Chaotic Time Series Research Articles

A modified artificial electric field algorithm and its application

Abstract As an efficient meta-heuristic technique, artificial electric field algorithm (AEFA) has been extensively applied to tackle various challenging tasks posed by practical scenarios. However, in the classical AEFA, the fitness function has a cumulative effect on the charge, resulting in limited search capability. To address this issue, a modified AEFA (MAEFA) is presented in this paper. More specifically, a novel charge calculation scheme is introduced to overcome the cumulative effect by gradually distinguishing the charges of particles during the evolutionary process. Further, an alternating search strategy is developed to calculate the total electrostatic force, thereby reinforcing the guiding effect of excellent individuals on the entire population. Subsequently, the performance of MAEFA is investigated using 42 well-benchmarked functions, two chaotic time series prediction problems, and two engineering design problems. Experimental results reveal that MAEFA is more competitive in comparison with several established AEFAs and 20 popular meta-heuristic techniques.

Prediction of optical chaos using a multi-stage extreme learning machine with data uncertainty

In this paper, we study the problem of predicting optical chaos for semiconductor lasers, where data uncertainty can severely degrade the performance of chaos prediction. We hereby propose a multi-stage extreme learning machine (MSELM) based approach for the continuous prediction of optical chaos, which handles data uncertainty effectively. Rather than relying on pilot signals for conventional reservoir learning, the proposed approach enables the use of predicted optical intensity as virtual training samples for the MSELM model learning, which leads to enhanced prediction performance and low overhead. To address the data uncertainty in virtual training, total least square (TLS) is employed for the update of the proposed MSELM’s parameters with simple updating rule and low complexity. Simulation results demonstrate that the proposed MSELM can execute the continuous optical chaos predictions effectively. The chaotic time series can be continuously predicted for a time period in excess of 4 ns with a normalized mean squared error (NMSE) lower than 0.012. It also demands much fewer training samples than state-of-the-art learning-based methods. In addition, the simulation results show that with the help of TLS, the length of prediction is improved significantly as the uncertainty is handled properly. Finally, we verify the prediction ability of the multi-stage ELM under various laser parameters, and make the median boxplot of the predicted results, which shows that the proposed MSELM continues to produce accurate and continuous predictions on time-varying optical chaos.

Grain storage temperature prediction based on chaos and enhanced RBF neural network

Grain storage has very strict temperature requirements. Aiming at the problems of nonlinear characteristics and poor prediction accuracy of temperature parameters in grain storage, a combination of chaos theory and enhanced radial basis neural network is proposed as a temperature prediction model for grain storage (C-ERBF). The model first determines the embedding dimension and time delay of the grain storage temperature sequence using chaos theory. It then calculates the Lyapunov exponent to confirm its chaotic properties and reconstructs the sequence in the phase space to extract the hidden dynamic information and structure behind the sequence. Furthermore, the q-Normalized Least Mean Square Fourth (qXE-NLMF) algorithm is designed to enhance the radial basis function (RBF) neural network model for weight updating, to improve its prediction accuracy, and to accelerate the training speed of the model. As verified by the simulation experiments of Mackey–Glass chaotic time series prediction, the enhanced RBF (ERBF) network has faster convergence speed and lower steady-state error compared to the traditional RBF network. Finally, the optimized dataset from chaos theory is input into the model to achieve accurate predictions of grain storage temperature series. The experimental results show that the proposed C-ERBF model has higher prediction accuracy compared to other time series prediction methods. It can realize the grain pile temperature in advance, and take control measures in advance. This proactive approach significantly reduces the consumption of stored grain and prevents issues before they arise.

