A New Hybrid Prediction Method of El Niño/La Niña Events by Combining TimesNet and ARIMA

El Niño/La Niña events significantly impact human society, often resulting in considerable monetary losses. Accurate prediction has become crucial with triple La Niña events in this century. This study applied TimesNet to El Niño/La Niña event prediction for the first time. We proposed a hybrid prediction method based on extracting features from time series data and initially decomposing the time series data (Niño3.4) into several Intrinsic Mode Functions (IMFs) using the Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN). Based on the characteristics of each IMF, we used a hybrid method of TimesNet and ARIMA to make adaptive forecasts for them. We selected monthly data from 1950 to 2022, with the first 63 years used for training and shifted 12 periods (12 months) ahead to forecast the Niño3.4 index values for the next ten years. The experimental results of this study show that: 1) The pre-processing method using CEEMDAN can effectively extract the original time series data features and significantly improve the prediction performance; 2) proposed approach achieved good performance in predicting El Niño/La Niña events, particularly during the transition from El Niño to La Niña events (e.g., 2016, 2019-2020); 3) evaluation results indicate that the proposed model exhibits better predictive power (stability and accuracy of prediction results) than the current best single-order predictor, the ConvLSTM model, on the validation set of the last ten years.


I. INTRODUCTION
El Niño and La Niña events are often discussed together due to their similar occurrence characteristics and alternating nature [1].These events significantly impact the global climate, often causing natural disasters [2], [3], [4].For instance, the La Niña event can lead to a climate pattern of cold winters and hot summers, droughts in southern China [5], and floods in the north, resulting in substantial economic and property losses [6].Meanwhile, Indonesia, eastern Australia, and northeastern Brazil are more prone to flooding due to increased precipitation [7], [8], while equatorial Africa and the south-eastern United States experience droughts [9].In recent years, the frequency of La Niña events has increased due to severe global warming [10].With the recent occurrence of triple La Niña events since August 2020 [11], the hazards of El Niño/La Niña events are well known to the public, making long-term predictions of these events of great importance.
The associate editor coordinating the review of this manuscript and approving it for publication was Sajid Ali .
Today, El Niño forecasting methods are usually classified into three types: dynamical methods, statistical methods, and hybrid dynamical-statistical methods, which have proven to be feasible in predicting El Niño events [12].Nevertheless, the prediction of El Niño still faces great challenges: the uncertainty of ENSO in time and space [13], the decreasing accuracy of existing model prediction techniques with new data [14], the asymmetry of El Niño and La Niña occurrence [15], and the spring predictability barrier [16], [17], [18] all pose difficulties in predicting El Niño/La Niña events.
In recent years, with the widespread use of artificial intelligence techniques, several time series prediction tools such as deep learning have been applied to predict El Niño/La Niña events [19], providing more directions for statistical prediction methods.Among them, hybrid models have achieved better results.In a study by Wang et al. [20], the prediction of El Niño-related time series indices was performed using a variety of LSTM networks with different structures.The data were pre-processed with Empirical Mode Decomposition (EMD) techniques, which verified the high accuracy of the model in predicting 12-month ONI time series data.Guo et al. [21] incorporated Ensemble Empirical Mode Decomposition (EEMD) based on the CNN-LSTM network, which outperformed the traditional CNN-LSTM method.A prediction technique using ANN neural network to improve ARIMA proposed by Nooteboom [22] et al. obtained prediction results similar to the kinetic model in the short term.In addition, decision tree classification algorithms [23], image processing [24], [25], [26], support vector machines [25], [27], and Gaussian processes [27] have been incorporated into prediction studies with good results, which provide theoretical support for our research.
The large amount of antecedent data required by traditional kinetic models [20] tends to be computationally expensive.In contrast, deep learning prediction methods can be more efficient than traditional statistical methods [28].Although machine learning algorithms do not have the same prediction accuracy as kinetic models, deep learning can achieve more efficient multi-step prediction performance than statistical models and use fewer data [19], [20].
Our study focused on the temporal prediction of Niño3.4.Although there are many neural network models based on Spatio-temporal data that can provide longerterm, more accurate results: CNN [29], ENSO-ASC [30], ENSO-GTC [31], Attention-ConvLSTM [32], this often implies a more dimensional data dependence with training overhead.We want to propose a Niño3.4training model that relies only on one-dimensional data with low training overhead to reduce this study's large data dependency requirements and to address the practical need that historical climate data in a region are insufficient to support the construction of multidimensional data models.
This paper proposed an El Niño/La Niña index forecasting method based on the Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN) and TimesNet time series forecasting framework.First, the Niño3.4time series index is reconstructed using the CEEMDAN.Then the combined TimesNet-ARIMA forecasting method is used for each IMF.Finally, we obtain the Niño3.4forecast data by summarizing all IMF.By comparing the results with models such as ConvLSTM, which has the best results in the univariate Niño3.4 forecasting.According to the results, the proposed combined forecasting method achieves the most remarkable prediction capability for 2013-2022.
The structure of the remainder of this paper is summarized as follows: In Section II, we introduced the technical tools and datasets used in this paper; In Section III, we analyzed and compared the results of the model solution; In Section IV, we summarized the conclusions drawn; and In Section V, we further discussed future research.

