Sunspot Time Series Research Articles

An encoder–decoder architecture with Fourier attention for chaotic time series multi-step prediction

Multi-step prediction of chaotic time series has been a challenging problem. An encoder–decoder architecture based on novel Fourier attention was proposed, called FGNet, applied to chaotic time series multi-step prediction. Two perspectives were carefully re-considered in modeling, one was the integrated use of frequency and temporal domain information, the other was the fusion of channel information and series information. The encoder was a self-attention architecture with Fourier attention module for capturing dynamic information in time series. Correspondingly, the decoder used the Gated Recurrent Unit (GRU) module with identity mapping, and the decoder was used to model the representation based on temporal dependency. For the interpretability deficiency of Fourier attention in FNet, which was proposed by google, a novel Fourier attention module was proposed. Specifically, the frequency domain features were extracted using Fourier transform, series and channel features were integrated capture through channel swapping coupling. Then these features were fused in the frequency domain space to enhance feature representation and their interpretability is achieved through inverse Fourier transform. Simulated data (Mackey–Glass system and Lorenz) and real data (sunspot time series) were used to verify that FGNet proposed achieves pleasantly surprising multi-step prediction performance in practice.

Revealing Coupled Periodicities in Sunspot Time Series Using Bispectrum—An Inverse Problem

Sunspot daily time series have been available for almost two centuries providing vast and complicated information about the behavior of our star and especially the interaction of the motion of the planets and other possible interstellar phenomena and their effects on the surface of the Sun. The main result obtained from the sunspot time series analysis is the imprint of various periodicities, such as the planets’ orbital periods and the planetary synodic periods on the sunspots signature. A detailed spectrum representation is achieved by means of a periodogram and a virtual extension of the time length segments with zeroed samples for longer representations. Furthermore, the dependence or coupling of these periodicities is explored by means of a bispectrum. We establish the exact interdependencies of the periodic phenomena on the sunspot time series. Specific couplings are explored that are proved to be the key issues for the coupled periodicities on the sunspot time series. In this work, contrary to what has been presented in the literature, all periodic phenomena are limited within the time period of an 11-year cycle as well as the periodicities of the orbits of the planets. The main findings are the observed strong coupling of the Mercury, Venus, and Mars periodicities, as well as synodic periodicities with all other periodicities that appear on the sunspot series. Simultaneously, the rotation of the Sun around itself (25.6 to 33.5 days) provides an extensive coupling of all recorded periodicities. Finally, there is strong evidence of the existence of a quadratic mechanism, which couples all the recorded periodicities, but in such a way that only frequency pairs that sum up to specific periods are coupled. The justification for this kind of coupling is left open to the scientific community.

Data-Driven Forecasting of Sunspot Cycles: Pros and Cons of a Hybrid Approach

Understanding the number of sunspots is crucial for comprehending the Sun’s magnetic-activity cycle and its influence on space weather and the Earth. Recent advancements in machine learning have significantly improved the accuracy of time-series predictions, revealing a compelling approach for sunspot forecasts. Our work takes the pioneering work by proposing a hybrid forecasting approach that combines the Seasonal Autoregressive Integrated Moving Average (SARIMA) with machine-learning algorithms like Random Forest and Support Vector Machine, delivering high prediction accuracy. Despite its high accuracy, we highlight the need for caution in deploying machine-learning-based methods for sunspot-number prediction, demonstrated through a detailed case study with only three extra time stamps leading to a dramatic change. More specifically, when making a forecast of monthly averaged sunspot numbers from 2023–2043 based on data from 1749–2023, we found that the observations in June, July, and August 2023 have a significant impact on the forecast, particularly in the long term. Given the multiseasonal and nonstationary nature of the sunspot time series, we conclude that this kind of phenomenon cannot be simply captured by a pure data-driven model, which can be highly sensitive in the forecast in the long term, and requires a more comprehensive approach, possibly with a model that includes physics.