A chaotic time series prediction model based on the improved dung beetle optimizer and echo state network

Echo state network (ESN) possesses advantages such as simple network structure, ease of training, and reliable prediction performance, making them widely applied in the field of time series prediction. Selecting the optimal reservoir parameters is a key issue in ESN research, as it determines the effectiveness of the network prediction, and it is crucial to design an efficient optimization method for parameter optimization. This paper introduces an improved version of the dung beetle optimizer (IDBO), which employs various strategies to enhance population initialization, algorithm optimization capability, and convergence speed. For theoretical function optimization problems, comparing IDBO with other commonly used optimization methods validated its effectiveness and feasibility. Subsequently, combining IDBO with ESN to construct a new model, IDBO-ESN, and conducting time series prediction experiments, the superior performance of this model is verified on two benchmark datasets and one real dataset.

Cleaning method for abnormal energy big data based on sparse self-coding

In order to reduce the interference of abnormal energy big data and improve the accuracy of anomaly cleaning and detection, this paper proposes a method for cleaning abnormal energy big database based on sparse self-coding. Firstly, the abnormal data detection method based on multi-criteria evaluation is used to analyze the spectral feature distribution of abnormal energy big data. By using chaotic time series reconstruction model, robust local weighted regression analysis and sparse self-coding method, the feature decomposition of time series of energy data is realized. Then, according to the periodicity of the original sequence, the original sequence is segmitated adaptively to represent the morphological characteristics of the abnormal energy data sub-sequences driven by dynamic carbon emissions, and the anomaly index of each sub-sequence is obtained by AFCM algorithm. Secondly, an energy anomaly evaluation model based on LOF value is established. Finally, the output decision function of data cleaning is constructed to realize the abnormal energy big data cleaning. The test results show that the error detection rate of this method is 0.24%, the missing detection rate is 0.27%, the cleaning rate can reach 99.49%, and the cleaning time is less than 2s.

High- accuracy chaotic time series prediction of the flexible beam-ring model based on P$$\_$$CNN-BiLSTM ED network

High- accuracy chaotic time series prediction of the flexible beam-ring model based on P$$\_$$CNN-BiLSTM ED network

Chaotic laser time series prediction based on an improved logistic mapping algorithm echo state network

Chaos synchronization plays vital functions in the fields of optical chaos secure communication. The synchronization performance can be significantly degraded by parameter mismatches between the chaotic transmitter and receiver. In this paper, the Deep-Logistical Mapping Echo State Network (D-LMESN) is proposed to enhance the performance of chaos synchronization. The network is upgraded by using an improved logical mapping algorithm and a deep reserve pool structure with phase space reconstruction. Results show that D-LMESN exhibits better performance in the prediction of chaotic time series, thanks to the adaptive parameter adjustment, which increases the ability to capture the dynamic characteristics of complex systems. Compared with ESN, the mean square error of this model is reduced by 55% and 72%, respectively, in chaotic laser simulation and actual data experiments. This provides a new possibility, to our knowledge, for the development of chaotic secure communication.

Quantifying the Uncertainty of Reservoir Computing: Confidence Intervals for Time-Series Forecasting

Recently, reservoir computing (RC) has emerged as one of the most effective algorithms to model and forecast volatile and chaotic time series. In this paper, we aim to contribute to the understanding of the uncertainty associated with the predictions made by RC models and to propose a methodology to generate RC prediction intervals. As an illustration, we analyze the error distribution for the RC model when predicting the price time series of several agri-commodities. Results show that the error distributions are best modeled using a Normal Inverse Gaussian (NIG). In fact, NIG outperforms the Gaussian distribution, as the latter tends to overestimate the width of the confidence intervals. Hence, we propose a methodology where, in the first step, the RC generates a forecast for the time series and, in the second step, the confidence intervals are generated by combining the prediction and the fitted NIG distribution of the RC forecasting errors. Thus, by providing confidence intervals rather than single-point estimates, our approach offers a more comprehensive understanding of forecast uncertainty, enabling better risk assessment and more informed decision-making in business planning based on forecasted prices.