II. METHODOLOGY A. FRAMEWORK OF CEEMDAN-TIMESNET-ARIMA
The model framework designed in this paper is shown in Figure 2 and can be summed up as the following Using the Niño3.4index from 1950-2012 for the mode decomposition (Figure 1), we slid the forecast backward for ten years using 12 periods as the forecast length (sliding window length).Each time's sliding will expand the training set's length by 12 original data nodes, and similarly, the IMF and residual after the mode decomposition will also change.The data normalization is handled as follows: Detailed descriptions of CEEMDAN, TimesNet, ARIMA can be found in the following sections.

FIGURE 1.
The prediction method adopted in this article: given 12 periods as the length of the sliding prediction window, after completing one round of prediction, the training data are obtained from the actual data complementary, and a total of 10 rounds of prediction are conducted in this paper, i.e., predicting the Niño3.4index from 2013-2022.

B. COMPLETE EEMD WITH ADAPTIVE NOISE
Empirical Mode Decomposition (EMD) was first proposed in 1998 [33] to decompose nonlinear, non-smooth data into several Intrinsic Mode Functions and separate the local features of the data.The criteria for identifying an IMF can be summarized as follows: 1) In the entire series, the number of extreme value points is equal to, or at most, one different from the number of zero-crossings; 2) At any point, the mean value of the envelope defined by the local extreme value points and the envelope defined by the local minimal value points is zero.In practical use, the EMD algorithm may encounter the problem of modal confusion, where the IMFs contain significantly different time scales, making it difficult to use them further.To address this issue, improvement algorithms, such as Ensemble Empirical Mode Decomposition (EEMD) [34], complementary EEMD (CEEMD) [35] and Variational Mode Decomposition (VMD) [36], were proposed successively to alleviate the problem of modal mixing by adding positive and negative Gaussian white noise.However, a certain amount of white noise remains in the IMFs, and the separation effect is not optimal.
In 2011, Torres et al. [37] proposed the ''Complete EEMD with Adaptive Noise'' (CEEMDAN) method, which effectively addresses the problem of transferring white noise from high to low frequencies.This method has been widely used in time series prediction [38], [39], [40].CEEMDAN distinguishes itself from other signal decomposition algorithms by incorporating an adaptive process and multiple iterations.These enhancements serve to enhance the stability and reliability of the computation, while mitigating the impact of local extreme points.The IMF components obtained by the CEEM-DAN method are reused as input for subsequent prediction models, resulting in a good performance.The algorithmic flow of CEEMDAN is summarized in Appendix A.

C. TIMESNET
Rapid development in various fields has led to increased temporal prediction research.Recently, Transformer net-works, which rely on self-attention mechanisms, have received considerable attention, distinguishing them from traditional networks that rely on convolution or recursion [41].In addition, Informer [42], which is based on a multi-headed self-attention mechanism, Autoformer, forecasting tools have achieved satisfactory results in terms of long-term forecasting performance [43], [44], [45], providing a new direction for El Niño/La Niña forecasting.
In 2023, a novel time series prediction model, TimesNet, was proposed based on ''TimesBlock'' [46], which achieved state-of-the-art (SOTA) results in five major areas, including series prediction, attracting various attention.TimesNet differs from traditional one-dimensional feature extraction methods.It transforms a one-dimensional time series into a two-dimensional tensor set based on multiple cycles, allowing it to capture relative changes within and between corresponding cycles.Given the high degree of variation and complex periodicity of the Niño3.4index, TimesNet can effectively identify changes within and between periods, demonstrating the significant potential for accurate forecasting.TimesNet primarily utilizes Symmetric Mean Absolute Percentage Error (SMAPE) as the model training and evaluation loss function.This choice allows for assessing the model's goodness-of-fit while considering its complexity.
The core technologies of TimesNet can be summed up as follows.

