A Hybrid Model Based on CEEMDAN-GRU and Error Compensation for Predicting Sunspot Numbers

Abstract

To improve the predictive accuracy of sunspot numbers, a hybrid model was built to forecast future sunspot numbers. In this paper, we present a prediction model based on complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN), gated recurrent unit (GRU), and error compensation for predicting sunspot numbers. CEEMAND is applied to decompose the original sunspot number data into several components, which are then used to train and test the GRU for the optimal parameters of the corresponding sub-models. Error compensation is utilized to solve the delay phenomenon between the original sunspot number and the predictive result. We compare our method with the informer, extreme gradient boosting combined with deep learning (XGboost-DL), and empirical mode decomposition combined long short-term memory neutral network and attention mechanism (EMD-LSTM-AM) methods, and evaluation metrics, such as RMSE and MAE, are used to measure their performance. Our method decreases more than 2.2813 and 3.5827 relative to RMSE and MAE, respectively. Thus, the experiment can demonstrate that our method has an obvious advantage compared to others.

1. Introduction

Sunspots refer to the dark spots, which often appear on the sun’s photosphere, and they reflect solar activity [1]. Their basic indication is the sunspot number, which is closely related to climate anomalies and the weather on earth [2,3]. When these clusters of sunspots reach their peak, they trigger solar flares and coronal mass ejections (CMEs), in which energetic particles released via solar activity flow toward the earth and interact with the earth’s magnetic field, which in turn leads to geomagnetic storms. As geomagnetic storms affect the earth’s ionosphere, radio signal propagation is impeded, which not only affects terrestrial radio communications, such as electronic equipment and radio and television transmissions, but can also lead to the disruption of satellite signals, affecting the navigation systems of aircraft and ships, as well as the normal operation of artificial satellites. In extreme cases, these phenomena may also affect the power networks, leading to disruptions in the supply of electricity. This has important implications for decision making and the reduction in accidents in electronic communications, precision manufacturing, and everyday life. Therefore, the accurate prediction of sunspot number is very important [4]. The sunspot number is non-linear and non-stationary, which makes it very difficult to predict, and it belongs to a typical time series [5,6]. Many scholars pay more attention in building different sunspot number prediction models to improving the prediction accuracies. There are two main class methods for predicting the sunspot number: one is based on mathematical statistics, and the other is based on machine learning.

In recent years, statistical models have commonly been applied to predict the sunspot number, such as exponential smoothing [7,8], Kalman filters [9], autoregressive moving average [10], etc. The autoregressive moving average (ARMA) is a well-known classical statistical method [11]. Kapoor and Wu [12] fitted 200 years of sunspot data using the ARMA model and predicted the yearly average (long-term) and daily average (short-term) sunspot numbers. Due to the insufficiency of the ARMA model, the difference operation was blended with ARMA—called the autoregressive integrated moving average model (ARIMA)—to process the complex time series. To remedy these shortcomings, Sun et al. proposed a combinatorial prediction framework, which uses a grid search to select appropriate sub-model hyperparameters [13]. To predict sunspot numbers, Chattopadhyay [14] used an autoregressive neural network, ARMA, and ARIMA. Khalid et al. [15] forecasted the sunspot number of Solar Cycle 24 using the Fourier series approach and compared it with the ARIMA method. Chen et al. [16] used a Bayesian approach for optimizing multiple hyperparameters of extreme gradient boosting trees to improve the prediction accuracy. Akhter et al. [17] predicted different solar activities, including the sunspot number, coronal index, etc., using the ARIMA(p,d,q) model. The data from 1944 to 2009 comprised the test dataset, and the values of p, d, and q were set to 1, 0, and 1. The result showed that ARIMA exhibited better performance in forecasting the sunspot number.