Realizing submeter spatial resolution for Raman distributed fiber-optic sensing using a chaotic asymmetric paired-pulse correlation-enhanced scheme

The sensing spatial resolution and signal-to-noise ratio (SNR) of Raman distributed optical fiber sensors are limited by the pulse width and weak Raman scattering signals. Notably, the sensing spatial resolution cannot exceed the order of meters at several kilometers sensing distances. To break through this physical bottleneck, a novel, to our knowledge, Raman scattering model based on paired-pulse sensing is constructed. The fundamental origins of the observed limited spatial resolution of conventional schemes are analyzed, and a chaotic asymmetric paired-pulse correlation-enhanced scheme for Raman distributed fiber-optic sensing is proposed and experimentally demonstrated. The proposed scheme uses a chaotic asymmetric paired-pulse as the sensing signal and extracts the light intensity information of each data point of the sensing fiber, which carries the random undulation characteristics of chaotic time series, based on the time-domain differential reconstruction method. This scheme overcomes the pulse width limitation of spatial resolution via correlation and demodulation, enhances the correlation characteristics between the temperature-modulated Raman scattered light field and detection signal, and improves the SNR. Finally, a sensing performance of 10 km, a spatial resolution of 30 cm, and an SNR of 6.67 dB are realized in the experiment. This scheme provides a new research idea for a high-performance Raman distributed optical fiber sensing system.

The Time Series Classification of Discrete-Time Chaotic Systems Using Deep Learning Approaches

Discrete-time chaotic systems exhibit nonlinear and unpredictable dynamic behavior, making them very difficult to classify. They have dynamic properties such as the stability of equilibrium points, symmetric behaviors, and a transition to chaos. This study aims to classify the time series images of discrete-time chaotic systems by integrating deep learning methods and classification algorithms. The most important innovation of this study is the use of a unique dataset created using the time series of discrete-time chaotic systems. In this context, a large and unique dataset representing various dynamic behaviors was created for nine discrete-time chaotic systems using different initial conditions, control parameters, and iteration numbers. The dataset was based on existing chaotic system solutions in the literature, but the classification of the images representing the different dynamic structures of these systems was much more complex than ordinary image datasets due to their nonlinear and unpredictable nature. Although there are studies in the literature on the classification of continuous-time chaotic systems, no studies have been found on the classification of discrete-time chaotic systems. The obtained time series images were classified with deep learning models such as DenseNet121, VGG16, VGG19, InceptionV3, MobileNetV2, and Xception. In addition, these models were integrated with classification algorithms such as XGBOOST, k-NN, SVM, and RF, providing a methodological innovation. As the best result, a 95.76% accuracy rate was obtained with the DenseNet121 model and XGBOOST algorithm. This study takes the use of deep learning methods with the graphical representations of chaotic time series to an advanced level and provides a powerful tool for the classification of these systems. In this respect, classifying the dynamic structures of chaotic systems offers an important innovation in adapting deep learning models to complex datasets. The findings are thought to provide new perspectives for future research and further advance deep learning and chaotic system studies.

Multi-Step-Ahead Prediction of Chaotic Time Series: Self-Healing Algorithm for Restoring Values at Non-Predictable Points

Multi-Step-Ahead Prediction of Chaotic Time Series: Self-Healing Algorithm for Restoring Values at Non-Predictable Points

Discussion on optical solitons, sensitivity and qualitative analysis to a fractional model of ion sound and Langmuir waves with Atangana Baleanu derivatives

Abstract This research explores new soliton solutions to the Atangana–Baleanu derivative (ABD) fractional system of equations for ion sound and Langmuir waves (FISLW). We utilize the fractional ABD operator to convert our system into an ordinary differential equations. In recent years, machine learning (ML) evolves significantly in the context of data analysis and computing different solutions, which typically enables systems to operate wisely. Now, we are going to use numerous ML tools including matplotlib. pyplot as plt, scipy.integrate, mpl toolkits.mplot3d, and Axes3D to generate various types of optical solutions by using complete discriminant of the polynomial method. We will also analyze solutions for the hyperbolic function, trigonometric function, Jacobian elliptic function (JEF), and other solitary wave solutions. Solitons have extensive uses in pure and applied mathematics, including nonlinear partial differential equations: the Boussinesq equation, the nonlinear Schrödinger equation, and the sine-Gordon equation, Lie groups, Lie algebras, and differential and algebraic geometry. In addition, we study the chaotic behaviour, i.e., 2D, 3D, time series, Poincarè maps, and sensitivity analysis of our governing model. Sensitivity analysis explores how changes in a system’s variables affects its behaviour.