1) CONVERTING 1D VARIATIONS INTO 2D VARIATIONS
TimesNet analyzes the correlated changes of time series periods by Fast Fourier Transform (FFT).For a time series X 1D ∈R L×C of length L and a number of variables C, the amplitude (A) of the periodic basis function can be described as: where FFT(•) and Amp(•) denote the FFT and amplitude values calculation.Only the first k amplitude values (f) are selected to avoid high-frequency noise.
Accordingly, the period length of the sequence (p) can be determined: Eq.( 1)-Eq.( 3) can be summarized as: Using the derived sequence frequency and period length, the original sequence can be regularized into a twodimensional tensor: where Padding(•) is to pad the time series backward with numerical zeros to make it compatible with Reshape p i ,f i (•).p i and f i denote the number of rows and columns of the transformed 2D tensor, respectively.

2) TIMESBLOCK
The main framework of TimesNet, which is organized by the residual way, and for the lth layer of TimesNet, its input-output process can be expressed as: In particular, when l = 0, there is: In TimesBlock, various temporal two-dimensional variations from k different re-shaping tensors are captured by a Parameter-efficient Inception block in two-dimensional space.They are fused according to the normalized amplitude values (Figure 3).

FIGURE 3.
The main structure of TimesNet: TimesNet comprises several TimesBlock connected in the way of residuals, which need to go through four main steps of reshaping, information extraction, anti-reshaping, and aggregation inside TimesBlock [46].

3) PARAMETER-EFFICIENT INCEPTION BLOCK
The information representation of the sequence can be easily obtained by using the Parameter-efficient Inception block, which is defined as: This approach originates from the visual processing mechanism [47], where the sequence needs to be reshaped after processing for the next step, here using Trunc(•) to truncate the sequence X l,i 2D to the original length L:

4) ADAPTIVE AGGREGATION
After a parametric effective initial fast a series of 1D sequences {X l,1 1D , . . ., X l,k 1D }, weighted using the amplitude A. This measure is inspired by Autoformer [48] (the amplitude can reflect the relative importance of the selected frequencies and periods).In summary, the output sequence can be obtained as follows: D. ARIMA The ARIMA model is one of the most commonly used time series forecasting tools [49], [50], with the excellent performance of short series trend forecasting and the simplicity of the model.For a stationary sequence y t , ARIMA(p,d,q) can be described as: where, ϕ i , θ i are Coefficient terms, L is lag operator, ε t is random error, p, d and q are positive integers, referring to the order of the autoregressive, integrated, and moving average parts, respectively.

E. EVALUATION METRICS
The evaluation metrics used in this study are the root mean square error (RMSE) and mean absolute error (MAE), which reflect the model's predictive ability.A smaller RMSE and MAE indicate more vital predictive power and lower prediction errors, as mathematically defined below: where, ŷi denotes the predicted value and y i denotes the true value.

FIGURE 4.
The prediction method adopted in this article: given 12 periods as the length of the sliding pre-diction window, after completing one round of prediction, the training data are obtained from the actual data complementary, and a total of 10 rounds of prediction are conducted in this paper, i.e., predicting the Niño3.4index from 2013-2022.

III. EXPERIMENT AND ANALYSIS A. DATASET
This paper uses the Niño3.4index, which follows the ENSO event identification criteria developed by the China Meteorological Administration, National Climate Center, to describe the onset, end, and amplitude of El Niño (La Niña) events [51].Data for this index can be obtained directly from the National Oceanic and Atmospheric Administration (NOAA) and are calculated from the Niño3.4region (5 • N-5 • S; 170-120 • W) in the monthly NOAA ERSST V5 report [52].
According to the definition, an El Niño/La Niña event occurs when the absolute value of the three-month sliding average of the Niño3.4index is greater than or equal to 0.5 • C (Niño3.4 ≥ 0.5 • C is an El Niño event and Niño3.4 ≤ −0.5 • C is a La Niña event).The Niño3.4 index can uniquely determine the event's intensity weak(0.5 The data used in this study span from January 1950 to December 2022, totaling 876 data points.Each data point represents the average Niño3.4index (three-month sliding average) for two-time issues before and after its inclusion.The trend of the time series data has shown in the plotted Niño3.4 index (Figure 4).Specifically, Figures 4-b and 4-c show the time-zoned data for the longer duration of La Niña events in the last two decades.
The division between the training data and testing data is given in Table 1, and the statistical properties of the data are similar, enhancing the confidence level when vali-dating the model.

