the hierarchical characteristics of neural networks, a heterodimensional multifactor neural evolution algorithm HD‐MFEA neuro‐evolution is proposed to train multiple neural networks simultaneously. Through experiments on chaotic time series data sets, we find that HD‐MFEA neuro‐evolution algorithm is far superior to other evolutionary algorithms, and its convergence speed and accuracy are better than the gradient algorithm commonly used in neural network training.

Data driven forecasting of aperiodic motions of non-autonomous systems.

In the present effort, a data-driven modeling approach is undertaken to forecast aperiodic responses of non-autonomous systems. As a representative non-autonomous system, a harmonically forced Duffing oscillator is considered. Along with it, an experimental prototype of a Duffing oscillator is studied. Data corresponding to chaotic motions are obtained through simulations of forced oscillators with hardening and softening characteristics and experiments with a bistable oscillator. Portions of these datasets are used to train a neural machine and make response predictions and forecasts for motions on the corresponding attractors. The neural machine is constructed by using a deep recurrent neural network architecture. The experiments conducted with the different numerical and experimental chaotic time-series data confirm the effectiveness of the constructed neural network for the forecasting of non-autonomous system responses.

Classification of Chaotic Signals of the Recurrence Matrix Using a Convolutional Neural Network and Verification through the Lyapunov Exponent

This study classified chaotic time series data, including smooth and nonsmooth problems in a dynamic system, using a convolutional neural network (CNN) and verified it through the Lyapunov exponent. For this, the classical nonlinear differential equation by the Lorenz model was used to analyze a smooth dynamic system. The vibro-impact model was used for the nonsmooth dynamic system. Recurrence is a fundamental property of a dynamic system, and a recurrence plot is a representative method to visualize the recurrence characteristics of reconstructed phase space. Therefore, this study calculated the Lyapunov exponent by parametric analysis and visualized the corresponding recurrence matrix to show the dynamic characteristics as an image. In addition, the dynamic characteristics were classified using the proposed CNN model. The proposed CNN model determined chaos with an accuracy of more than 92%.

Data-driven inference modeling based on an on-line Wang-Mendel fuzzy approach

To address the modeling of continuous production process with dynamic and nonlinear characteristics, an on-line Wang-Mendel fuzzy inference model is proposed in this paper, which extracts the fuzzy rules from the raw data without prior knowledge. Considering the state of the industrial system changes dynamically online, typical data samples with support degrees are used to describe the sample distribution characteristics in each fuzzy region. With respect to online dynamic learning process, a self-evolutionary strategy with adaptive memory factors is proposed, and the fuzzy rule structures are updated gradually by a stochastic gradient descent method. For on-line updating, a loss function is constructed by considering the inference errors and the previous rule structure, in order to achieve continuous learning without forgetting the knowledge learned before. In the reasoning process, a sparse fuzzy reasoning approach is designed for extrapolating the knowledge of the fuzzy regions without sample data. To verify the effectiveness of the proposed method, chaotic time series data with noises and industrial practical data coming from a steel plant are employed for experimental analyses. The experimental results show that, the proposed method is capable of describing a variety of dynamic features and exhibiting high accuracy for the industrial data.

Chaotic Image Encryption Using Hopfield and Hindmarsh-Rose Neurons Implemented on FPGA.

Chaotic systems implemented by artificial neural networks are good candidates for data encryption. In this manner, this paper introduces the cryptographic application of the Hopfield and the Hindmarsh–Rose neurons. The contribution is focused on finding suitable coefficient values of the neurons to generate robust random binary sequences that can be used in image encryption. This task is performed by evaluating the bifurcation diagrams from which one chooses appropriate coefficient values of the mathematical models that produce high positive Lyapunov exponent and Kaplan–Yorke dimension values, which are computed using TISEAN. The randomness of both the Hopfield and the Hindmarsh–Rose neurons is evaluated from chaotic time series data by performing National Institute of Standard and Technology (NIST) tests. The implementation of both neurons is done using field-programmable gate arrays whose architectures are used to develop an encryption system for RGB images. The success of the encryption system is confirmed by performing correlation, histogram, variance, entropy, and Number of Pixel Change Rate (NPCR) tests.