Graph-let based approach to evolutionary behaviors in chaotic time series

In the Graph-let based time series analysis, a time series is mapped into a series of graph-lets, representing the local states respectively. The bridges between successive graph-lets are reduced simply to a linkage with an information of occurrence. In the present work, we focus our attention on the bridge series, i.e., preserve the structures of the bridges and reduce the states into nodes. The bridge series can tell us how the system evolves. Technically, the ordinal partition algorithm is adopted to construct the graph-lets and the bridges. Results for the Logistic Map, the Hénon Map, and the Lorenz System show that the statistical properties for transition frequency network for the bridges, e.g., the number of visited bridges and the average out-entropy-degree, have the capability of characterizing chaotic processes, being equivalent with the Lyapunov exponent. What is more, the topological structure can display the details of the contributions of the transitions between the bridges to the statistical properties.

Analysis of human ambulation as a chaotic time-series: with nonlinear dynamics tools.

The aim of the present study is to investigate the complexity and stability of human ambulation and the implications on robotic prostheses control systems. Fourteen healthy individuals participate in two experiments, the first group run at three different speeds. The second group ascended and descended stairs of a five-level building block at a self-selected speed. All participants completed the experiment with seven inertial measurement units wrapped around the lower body segments and waist. The data were analyzed to determine the fractal dimension, spectral entropy, and the Lyapunov exponent (LyE). Two methods were used to calculate the long-term LyE, first LyE calculated using the full size of data sets. And the embedding dimensions were calculated using Average Mutual Information (AMI) and the False Nearest Neighbor (FNN) algorithm was used to find the time delay. Besides, a second approach was developed to find long-term LyE where the time delay was based on the average period of the gait cycle using adaptive event-based window. The average values of spectral entropy are 0.538 and 0.575 for stairs ambulation and running, respectively. The degree of uncertainty and complexity increases with the ambulation speed. The short term LyEs for tibia orientation have the minimum range of variation when it comes to stairs ascent and descent. Using two-way analysis of variance we demonstrated the effect of the ambulation speed and type of ambulation on spectral entropy. Moreover, it was shown that the fractal dimension only changed significantly with ambulation speed.

Strain-mediated reservoir computing with temporal and spatial co-multiplexing in multiferroic heterostructures

Physical reservoir computing (PRC), a brain-inspired computing method known for its efficient information processing and low training requirements, has attracted significant attention. The key factor lies in the number of computational nodes within the reservoir for its computational capability. Here, we explore co-multiplexing reservoirs that leverage both temporal and spatial strategies. Temporal multiplexing virtually expands the node count through the use of masking techniques, while spatial multiplexing utilizes multiple physical locations (e.g., Hall bars) to achieve an increase in the number of real nodes. Our experiment employs a strain-mediated reservoir based on multiferroic heterostructures. By applying a single voltage across the PMN-PT substrate (acting as global input) and measuring the output Hall voltages from four Hall bars (real nodes), we achieve significant efficiency gains. This co-multiplexing approach results in a reduction in the normalized root mean square error from 0.5 to 0.23 for a 20-step prediction task of a Mackey–Glass chaotic time series. Furthermore, the single input and four independent outputs lead to a fourfold reduction in energy consumption compared to the strain-mediated PRC with temporal multiplexing solely. This research paves the way for future energy saving PRC implementations utilizing co-multiplexing, promoting a resource-efficient paradigm in reservoir computing.