B. RESULTS OF CEEMDAN
We obtained the data with 9 IMFs and residual after applying CEEMDAN processing, where the decomposition parame-ters were set to a signal-to-noise ratio of 0.03 and 100 noise additions.Figure 5 displays the IMF components arranged in decreasing order of the decomposed sequence's frequency, with the residual presented in the last row.The residual reflect the overall sequence's trend in mode decomposition and act as monotonic functions under the decomposition conditions.Based on these conditions, we can assume that the Niño3.4index has increased in the last 70 years.The IMF obtained from the CEEMDAN decomposition has the property that its periodicity becomes smoother and simpler as the number of decompositions increases.The mean-variance and average of the decomposed data decreased significantly compared to the original data, as shown in Table 2.These changes in data quality improved the prediction accuracy of the subsequent model.

C. COMBINED PREDICTION OF TIMESNET-ARIMA MODEL
In this study, the expert modeller [53] available in SPSS was employed to ascertain the ARIMA parameters, namely p and q.The original series exhibited smoothness following the first-order differencing, thereby enabling the identification of the appropriate value of d as 1.To enhance the model's performance, the dependent variable was initially transformed using its natural logarithm, and subsequently, the most suitable parameters were determined through the Normalized Bayesian Information Criterion (BIC).Additionally, we conducted Box-Jenkins modelling steps, and a comparative analysis revealed a minimal disparity in the prediction accuracy between the two approaches.
To further investigate the prediction effect of TimesNet and ARIMA on each IMFs and residual, we made predictions on them separately (2012 data were selected for this purpose).In the prediction effect of IMF1-IMF3, TimesNet has better prediction ability because it is more suitable for complex multi-period series (Figure 6-a).
Furthermore, in the prediction effects of IMF4-IMF9 and residual, ARIMA model is more prominent (Figure 6-b,c).ARIMA can capture its periodic features well when the IMFs features become more stable and smoother.
The main difference between the TimesNet and ARIMA methods is that the TimesNet model constructs a two-dimensional tensor for forecasting by decomposing complex time variation into multiple cycles.On the other hand, ARIMA relies only on one-dimensional historical data for regression forecasting, which requires a higher degree of data smoothness and essentially captures linear relationships.
The article adopts the CEEMDAN approach to accurately decompose the raw data based on considering the predicted series' properties.Niño3.4 exhibits an overall periodicity with the randomness and uncertainty of the changing trend, which is the main difficulty in predicting El Niño/La Niña events, and by decomposing the raw time series, the signals at different frequencies can be obtained, which can make the Nino3.4's each IMF in different frequency domains can be regarded as a sequence of the degree of influence of different complex events on the Niño3.4index, which can make the prediction effect more accurate and interpretable.
TimesBlock can adaptively detect multi-periodicity so that TimesNet can show excellent predictive performance for the first few complex variation IMFs.For the remaining IMFs as well as the residual series, the decomposed series has better stability, and the ARIMA model can accurately predict the series within a prediction order of 12 periods despite the reduced periodicity pattern of the series variation.
Based on this characteristic, we adopted a combined model approach.Specifically, we trained IMF1-IMF3 separately using the TimesNet model and the ARIMA model to train the remaining parameters.We validated this idea in the 2012 prediction (Figure 6-d  Compared to the TimesNet model, which had more volatile prediction results, the ARIMA model had more moderate prediction results, and its prediction effect was better when Niño3.4 exhibited minimal fluctuations.However, when Niño3.4 fluctuated significantly (June-October), ARIMA failed to capture this feature, while TimesNet was better at capturing this ''sudden change.''Nonetheless, TimesNet's early fluctuations limited the stability of its predictive power.By combining the advantages of both models, our proposed approach achieved a closer approximation to the actual value.TimesNet and ARIMA models have different predictive capabilities for different IMFs, with TimesNet being better at capturing the effects caused by the short-term, unstable factors of Niño3.4.ARIMA, on the other hand, is much better at capturing long-term, stable influences.The El Niño/La Niña phenomenon develops and evolves under the multiple superpositions of these two factors, and the hybrid TimesNet-ARIMA forecast method can extract and forecast these features more comprehensively.The hybrid prediction model introduced in this paper leverages the predictive capabilities of the ARIMA model for stationary sequences and TimesNet for complex and unstable sequences.By decomposing the original sequences into distinct components suitable for prediction using the CEEMDAN method, the hybrid model is engineered to achieve optimal predictive performance.