With the rapid development of machine learning, it is now applied to predict sunspot numbers—for example, linear regression (LR) [18], random forest (RF) [19], radial basis function (RBF) [20], the Bayesian method [21,22], etc. The support vector machine (SVM) was a very popular method in past time, and it was used to forecast the sunspot number [23]. Li Rong, Yanmei, and Cui et al. [24] merged the K-nearest neighbor and SVM, or KNN-SVM. According to their experiment, the KNN-SVM model has the potential to offer more precision than SVM. Four machine-learning techniques—LR, RF, RBF, and SVM—were employed by Dani, T. and Sulistiani, S. [25] to forecast the greatest amplitude of sunspot numbers in Solar Cycle 25. They discovered that the results of all four approaches indicated that Solar Cycle 25 would begin in late 2019 or early 2020. Regarding the images and data, deep learning exhibits exceptional capabilities [26]. The convolutional neural network (CNN) is the classical deep-learning method, and it has been applied to different fields, such as image processing [27], text understanding [28], time series forecasting [29], etc. Reza and Ruhila et al. [30] utilized the recurrent neural network (RNN) with the best optimized parameters to forecast the sunspot number and simulated the prediction process with MATLAB 7. CNN and RNN can extract the feature of the time series, but they cannot learn the dependency relationship between the data. Thus, the long short-term memory (LSTM) neural network, which was proposed by S. Hochreiter and J. Schmid Huber [31], was used to overcome insufficiencies [32]. Kumar Abhijeet and Kumar Vipin [33] presented a multi-layer deep-learning architecture for effective prediction by improving the one-dimensional convolutional LSTM network with a stacked network and then forecasted the four types of sunspot numbers with different frequencies, including yearly and monthly data and 13-month smoothing. They predicted that the peak values would appear in 2024. To decrease the complexity of LSTM and the number of super parameters and promote the performance of deep learning [34,35], the gated recurrent unit (GRU) not only ameliorates these problems but also eases the probability of the occurrence of a vanishing gradient [36,37]. Zhou and Zhang et al. [38] utilized the informer for long sequence time series. Their methods could precisely capture the dependency coupling relationship in a long-time range between the outputs and inputs, and the experimental results showed that the model had obvious advantages over existing methods.

To further promote the accuracy of sunspot number prediction with a single machine-learning model, some hybrid methods are proposed by many scholars [39]. For example, Lee and Taesam [40] used empirical mode decomposition (EMD) and LSTM to construct a new model in order to predict the sunspot number in Solar Cycle 25. Yang and Fu et al. [41] fused the attention mechanism with the EMD-LSTM model to capture abrupt points in sequence data, and their model had higher precision [42]. Nghiem and Le et al. [43] presented a CNN–Bayesian LSTM method to predict time series with respect to, e.g., weather and sunspot datasets. To measure the uncertainty of the predicted model, the Bayesian method was adopted. The result showed that their model was more effective than other models for uncertain quantification. The typical statistical ARIMA was also combined with deep-learning methods for predicting the sunspot number, such as ARIMA-ANN [44], ARIMA-LSTM [45], and βSARMA-LSTM [46]. Panigrahi and Pattanayak et al. [47] applied the hybridization of the ARIMA, exponential smoothing, and SVM to predict sunspot number time series. Dai and Liu et al. [48] adopted the phase space reconstruction method to transform the form of the sunspot dataset for adapting the neural network; then, the reconstructed data were input into a temporal convolutional neural (TCN) network. They predicted the sunspot number from January 2020 to December 2030. The GRU in Ref [49] was leveraged to predict the sunspot number. The MAPE value reached the smallest value at 7.17% when the dataset was divided into 70% training data and 30% testing data. In Ref [50], the DHNN model, which comprised CNN, GRU, and attention mechanisms, was used to predict different chaotic time series. Two datasets containing the theoretical Lorenz dataset and a coal-mine gas concentration dataset were used to verify the performance of their proposed model. Dang and Chen [51] compared the prediction effect between the non-deep-learning methods and deep-learning methods; then, they used hybrid methods on the XGBoost and deep-learning model, denoted as XGBoost-DL, to predict the sunspot number.