Hemispheric Prediction of Solar Cycles 25 and 26 From Multivariate Sunspot Time-series Data via Informer Models

Hemispheric Prediction of Solar Cycles 25 and 26 From Multivariate Sunspot Time-series Data via Informer Models

Stacked 1D Convolutional LSTM (sConvLSTM1D) Model for Effective Prediction of Sunspot Time Series

Stacked 1D Convolutional LSTM (sConvLSTM1D) Model for Effective Prediction of Sunspot Time Series

Multiple Nyström Kernel Adaptive Filter Under Minimum Generalized Cauchy Loss Criterion

The multikernel adaptive filters (MKAFs) have been successfully applied to resolve the issue of kernel parameter selection in traditional single kernel adaptive filters. However, owing to the linear growing network structures, conventional MKAFs commonly suffer a lot from large computational and memory burdens. To solve this problem, a multiple Nyström approximation is proposed to curb the computational complexity of MKAFs in this brief. More concretely, the multiple Nyström method is incorporated into the kernel generalized Cauchy conjugate gradient algorithm, generating a novel multiple Nyström kernel generalized Cauchy conjugate gradient algorithm (MNKGCCG). It is noted that the MNKGCCG can achieve the desirable filtering performance with low computational cost in the fixed-dimensional feature space. Experimental results on Mackey-Glass time series and sunspots time series predictions in non-Gaussian noise environments demonstrate the superiorities of the proposed MNKGCCG algorithm in terms of filtering accuracy and robustness.

Predicting the Functional Dependence of the Sunspot Number in the Solar Activity Cycle Based on Elman Artificial Neural Network

The possibility of predicting the function of the time dependence of the sunspot number (SSN) inthe solar activity cycle is analyzed based on the application of the Elman artificial neural network platform tothe historical series of observational data. A method for normalizing the initial data for preliminary trainingof the ANN algorithm is proposed, in which a sequence of virtual idealized cycles is constructed using scaledduration coefficients and the amplitude of solar cycles. The correctness of the method is analyzed in a numericalexperiment based on modeling the time series of sunspots. The intervals of changing the adaptableparameters in the ANN operation are estimated and a mathematical criterion for choosing a solution is proposed.The significant asymmetry of its ascending and descending branches is a characteristic property of theconstructed functional dependence of the sunspot number cycle. A forecast of the time course for the current25th cycle of solar activity is presented and its correctness is discussed in comparison with other forecastresults and the available data of solar activity status monitoring

Characterizing the Solar Activity Using the Visibility Graph Method

In this paper, the Sun and its behavior are studied by means of complex networks. The complex network was built using the Visibility Graph algorithm. This method maps time series into graphs in which every element of the time series is considered as a node and a visibility criterion is defined in order to connect them. Using this method, we construct complex networks for magnetic field and sunspots time series encompassing four solar cycles, and various measures such as degree, clustering coefficient, mean path length, betweenness centrality, eigenvector centrality and decay exponents were calculated. In order to study the system in several time scales, we perform both a global, where the network contains information on the four solar cycles, and a local analysis, involving moving windows. Some metrics correlate with solar activity, while others do not. Interestingly, those metric which seem to respond to varying levels of solar activity in the global analysis, also do in the moving windows analysis. Our results suggest that complex networks can provide a useful way to follow solar activity, and reveal new features on solar cycles.

Charting the solar cycle

Sunspot records reveal that whilst the Sun has an approximately 11 year cycle of activity, no two cycles are of the same duration. Since this activity is a direct driver of space weather at Earth, this presents an operational challenge to quantifying space weather risk. We recently showed that the Hilbert transform of the sunspot record can be used to map the variable cycle length onto a regular “clock” where each cycle has the same duration in Hilbert analytic phase. Extreme geomagnetic storms rarely occur within the quiet part of the cycle which is a fixed interval of analytic phase on the clock; there is a clear active-quiet switch-off and quiet-active switch-on of activity. Here we show how the times of the switch-on/off can be determined directly from the sunspot time-series, without requiring a Hilbert transform. We propose a method-charting-that can be used to combine observations, and reports of societal impacts, to improve our understanding of space weather risk.

Prediction of Fuzzy Sunspot Time Series By Using RBFANN

We are present in this paper a new modified method of prediction of Sunspot time series after Fuzzify the data by using Fuzzy C-means clustering method (FCM). Our method consist of forecasting as first phase and prediction as a second phase. The accuracy of the results that we obtained is measured by M.S.E and we got a good results.