D. PREDICTIVE EFFECT OF THE TIMESNET-ARIMA MODEL
In order to test the effectiveness of the proposed approach, the Niño 3.4 index from 1950 to 2012 was used as the training data and input to the model for 12-period (12-month) forecasting, Figure 7 shows the forecasting results of the combined model for the period 2013-2022.
According to the data presented in the figure, the comparison between predicted and actual values shows that the proposed approach achieved better predictions in 2013, 2016, 2017, 2019, and 2020.In the forecasts of these years, the evolution trend of the annual Niño3.4 index can be better predicted, but in the year with a higher degree of variability in the Niño3.4 index (2015), the predictive power of the model decreases.It is still difficult to accurately predict highintensity El Niño/La Niña events in statistical models [20], especially when this event occurs at the end of the model prediction cycle.However, the changes in this feature can be better captured when it occurs in the first and middle part of the prediction cycle (2016).
To compare the predictive ability of our proposed approach, we selected ConvLSTM, which currently has the best prediction for single-order Niño3.4 as a reference.(Further references to the literature [20] can be made for the structure of the contrasting models.)We also conducted experiments using TimesNet with the ARIMA model, and the input data for all compared models were processed by CEEMDAN decomposition.Table 3 presents the combined prediction indexes.
The results in the table show that the proposed approach has the best predictive power in 10 years (MAE = 0.Moreover, in terms of stability of forecast errors, the combined model has the best variance values for MAE (0.088 • C) and RMSE (0.0077 • C) either, indicating that the proposed approach is more robust in its forecasting ability and has less probability of extreme deviations in forecasts compared to other models.The performance of the proposed approach can be summarized as the best overall result, with a few years differing little from the optimal prediction.FIGURE 7. results of the proposed model for 2013-2022: the red curve is the actual value, the blue curve is the forecast value of the proposed approach, and the histogram shows the average error of the top and bottom three months of the month (the empty value of the first and last month is recorded as zero, and only the average of the two months is calculated).
In 2017 (nodes 48-60) and 2021 (nodes 96-108), the ARIMA model achieves better results in terms of forecasting effectiveness, which is explained by the more moderate development of the Niño3.4index during this period and the absence of large fluctuations in the indicator.Although both the statistical and dynamical models showed lower predictability when forecasting before and throughout the spring (February to May) [16].However, according to the prediction results of the models (Figure 8), no significant decrease in prediction accuracy was observed using both ConvLSTM and TimesNet techniques when faced with spring predictability barrier.This indicates that both the proposed model and ConvLSTM can predict El Niño/La Niña better than one year in advance.
For the 2019 forecast, ConvLSTM shows a relatively large deviation (Figure 9), which is caused by the presence of a lag of about four periods (the actual value in August 2019 is equal to the predicted value in November), and ConvLSTM incorrectly captures the downward trend of Niño 3.4.While TimesNet captures this trend (July and October 2019), the proposed combined model achieves a better prediction (MAE = 0.1483 • C).      also calculated the predictive power of TimesNet, ConvLSTM, and the proposed approach in terms of the prediction lag order performance (using the RMSE metric, see Figure 10), respectively; ConvLSTM has a relatively low RMSE in the first three periods and then maintains a relatively fast growth; TimesNet has a relatively large error in the first three periods, but then the RMSE is maintained in the middle of ConvLSTM and the proposed approach.The RMSE of the proposed approach increases slowly with the growth of the prediction lag order and maintains the trend of minimum error.It is noteworthy that TimesNet has a minor lag error at orders 4-8, which is even lower than the RMSE of the first three periods, it is probably related to the weak learning ability of TimesNet in the early stage.
In addition, we calculated the cross-correlation coefficients between the predicted and true values of these models over the 12-month period as follows: where y is the true value, y p is the predicted value, N is 12.
According to the data in the table (Table 4), the proposed method has the slowest decline in the cross-correlation coefficient between predicted and true values, which again confirms the superiority of the TimesNet-ARIMA model in terms of predictive power.In addition, the proposed model achieved significant predictive stability over twelve periods, significantly outperforming the ConvLSTM.In addition, we also compared the differences in the prediction models between El Niño and La Niña events (Table 5).It has showed that both TimesNet and the proposed model perform better in predicting La Niña events, probably due to the characteristics of La Niña events in Niño3.4 values, which are less than −0.5 • C compared to the more significant fluctuations of El Niño, interval maintains a more moderate trend.