In this paper, we present a combined prediction algorithm based on CEEMDAN-GRU and error compensation. Because the sunspot number is a chaotic times series and may contain some noise, CEEMDAN can decompose it to several stable sub-sequences and simultaneously eliminate the noise in it. It is well known that GRU is very suitable for predicting long-time series with higher performance [52]; therefore, different sub-models based on GRU are adopted to predict different sub-sequences with the corresponding optimal parameters here. No matter what type of deep-learning neural network prediction model is used, there are always some delays in the prediction results compared with the true observed values [53,54]. Error compensation is added to mitigate this problem.

The main contributions of this paper are summarized as follows:

(1)

According to the chaotic and unstable features of the sunspot number and the delay phenomenon between predictive values and true values, different strategies or methods are adopted to overcome these problems. Therefore, we propose a prediction algorithm based on CEEMDAN-GRU and error compensation.

(2)

Because the sunspot number is decomposed into several components with different properties, such as frequencies and amplitudes, the sub-models are trained using different components to obtain their corresponding optimal parameters; then, they are used to predict further values to enhance the precision of the model.

(3)

We compare our method with a single deep-learning method, such as the LSTM, GRU, and informer, and some hybrid models, such as EMD-LSTM and XGBoost-DL. The experimental result demonstrates the advantages of our method.

(4)

More values are predicted using our method. We depict further predictive results obtained using different methods, including predictions based on machine learning (prediML), Kalman filtering, and machine learning (KFprediML), and the results of the Space Weather Prediction Center (SWPC). The trend of our results is similar to rediML and KFprediML, but it is different compared to the result of the SWPC.

The rest of the paper is organized as follows. Section 2 introduces our method. Section 3 presents the description of sunspot numbers, our experimental results, analyses, and comparisons with other methods. Section 4 provides the conclusions.

2. Method

2.1. Data Preprocessing and Reconstruction

In the early period, there are some special values, with zero and null in the sunspot number (SSN), due to observational equipment limitations and other factors. We process special values via the kriging interpolation method and then normalize the data:

$$\mathit{Y}=(\mathit{Y}-y_{\mathrm{m}\mathrm{i}\mathrm{n}})/(y_{\mathrm{m}\mathrm{a}\mathrm{x}}-y_{\mathrm{m}\mathrm{i}\mathrm{n}})$$

(1)

where y_min and y_max are the corresponding maximum and minimum of sequence $\mathit{Y}$.

2.2. The Principle of CEEMADN

Empirical mode decomposition (EMD) is an adaptive time–frequency decomposition method, which can decompose a linear signal or non-linear signal into a finite intrinsic model function (IMF), and each IMF presents a different time-scale feature. Aiming at the mode mix problem in EMD, CEEMDAN can solve it by adaptively adding white noise [49].

Suppose y(t) is the value of the original signal Y at the t moment; v^j(t) is the white noise, which satisfies the standard normal distribution; E_k(•) is the kth mode component via EMD decomposition; and IMF_k is the kth mode component via CEEMDAN decomposition. The detailed decomposition process of the CEEMDAN algorithm is as follows:

(1)

First, the Gaussian white noise v^j(t) is added to the original signal y(t). The time series y^j(t) can be obtained after adding Gaussian white noise for the jth time. Then, the first intrinsic mode component IMF₁(t) of CEEMDAN is obtained via the following equation:

$${IMF}_1\left(t\right)=\frac{1}{N}{\sum }_{j=1}^N{\mathrm{E}}_1\left(y^j\left(t\right)\right)=\frac{1}{N}{\sum }_{j=1}^N{\mathrm{E}}_1(y(t)+{\alpha }_0{v}^j(t))$$

(2)

where j is the number of times the Gaussian white noise is added, j = 1, 2, …, N; and α₀ is the noise coefficient.