Adaptation of dynamical properties of time series data and its applications in deep brain stimulation

The chaotic nature of the brain can be observed by electroencephalogram signals. This chaotic behavior can be affected by the progressive nature of neurodegenerative disorders like Parkinson disease. The gradual changes in dynamical behavior of brain can be tracked to facilitate effective and timely treatment. Deep brain stimulation (DBS) is used in therapy when the medication stops working. We investigate the use of chaotic signal as stimulus in DBS. We stimulate a simulated model of isolated neuron with different types of stimuli to see if periodicity in neuronal spiking can be disrupted and show that neuron, when stimulated with chaotic signal, does fire up in non-periodic/chaotic manner. Furthermore, as a step toward the development of our system for estimation of chaotic behavior of brain, we investigate the use of recurrent neural networks to adapt the chaotic characteristics of a chaotic time series in this research work. We explore two different setups of long short-term memory (LSTM). In first setup, we define three unique topologies of LSTM network and analyze those for chaotic parameter estimation in seven different test cases for shallow and deep networks. We show that the deep LSTM networks are capable of modeling the chaotic behavior of a wide range of parameters and that the network performs the best when the architecture is driven by chaotic attributes of the time series data. In second setup, we use LSTM network in a traditional configuration to predict the chaotic time series data and demonstrate that the LSTM network can make prediction over a range of chaotic parameters with adequate accuracy. This provides the basis for the generation of chaotic stimulation signals when required.

High-Performance Time Series Prediction With Predictive Error Compensated Wavelet Neural Networks

Machine learning (ML) algorithms have gained prominence in time series prediction problems. Depending on the nature of the time series data, it can be difficult to build an accurate ML model with the proper structure and hyperparameters. In this study, we propose a predictive error compensation wavelet neural network model (PEC-WNN) for improving the prediction accuracy of chaotic and stochastic time series data. In the proposed model, an additional network is used for the prediction of the main network error to compensate the overall prediction error. The main network takes as inputs the time series data through moving frames in multiple-scales. The same structure and hyperparameter sets are applied for quite distinct four types of problems for verification of the robustness and accuracy of the proposed model. Specifically, the Mackey-Glass, Box-Jenkins, and Lorenz Attractor benchmark problems, as well as drought forecasting are used to characterize the performance of the model for chaotic and stochastic data cases. The results show that the PEC-WNN provides significantly more accurate predictions for all compared benchmark problems with respect to conventional machine learning and time series prediction methods without changing any hyperparameter or the structure. In addition, the time and space complexity of the PEC-WNN model is less than all other compared ML methods, including long short-term memory (LSTM) and convolutional neural networks (CNNs).

Research of Mapping Element Space Method for Uncertain Lurie System Stability Diagnosis

A new diagnosis method of Lurie system stability by using Chaotic time series data was proposed in order to solve stability analysis of flight control system with fault and uncertain. Firstly, the reasons for the instability of the nonlinear system of Ruri leaf was analyzed by using the small gain theory and linear matrix inequalities in the robust control theory. And the stability conditions of Rui leaf system under the condition of uncertainty and failure were proved theoretically. In order to quantify the stability of nonlinear systems in complex Ruri leaves, based on the theory of phase space reconstruction, the nonlinear Ruri system with continuous discrete characteristics was converted into an approximate time discrete equation, mapping it to a low dimensional primitive space, by introducing Q Gauss function into Kernel function, the generalization ability of neural networks are enhanced, realizing the stability analysis method based on the characteristic of primitive migration, which is suitable for various fault and uncertainty conditions, and the relative quantized stability norm can be given. The simulation shows that the present method can effectively solve the stability analysis and determination of flight control system under various factors.

Characterizing time series by extended complexity-entropy curves based on Tsallis, Rényi, and power spectral entropy.

In this paper, we create three different entropy curves, Tsallis q-complexity-entropy curve, Rényi r-complexity-entropy curve, and Tsallis-Rényi entropy curve via extending the traditional complexity-entropy causality plane and replacing the permutation entropy into power spectral entropy. This kind of method is free of any parameters and some features that are obscure in the time domain can be extracted in the frequency domain. Results from numerical simulations verify that these three entropy curves can characterize time series efficiently. Chaotic and stochastic time series can be distinguished based on whether the q-complexity-entropy curves are opened or closed. The unrelated stochastic process has a negative curvature associated with the Rényi r-complexity-entropy curve, whereas there are positive curvatures for related cases. In addition, the Tsallis-Rényi entropy curve can display the relationship between two entropies. Finally, we apply this method to sleep electrocardiogram and electroencephalography signals. It is proved that these signals possess similar features with long-range correlated 1/f noise. It is robust enough to exhibit different characteristics for each sleep stage. By using surrogate data sets, the nonlinearity of simulated chaotic time series and sleep data can be identified.