Sparse compressed deep echo state network with improved arithmetic optimization algorithm for chaotic time series prediction

A sparse compressed deep echo state network (SCDESN) incorporating sparse input units, compressed sampling and principal component analysis (PCA) units is proposed in this paper, and theoretically proved the sufficient and necessary conditions to ensure the echo state characteristics of the proposed scheme. In addition, a hybrid arithmetic optimization algorithm based on matrix design strategy and boundary selection strategy (SVD-HAOA) was proposed to optimize the hyper-parameters of the SCDESN model, taking into account the characteristics of the SCDESN model. The steps and process for optimizing the hyper parameters of the SCDESN model using SVD-HAOA were presented. Finally, the SCDESN model optimized by SVD-HAOA was experimentally and summarized on a classic benchmark dataset and two real-world chaotic time series. The outcomes of the simulation indicated that the suggested model surpassed alternative models in performance and enhanced computational efficiency while maintaining prediction accuracy. This model may serve as a viable alternative for real-world applications. Noteworthy, the source code of SCDESN-SVD-HAOA is publicly available at https://github.com/qiuqiuyaaa/SCDESN-SVD-HAOA.

A novel chaotic time series wind power point and interval prediction method based on data denoising strategy and improved coati optimization algorithm

Wind power prediction plays a pivotal role in increasing the power grid stability and mitigating market transaction risks. To enhance the prediction accuracy of wind power, this study presents a novel chaotic time series wind power point and interval prediction approach, focusing on data processing, as well as an improved coati optimization algorithm. First, the improved wavelet threshold denoising (IWTD) technique is constructed to eliminate the noise from the original wind power data. Then, the maximal information coefficient (MIC) is adopted to determine the optimal inputs of the prediction model. Subsequently, the improved coati optimization algorithm (ICOA) is developed to optimize the hyperparameters of the bidirectional long short-term memory (BiLSTM), deep belief network (DBN), and gated recurrent unit (GRU) models, along with the linear weight coefficients of the combined model, thereby enhancing the point prediction accuracy. Finally, kernel density estimation (KDE) is employed to obtain interval predictions for wind power with different confidence levels. Results show that the proposed prediction method effectively improves the prediction accuracy compared with other popular prediction models.

Chaotic time series prediction based on physics-informed neural operator

This paper investigates the prediction of chaotic time series using physics-informed neural operator (PINO) with different driven methods, such as data-driven method, physics-driven method, and hybrid data-physics-driven method. Here, the chaotic time series are generated from two classical time delayed chaotic systems, including Mackey–Glass equation (MG) and Optoelectronic Oscillator (OEO). The simulation results from these two chaotic systems or experimental results from Optoelectronic Oscillator demonstrate that when there is enough data with good quality, it is possible to train the model solely using data-driven method. Conversely, when possessing complete mastery of the physical prior knowledge, it is also available to train the model solely using physics-driven method, indicating there is no need to prepare label datasets as in data-driven method. However, when dataset quality is poor, for example, contaminated by noise, and precise physical prior knowledge is not fully acquired, the hybrid method will have a better performance.

Sparse reservoir computing with vertically coupled vortex spin-torque oscillators for time series prediction

Spin torque nano-oscillators possessing fast nonlinear dynamics and short-term memory functions are potentially able to achieve energy-efficient neuromorphic computing. In this study, we introduce an activation-state controllable spin neuron unit composed of vertically coupled vortex spin torque oscillators and a V–I source circuit is proposed and used to build an energy-efficient sparse reservoir computing (RC) system to solve nonlinear dynamic system prediction task. Based on micromagnetic and electronic circuit simulation, the Mackey–Glass chaotic time series and the real motor vibration signal series can be predicted by the RC system with merely 20 and 100 spin neuron units, respectively. Further study shows that the proposed sparse reservoir system could reduce energy consumption without significantly compromising performance, and a minimal response from inactivated neurons is crucial for maintaining the system’s performance. The accuracy and signal processing speed show the potential of the proposed sparse RC system for high-performance and low-energy neuromorphic computing.

Prediction of Multivariate Chaotic Time Series using GRU, LSTM and RNN

Prediction of Multivariate Chaotic Time Series using GRU, LSTM and RNN