(2)

The first-order residual r₁(t) is the difference between the source signal y(t) and the first-order mode component IMF₁(t).

$$r_1\left(t\right)=y\left(t\right)-{IMF}_1\left(t\right)$$

(3)

(3)

On the basis of the first-order residual r₁(t), white Gaussian noise v^j(t) is continued to be added to obtain a new signal $r_1^j(t)$. The $r_1^j(t)$ is also decomposed by EMD. The second IMF mode component IMF₂(t) obtained through EMD decomposition is

$${IMF}_2\left(t\right)=\frac{1}{N}{\sum }_{j=1}^N{E}_2\left(r_1^j\left(t\right)\right)$$

(4)

(4)

The second-order residual r₂(t) is the difference between the new source signal r₁(t) and the IMF₂(t).

$$r_2\left(t\right)=r_1\left(t\right)-{IMF}_2\left(t\right)$$

(5)

(5)

Steps (3)–(4) are repeated until the residual signal r_k(t) is monotone and cannot be decomposed. If the number of IMF components is K, the original signal can be re-calculated as follows:

$$y\left(t\right)={\sum }_{i=1}^K{IMF}_i\left(t\right)+r_K(t)$$

(6)

2.3. GRU

GRU, which was designed by Chung and Cho et al. in 2014 [50], is a type of RNN. It is similar to LSTM and is used for solving problems in neural networks, such as long-term memory, gradient explosion in the back propagation process, etc.

The structure of the GRU neural network is shown in Figure 1. The r(t) is the reset gate; z(t) is the update gate; y(t) is the input at the t moment; p(t) is the output at the t moment; and h(t − 1) is the hidden state from the previous moment. The principle of GRU is as follows:

(1)

Reset gate: the gating signal with a scope of [0, 1] can be output from the gate:

$$r\left(t\right)=\mathsf{\sigma }\left(W_{r}h\right(t-1)+W_{r}y(t\left)\right)$$

(7)

where W_y is a weight, and σ is a sigmoid function.

After the gating signal is computed, the scope of h(t) can be transferred to [−1,1] according to Formula (8):

$$h(t)=tanh[(r(t)\mathrm{☉}(W_h^{\prime }h(t-1))+W_y^{\prime }y(t))]$$

(8)

where $W_h^{\prime }$ and $W_y^{\prime }$ are the training weights of h(t − 1) and y(t), respectively. ☉ denotes Hadamard multiplication. tanh() is a non-linear activation function.

(2)

Update gate: the update gates are computed in the same form as reset gates. For each gate, the results of the weight matrix, which is multiplied with the input state or hidden state, are unique, meaning that the final vector is different:

$$u\left(t\right)=\sigma \left(W_{z}h\right(t-1)+W_{z}y(t\left)\right)$$

(9)

where W_z is a weight matrix of the update gate.

Then, h(t) can be updated using Formula (10):

$$h\left(t\right)=u\left(t\right)\mathrm{☉}h(t-1)+(1-u(t\left)\right)\mathrm{☉}h\left(t\right)$$

(10)

where u(t) denotes selective memorization, and 1 − u(t) denotes selective forgetting.

The advantage of GRU neural networks compared with LSTM networks is that they only perform selective memorization and selective forgetting at the same time, but LSTM requires several gates. The output structure of a GRU is the same as a standard RNN, but the inner flow is similar to LSTM. GRU can reach a comparable effect, although it requires fewer parameters and gates than LSTM.

2.4. Error Compensation Part

Since predicting the next value requires utilizing the current predicted value, error accumulation and drifting cannot be avoided. To reduce prediction errors, we use the dynamic error compensation strategy for the prediction result of the neural network. The error e(t) between the source data and prediction values can be computed as follows:

$$e\left(t\right)=y\left(t\right)-p\left(t\right)$$

(11)

where p(t) denotes the predictive values of the GRU, and t is the t moment.

Let e(t) be the input data for the GRU neural network; the prediction output is e’(t). The error compensation $\widehat{e}\left(t\right)$ can be worked out using Formula (12):

$$\widehat{e}\left(t\right)=\alpha \times e\left(t\right)+\left(1-\alpha \right)\times {\mathrm{e}}^{\prime }(t)$$

(12)

where $\mathit{\alpha }$ is a coefficient of adjustment, and its value is assigned 0.2 in our experiment.