Chaotic time series prediction using DTIGNet based on improved temporal-inception and GRU

To improve the prediction accuracy of chaotic time series, deep extraction of the system evolutionary patterns is a key problem in modeling. In this paper, we propose a deep learning model of automatic multi-scale feature extraction for chaotic time series prediction. A hybrid deep neural network named deep temporal-inception module and gated recurrent unit network (DTIGNet) is designed. The improved temporal-inception module stacks dilated causal convolution of different depth to increase the network adaptability to different scales and improve the network nonlinear characterization ability, and an optional 1 × 1 convolutional kernel as shared residual connection. The model is applied to the Mackey-Glass system, Rossler system, Lorenz system and sunspots time series to verify the applicability and effectiveness in chaotic time series prediction. The results show that the DTIGNet proposed has higher accuracy and better performance compared with other methods according to the 6 prediction evaluation metrics adopted.

Two Stage Prediction Model of Sunspots Monthly Value Based on CEEMDAN and Particle Swarm Optimization ELM

Sunspot number is the most basic parameter to describe the level of solar activity. The accurate prediction of sunspot number can reflect the electromagnetic disturbance level of the electromagnetic layer, ionosphere and the middle and high layers in the future in advance, so as to provide important reference information for navigation, positioning, communication and the prediction of orbital attenuation of LEO satellites. Aiming at the characteristics of sunspot time series such as non-stationary, chaotic and difficult to predict, this paper proposed a two-stage combined prediction model based on complete Ensemble Empirical mode decomposition with adaptive noise (CEEMDAN), particle swarm optimization (PSO) and extreme learning machine(ELM) network. In the first stage of prediction: firstly, the original sunspot monthly mean series is decomposed by CEEMDAN to reduce the non-linearity and non-stationary of the original series. Then, an ELM prediction model is established for the sub-sequences decomposed by CEEMDAN, and the input layer dimension, hidden layer dimension, input layer weight and hidden layer bias of ELM are optimized by PSO algorithm. Finally, the prediction results of the first stage are obtained by superimposing the prediction results of each sub-sequence. The second stage is the error self-correction stage. Firstly, the prediction error sequence of the first stage is obtained. Then, the CEEMDAN-PSO-ELM prediction model is used to self-correct the prediction error of the first stage. Finally, the prediction results of the first stage and the self-correction results of the second stage are superimposed to obtain the final prediction value of the monthly sunspot number.In this paper, CEEMDAN is used to reduce the non-linearity and non-stationary of the sunspot series, and PSO is used to determine the best parameters of ELM network, and the useful information in the prediction error is fully considered, which effectively improves the prediction accuracy of sunspot monthly mean series. The prediction experiment is carried out by using the measured sunspot monthly mean series, and the proposed model is compared with wavelet neural network (WNN), back propagation neural network (BPNN), ELM, CEEMDAN-ELM and CEEMDAN-PSO-ELM. The experimental results show that the prediction accuracy of the proposed two-stage prediction model has been significantly improved, and has better prediction stability.

Symbolic Regression Based Feature Extraction of Shallow Neural-Networks for Identification and Prediction

This paper proposes a feature extraction method to improve the performance of shallow neural-network models with less number of parameters to apply especially on embedded system design at remote applications. Feature extraction method is designed using fuzzy c-means clustering based fuzzy system design cascaded a layer of symbolic operators and functions, respectively. During the training stage of neural-networks, symbolic operators and functions are selected using random-learning theory with the unity internal weights such that based on the prediction performance, optimal sequences are recorded for feature extraction to be utilized on testing phase. Extracted features are here used to empower the single-layer neural-network (SLNN) with sigmoid hyperbolic activation functions, functional-link neural- network (FLNN) with Chebyshev polynomials and Pi-Sigma higher-order neural-network (PSNN) with sigmoid activation functions, respectively. The internal and output parameters of the appended shallow neural-networks are optimized using batch optimization methods. Proposed regression models are first tested on identification of an artificial discrete-time dynamic system and real-time inverted pendulum then also for prediction of the sunspot time-series and traffic density estimation. As a result, the prediction performance of shallow neural networks is improved to be used in future applications.

Does Machine Learning reconstruct missing sunspots and forecast a new solar minimum?