The final predictive result $\widehat{y}\left(t\right)$ of the whole model is calculated using Formula (13):

$$\widehat{y}\left(t\right)=\widehat{e}\left(t\right)+p(t)$$

(13)

2.5. The Multi-Forecasting Model

The flowchart of our algorithm, which is based on CEEMDAN-GRU and error compensation for predicting sunspot numbers in this study, is shown in Figure 2. The detailed process of the prediction part is shown in Figure 3. Our algorithm first utilizes the CEEMDAN to decompose the SSN into several sub-sequences, including some IMFs and a residual. To improve the prediction accuracy, each sub-sequence—supposing the index is i (I = 1, …, K)—is reconstructed into a training set (TA_i) and a test set (TN_i). The corresponding prediction sub-model (called sub-model i) based on the GRU network is trained by TA_i for its optimal parameters. Then, sub-model i is validated by predicting its TN_i. To reduce the error between the prediction result and SSN, the error compensation method based on the GRU network is used. We use SSN and the e(t) as inputs and outputs to train the GRU network; then, the error e(t) is taken as the input of the trained GRU network to predict a new error $e^{\prime }\left(t\right)$. The final prediction result $\widehat{y}\left(t\right)$ is generated by adding the prediction result $p\left(t\right)$ and the $\widehat{e}\left(t\right)$.

The process of our algorithm is summarized as follows:

Step 1: Several components, contain the IMFs and a residual, are obtained by decomposing SSN via CEEMDAN.

Step 2: The different sub-models are trained using the corresponding training sets, denoted as TN_i (i = 1, …, K), for their optimal parameters; then, they are used to predict the homologous testing sets, denoted as TA_i (i = 1, …, K). The predictive result p(t) is the sum of the prediction values of each component.

Step 3: The error e(t), which is calculated by Formula (11), is also predicted based on the GRU using the same splitting method to obtain the new predictive error e′(t). Thus, the $\widehat{e}(t)$ for error compensation, which is reconstructed in Section 2.4, can be obtained.

Step 4: The final prediction result is computed using Formula (13).

2.6. Performance Criteria

To verify the model’s accuracy, three popular criteria, i.e., root mean square error (RMSE), mean absolute error (MAE), and correlation coefficient (Corr), are utilized to evaluate the validity of our algorithm in this study. As everyone knows, if RMSE and MAE are smaller, this means that the prediction model is better. However, the Corr is the opposite. If the Corr is closer to 1, this means that the prediction effect of the model is better. The computational formulae of these three performance indices are shown as follows:

$$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{\frac{\sum _{\mathrm{t}=1}^{\mathrm{n}}{(y\left(t\right)-\widehat{y}\left(t\right))}^2}{\mathrm{n}}}$$

(14)

$$\mathrm{M}\mathrm{A}\mathrm{E}=\frac{1}{\mathrm{n}}\sum _{t=1}^{\mathrm{n}}\left|y\left(t\right)-\widehat{y}(t)\right|$$

(15)

$$\mathrm{C}\mathrm{o}\mathrm{r}\mathrm{r}=\frac{\sum _{\mathrm{k}=1}^{\mathrm{n}}(y(t)-\overline{y})(\widehat{y}(t)-\overline{\widehat{y}}(t))}{\sqrt{\sum _{t=1}^{\mathrm{n}}{(y(t)-\overline{y})}^2}\sqrt{\sum _{t=1}^{\mathrm{n}}{(\widehat{y}(t)-\overline{\widehat{y}})}^2}}$$

(16)

where y(t) represents a value at the t moment of the observed sequence Y; and $\widehat{y}\left(t\right)$ is a prediction value at the t moment of the predicted sequence $\widehat{\mathit{Y}}$. The $\overline{y}$ and $\overline{\widehat{y}}$ correspond to the average value of Y and $\widehat{\mathit{Y}}$.

3. Experimental Results and Analysis

3.1. Data Description

The SSN is downloaded from SIDC (website: “https://www.sidc.be/silso/home” accessed on 5 June 2023; source: WDC-SILSO, Royal Observatory of Belgium, Brussels). There are 2547 records of the monthly mean total sunspot numbers from January 1811 to March 2023 (see Figure 4) in the dataset, and other statistical descriptions, including the maximum, minimum, and standard deviations, are presented in

Table 1. Statistical description of the dataset.

Dataset	Time Period	Type	Size	Standard Deviation	Minimum	Maximum
Sunspots	1811.1–2023.3	Monthly	2547	63.45	0.2	285

. The maximum and minimum of the SSN are 285 and 0.2, respectively, and they are more than 1400 times apart. We can observe that the SSN is a chaotic sequence and does not have obvious regular periods [16]. The dataset is decomposed into nine IMF components and a residual component (RES) (see the sub-figure on the left in Figure 5) by CEEMDAN. In the figure, we can see that some components, such as the first four components, are high-frequency signals, and they do not have an obvious time period. The remaining components are low-frequency signals. We also compute their center frequencies from IMF_1 to IMF_9 and RSE. Their center frequencies are 840.66, 382.26, 140.5, 87.91, 16.84, 15.69, 7.06, 5.49, and 0.784, respectively. The center frequencies of IMF_1, IMF_2, IMF_3, and IMF_4 are relatively large; thus, they are also difficult to predict. Therefore, the degree of prediction difficulty ranges from high to low. The sub-figure on the right in Figure 5 is the corresponding frequency of each component, and it shows that the first several components are relatively complex.

To facilitate the training of the network, all components are split under the same proportion in chronological order into a training set (1632 records from January 1811 to December 1946), validation set (408 records from January 1947 to December 1980), and testing set (507 records from January 1981 to March 2023). The ratio of the training and validation sets is 8/2. To obtain the optimal parameters of different sub-models, each sub-model is trained by its corresponding training set. The validation set is used to monitor the hyperparameter tuning process, and the testing set is used to assess the performance of each trained prediction model.

3.2. Training Set

In this paper, we use CEEMDAN to decompose the SSN, and then, several sub-sequences (IMFs) can be obtained. To improve the predictive accuracy of the model, we adopt single-step prediction. Because 1 year has 12 months, the input size and output size are set as 12 and 1, respectively. Because the data features, such as frequency, trend, and amplitude, contained in different sub-sequences should be different, different sub-model-based GRU neural networks with varying parameters are used to train and predict the corresponding sub-sequences. We select Adam and SGD as the candidate activation functions and 0.01, 0.001, and 0.001 as the candidate values of the learning rate. The value of the epoch is set from 500 to 5000 with different step sizes. Different combinations of neural network parameters are tested, and the best one is found for each sub-model when we train the sub-model on its corresponding training set. Extensive experiments are applied to verify the optimal values of the training parameters of different sub-models. We observe that the predictive results are best when the activation function is Adam, and the learning rate is assigned 0.001. Detailed values of some parameters, including the input size, number of layers, output size of 1, activation function, and loss function, are obtained: 12, 64, 1, Adam, and RMSE, respectively. Detailed values of different parameters with these values are listed in

Table 2. The training parameter setting.

No.	Training Parameter	Setting
1	Input size	12
2	Number of layers	64
3	Output size	1
4	Activation function	Adam
5	Loss function	RMSE
6	Learning rate	0.001
7	Batch size	100
8	Dropout	0.2

. However, the optimal epochs are not the same. The epoch size is 700 for sub-models 1 and 2. The epoch of sub-models 3, 4, and 7 is 4000. The epoch of sub-models 5 and 9 is 2000. The epoch of sub-models 6 and 8 is 5000.

3.3. Error Analysis

To analyze the error between the predicted result and the original data, the predictive error sequences can be calculated using $y\left(t\right)-\widehat{y}(t)$, and they are denoted as PE. The error result PE is shown in Figure 6. In the figure, we can observe that many predictive errors are very close to 0. Figure 7 shows the histogram of PE, the amount of PE with ranges at [0, 1], [1, 2], [2, 3], [3, 4], and their counts: 320, 115, 37, 19, and 16. Therefore, this shows that most errors are clustered around 1.