The retrodiction and prediction of solar activity are two closely-related problems in dynamo theory. We applied Machine Learning (ML) algorithms and analyses to the World Data Center’s newly constructed annual sunspot time series (1700–2019; Version 2.0). This provides a unique model that gives insights into the various patterns of the Sun’s magnetic dynamo that drives solar activity maxima and minima. We found that the variability in the ~ 11 -year Sunspot Cycle is closely connected with 120-year oscillatory magnetic activity variations. We also identified a previously under-reported 5.5 year periodicity in the sunspot record. This 5.5-year pattern is co-modulated by the 120-year oscillation and appears to influence the shape and energy/power content of individual 11-year cycles. Our ML algorithm was trained to recognize such underlying patterns and provides a convincing hindcast of the full sunspot record from 1700 to 2019. It also suggests the possibility of missing sunspots during Sunspot Cycles −1, 0, and 1 (ca. 1730s-1760s). In addition, our ML model forecasts a new phase of extended solar minima that began prior to Sunspot Cycle 24 (ca. 2008–2019) and will persist until Sunspot Cycle 27 (ca. 2050 or so). Our ML Bayesian model forecasts a peak annual sunspot number (SSN) of 95 with a probable range of 80–115 for Cycle 25 between 2023 and 2025.

Time Series Prediction Method Based on E-CRBM

To solve the problems of delayed prediction results and large prediction errors in one-dimensional time series prediction, a time series prediction method based on Error-Continuous Restricted Boltzmann Machines (E-CRBM) is proposed in this paper. This method constructs a deep conversion prediction framework, which is composed of two E-CRBMs and a neural network (NN). Firstly, the E-CRBM models of the original input sequence and the target prediction sequence are trained, respectively, to extract the time features of the two sequences. Then the NN model is used to connect and transform the time features. Secondly, the feature sequence H1 is extracted from the original input sequence of test data through E-CRBM1, which is used as input of NN to obtain feature transformation sequence H2. Finally, the target prediction sequence is obtained by reverse reconstruction of feature transformation sequence H2 through E-CRBM2. The E-CRBM in this paper introduces the residual sequence of NN feature transformation in the hidden layer of CRBM, which increases the robustness of CRBM and improves the overall prediction accuracy. The classical time series data (sunspot time series) and the actual operation data of reciprocating compressor are selected in the experiment. Compared with the traditional time series prediction method, the results verify the effectiveness of the proposed method in single-step prediction and multi-step prediction.

Forecasting of Sunspot Time Series Using a Hybridization of ARIMA, ETS and SVM Methods

Solar activity directly influences the heliospheric environment and lives on the Earth. Sunspot number (SN) is one of the most crucial and commonly predicted solar activity indices. The prediction of SN time series is a challenging problem owing to its non-stationary, non-Gaussian and nonlinear nature. Therefore, improving the forecasting accuracy of SN time series is an important and challenging task. Motivated from this, in this paper, we have proposed a hybridization of the autoregressive integrated moving average (ARIMA); exponential smoothing with error, trend and seasonality (ETS); and support vector machine (SVM) to predict monthly and yearly SN time series. In this method, first ARIMA, ETS and SVM with linear kernel function are applied to the SN time series and the maximum of forecasts are determined to obtain the forecasts on linear component. Then the residual series is obtained by subtracting the forecasts on linear component from SN time series. The residual series is considered as nonlinear and modeled using SVM with Gaussian kernel function. Then the forecasts on linear component are added with the forecasts on nonlinear component to obtain the final forecasts. To evaluate the efficiency of the proposed method, three constituent models, one of the most popular deep learning models long short-term memory (LSTM), four hybrid methods, four ensemble methods are considered. Furthermore, two horizons, monthly and yearly sunspot time series are considered to evaluate the robustness of the proposed method. Results indicate the statistical superiority of the proposed methods over different horizons considering different accuracy measures.