3.4. Prediction Result Comparison with Other Methods

The classical time series prediction algorithms, including LSTM, GRU, and the composite algorithm EMD-LSTM, are compared with our algorithm. Figure 8 presents the predictive results of these algorithms. The lines in black, blue, cyan, green, and red correspond with the original data and the predictive results of LSTM, EMD-LSTM, GRU, and our algorithm. We can observe that all algorithms can reflect the entire trend and simulate the change in SSN. However, the results of two algorithms, including LSTM (blue line) and EMD-LSTM (cyan line), present relatively large differences compared to the original data. The result of the GRU is close to the original data, but the abrupt change points and the fluctuation parts exhibit obvious differences with the corresponding original data. Our algorithm can effectively overcome these problems, and the red line (our predictive result) is close to the black line (original data).

We compute three performance criteria, including MAE, RMSE, and Corr, to further demonstrate the advantage of our algorithm. The results of different criteria are listed in

Table 3. The indices of different methods.

Method	MAE	RMSE	Corr	Time (s)
LSTM	13.56	19.225	0.97342	30.221
EMD-LSTM	11.149	13.528	0.97883	51.224
GRU	2.414	3.176	0.98334	28.243
Our algorithm	0.91732	1.319	0.99972	61.934

. In the table, we find that the first two indices of LSTM and EMD-LSTM are all over 10, and the Corr of the two algorithms is less than 0.98. The GRU has better performance than LSTM and EMD-LSTM. In GRU, the values of MAE and RMSE decrease, although this two indices are all more than 2, and the value of Corr is 0.98334. The MAE and RMSE of our algorithm exhibit an obvious decrease compared to others, and the Corr is very close to 1. Therefore, our algorithm has better performance than others. The method proposed in this paper can improve performance in terms of accuracy and precision, which are within an acceptable time frame.

3.5. Ablation Experiment and Multi-Step Prediction Analysis

3.5.1. The Ablation Experiment of Different Modules in Our Model

To better analyze the effectiveness of the different parts in our algorithm, we design an ablation experiment. The CEEMDAN module and EC module are added into the GRU in sequence, and they are denoted as CEEMDAN-GRU and CEEMDAN-GRU-EC. Under the hyperparameters of the neural network, they are kept unchanged and are all trained by the same proportion of SSN. We compare the performance of the GRU, CEEMDAN-GRU, and CEEMDAN-GRU-EC using the metrics of RMSE and MAE.

Table 4. The results of the ablation experiment.

Method	RMSE	MAE	Corr
GRU	3.176	2.414	0.98331
CEEMDAN-GRU	1.833	1.459	0.99423
CEEMDAN-GRU-EC	1.319	0.91733	0.99971

presents the ablation experimental result. It can be observed that the RMSE and MAE of the single neural network GRU are above 2. When the CEEMDAN module is added, its metrics are 1.833 and 1.459, respectively, which decrease by 1.343 and 0.955 compared to the GRU. After adding the EC module into CEEMDAN-GRU, the metrics drop to 1.319 and 0.91733. Our algorithm carries out comparisons with the GRU and CEEMDAN-GRU, and the RMSE and MAE improve by 58.47% and 62%, and 28.04% and 37.13%, respectively. The Corr index of CEEMDAN-GRU-EC is higher compared to the GRU and CEEMDAN-GRU. Therefore, this indicates that both the CEEMDAN and EC are useful in our algorithm.

3.5.2. Multi-Step Prediction Analysis

To evaluate the generalization ability of our algorithm, we attempt to perform predictions for an additional 7 and 11 months. The prediction comparisons of the original SSN (black line), the next 1 month (blue line), the next 7 months (green line), and the next 11 months (red line) are shown in Figure 9. As the prediction step increases, the predicted results move further away from the true value. We can also note that these lines manifest differences for some values, especially fluctuating values.

Table 5. The predictive performance comparison of multi-step analysis.