An online self-organizing modular neural network for nonlinear system modeling

Modular neural network (MNN) has distinct advantage in many fields such as pattern recognition and pattern recognition. However it is still a challenge to dynamically adjust the MNN structure for dynamic nonlinear system modeling. This paper proposes a novel online self-organizing MNN (OSOMNN) for nonlinear system modeling. In OSOMNN, an online task decomposition algorithm and a self-organizing algorithm for subnetwork are introduced. Firstly, the task decomposition algorithm is implemented by the online clustering method based on distance and local density, which can online divide the original task into several simpler subtasks. Then subnetworks with single-layer feedforward neural network are built to learn the divided subtasks. Moreover, this paper develops a self-organizing algorithm for subnetwork, which can dynamically adjust its structure and is trained by the improved online gradient method with fixed memory mechanism (FMOGM). To demonstrate the effectiveness of OSOMNN for nonlinear system modeling, experimental investigations using four benchmark nonlinear systems and the monthly sunspots time series show that OSOMNN can automatically add or merge the subnetwork modules and optimize the structure of subnetworks for nonlinear system modeling with a better generalization performance than the established alternatives.

EMD and LSTM Hybrid Deep Learning Model for Predicting Sunspot Number Time Series with a Cyclic Pattern

The prediction of a time series such as climate indices and the sunspot number (SSN) with long-term oscillatory behaviors has been a challenging task due to the complex combination of oscillations. Frequency extraction algorithms have been developed to separate a time series into different oscillation components according to frequency, such as empirical model decomposition (EMD) and wavelet analysis. In the current study, the deep learning long short-term memory (LSTM) model was employed to predict the oscillation components extracted using EMD. The SSN series was modeled with the hybrid EMD-LSTM model. The simulation study results indicate that the LSTM model reproduces the smooth cyclic pattern of the sine function, and only a few hidden units are needed to model it. The EMD-LSTM model achieves better performance than does the LSTM model for mid-range SSN predictions while the LSTM achieves better performance within the first few time lags. However, the cyclic prediction of the SSN requires mid-range lags; thus, the superior performance of the EMD-LSTM model for these lags cannot be ignored. Furthermore, the remaining components from the significant EMD signals can be modeled to reveal the variability (or uncertainty) in the prediction. The summed residual is fitted by k-nearest neighbor resampling. The final SSN prediction results show that the EMD-LSTM model predicts a later and larger SSN for Solar Cycle 25 than does the LSTM model. Overall, the results lead to the conclusion that the EMD-LSTM model might be a suitable alternative for modeling complex sunspot time series with cyclic patterns.

Hetero-Dimensional Multitask Neuroevolution for Chaotic Time Series Prediction

Chaotic time series prediction has important research and application value, and neural network-based prediction methods have problems such as low accuracy and difficulty in determining the number of nodes in the hidden layer. In recent years, as a representative of evolutionary multitask optimization algorithms(EMTO), multifactorial evolutionary algorithms (MFEA) have achieved excellent results in Multitask Optimization (MTO) problems, but it does not work well in solving hetero-dimensional multitask problem. We analyzed this phenomenon and proposed the concept of harmful transfer. Then to solve these problems, in this paper, we propose the hetero-dimensional multitask neuroevolution algorithms (HD-MFEA neuroevolution) for chaotic time series prediction. First, we propose hetero-dimensional assortative mating algorithm and self-adaption elite replacement algorithm, so that the hetero-dimensional multitask evolutionary algorithm (HD-MFEA) can overcome the problems caused by harmful transfer on hetero-dimensional multitask problem. Second, in order to adapt to the hierarchical characteristics of neural networks, we propose the neural level segmental coding strategy. Third, we consider the optimization of neural networks with different hidden layer nodes as hetero-dimensional multitask problem and use the HD-MFEA neuroevolution to solve this problem. We verify the effectiveness of the HD-MFEA on six hetero-dimensional multitask benchmark problems. The Mackey-Glass, Lorenz and Sunspot time series are used to demonstrate the performance of the HD-MFEA neuroevolution. Experimental results have showed that the HD-MFEA neuroevolution performs well in predicting the chaotic time series problems.

FARIMA MODELING OF SUNSPOT HALE CYCLES TIME SERIES

The influence of solar activity on climate system at regional or global scales cannot be neglected. In this study, we will stimulate full-length sunspot time series (1932–2014) and all four Hale cycles by fractional auto regressive integrated moving average (FARIMA ([Formula: see text], [Formula: see text], [Formula: see text])) model. By minimizing Akaike information criterion (AIC), Bayesian–Schwarz information criterion (BIC) and Hannan–Quinn information criterion (HIC), the best FARIMA model for full-length sunspot time series is [Formula: see text] where [Formula: see text] is the back shift operator and [Formula: see text] is a white noise.