Predictive Step Size (Months)	RMSE	MAE	Corr
1	1.319	0.91732	0.99971
7	5.148	3.667	0.99344
11	6.468	4.908	0.99121

presents the values of the indicators. The longer the prediction step, the more clearly the indicator increases. The table shows that the 1-day predictive accuracy is advantageous than others.

3.6. Comparison with Other Studies

To verify the advantage of our method, we compare our method with some methods in other studies, including the following two categories: a single classical method (informer [38]) and hybrid methods (XGboost-DL [51] and EMD-LSTM-AM [42]). The single informer method has been proposed recently. XGboost-DL is an integrated regression model utilizing deep learning. The EMD-LSTM attention mechanism is a combination of signal decomposition and deep learning. Therefore, we select the comparative literature according to aspects such as model complexity, timeliness, and feature fusion in order to reflect the performances of our method. The XGboost-DL and informer programs are downloaded from “https://github.com/yd1008/ts_ensemble_sunspot” accessed on 8 October 2023, and their parameters remain unchanged, with the exception that the dataset is updated. The results of different predictive methods are listed in

Table 6. Performance comparisons with other literature.

Method	RMSE	MAE	Time (s)
Informer [38]	29.904	22.351	44.481
XGboost-DL [51]	25.737	19.823	51.324
EMD-LSTM-AM [42]	3.642	4.523	53.264
Our algorithm	1.319	0.91731	61.933

. We can observe in the table that our method’s output increased to values above 2.281 and 3.583 relative to the RMSE and MAE, respectively. Therefore, the method of this paper provides a significant improvement in accuracy over an acceptable time frame compared to other methods.

3.7. Predicting Future Values

Because a substantial number of researchers also use different methods to predict future sunspot numbers, we download the predictive results of the well-known prediML, which is proposed by McNish and Lincoln [55,56], and the improved method using the Kalman filter (KFprediML) [9] from SIDC (Source: WDC-SILSO, Royal Observatory of Belgium, Brussels, Belgium). The predictive result is also downloaded from SWPC (Website: https://www.swpc.noaa.gov/ accessed on 21 July 2023) and denoted as SWPC. The predictive future results of different algorithms from April 2023 to March 2024 are shown in Figure 10. Our predictive results show that the sunspot number will continue to increase in the future, and its trend is similar to prediML and KFprediML. However, the result of SWPC exhibits an inverse trend. Due to the possible noise in the source data, the error between the observations and the real sunspots, and the generalizability of the model, our results can only reflect the development trend of sunspots. The change in sunspots may affect the actual operation of electronic systems and equipment. By predicting the change trend, the operation state of relevant electronic equipment can be adjusted in time.

4. Conclusions

The prediction of the sunspot number with high accuracy is very important. In this paper, we analyze these predictions from three aspects, including the features of the original data, model selection, and the difference between predictive results, using the deep-learning method and the original data, which may affect the predictive performance. Three problems examined in our analysis can be outlined as follows:

(1)

Data features: The original data comprise a chaotic and unstable time series. Thus, there is no periodicity, which is regularly manifested.

(2)

The relationship between datasets: Since sunspot numbers comprise a classical time series, there is an interdependence between them.

(3)

The delay phenomena between predictive data and original data: This problem generally exists, especially with respect to data with large fluctuations.

For the aforementioned reasons, we present our combined model for predicting the sunspot number, and we carry out comparisons with single and composite models. The experimental results can demonstrate that our model has a higher prediction accuracy. Finally, we predict sunspot numbers for the April 2023–March 2024 period. Further explorations of sunspot periodicity patterns will follow. The development and improvement of models for predicting sunspot activity cycles will continue. For example, network structures in the model can be optimized to reduce the running memory of models; the running time of the model can be reduced; and some novel mechanisms can be incorporated to improve the accuracy of the prediction. This is essential for understanding the long-term trends in solar activity. Simultaneously, for industries relying on satellite navigation and communications (e.g., aviation, maritime, etc.), it is possible to optimize the maintenance schedules and adjust the operational strategies during periods of high predicted sunspot activity. This will effectively prevent critical operations during periods of peak sunspot activity and reduce the navigation errors and communication problems caused by signal interference.