As mentioned above, we found that the model had better performance in fitting the downward trend curve, i.e. it worked well during this period of change from El Niño to La Niña events.It is clear that the Niño3.4index has a more considerable variance during an El Niño event, and its mean is also higher than the mean during a La Niña event.
To further compare the advantages of the proposed models in terms of prediction accuracy, the prediction results of the Dynamical and Statistical models were selected separately for comparison.
The selected models were all predicted from January, with a prediction order of 11, and each prediction resulted in a sliding three-month Niño3.4 index, with the relevant predictions of the models coming from IRI [54].In addition, we selected forecasts from 2013 to 2021 for comparison.The model forecasts for some years are unavailable due to missing data and have been additionally annotated in the table.
Tables 6-7 show the MAE/RMSE result, for the given comparison years, we can see that the deep learning models generally outperform the dynamic and numerical models in terms of prediction, which also indicates that our proposed hybrid prediction model can be a powerful tool for El Niño/La Niña events in a given year.Overall, the proposed model can achieve leading or close results to the optimal method, and the model has the lowest RMSE, MAE index variance, which further illustrates the stability of the proposed model.
The parameter settings for TimesNet are shown in Table 8.It is worth noting that due to stability considerations in the design of TimesNet, some filtering is performed, and when adjusting the corresponding control parameter k, we find that no significant increase in prediction accuracy occurs when k exceeds the set value.Therefore we believe that the appropriate filtering operation does not significantly affect the high-frequency outliers.However, determining the most appropriate value of k is still worthy of further research and can be explored more deeply in future work.We compared the training time spent using the TimesNet-ARIMA model with that of ConvLSTM (Figure 11).The TimesNet-ARIMA method obtained one-tenth of the training time spent compared to ConvLSTM.It further illustrates the overall lead of the proposed model in terms of prediction effectiveness and overhead and provides a basis for further extension of the TimesNet model to 2D data prediction.
The uncertainty of the model is also compared in Figure 11, where we selected five different lengths of training sets for training.In terms of the predictive power of the model, the RMSE index of the model did not show a significant increase as the length of the training set decreased, which further illustrates the excellent stability of TimesNet.
In the previous text, we employed a loss function that did not consider the model's complexity.To delve deeper into evaluating the model's performance while accounting for simplicity, we have opted for the Akaike Information Criterion (AIC) [55] as the loss function.AIC is the information loss between the fitted competing model and the true model.If the information loss is minimum for a model, then the model is superior and it tries to give a reality of the true model.The general AIC is defined as follows: where k is the number of parameters in the model, L is the likelihood.
As a loss function, the above equation can be written [59] as: where N is the number of samples, and RSS is short for residual square error.
The experimental results are presented in Table 9, where we conduct a comparison of TimesNet prediction outcomes utilizing two distinct loss functions: Mean Squared Error (MSE) and Akaike Information Criterion (AIC).The evaluation metrics employed for comparison are Root Mean Square Error (RMSE) and AIC.Since the parameters of the neural network are determined by the structure, we measured the number of parameters (k) of the TimesNet model under the optimal conditions to be 146030.
The experimental findings highlight that employing AIC as the loss function provides a degree of model parsimony and effectively mitigates the risk of model overfitting.One notable advantage of using AIC as the loss function lies in its ability to simplify the model structure.Furthermore, in terms of RMSE, the AIC-based approach yields similar results to the MSE method in certain years.The selection between prediction accuracy and model simplicity can be tailored to specific research requirements, granting flexibility in practical applications.
Finally, we conducted predictions of the Nino3.4index for 2023 using the proposed model, aiming to demonstrate its predictive performance further.The prediction process was carried out in two stages: Firstly, we forecasted the Nino3.4index for the entire year of 2023 by employing data from 1950 to 2022 as the training set (referred to as Proposed A).
106356 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Secondly, we specifically predicted the Nino3.4index for the second half of 2023 (referred to as Proposed B) using the training set data encompassing the period from 1950 to June 2023, and we selected four dynamical/statistical models for performance evaluation.The combined results of both prediction parts are presented in Figure 12. Analysis of the first part's prediction results demonstrates that the proposed model effectively captures the variations in the Nino3.4index during the initial six months.Moreover, when the training set is six months in advance, the proposed model achieves prediction outcomes comparable to the statistical model's.This finding further reinforces the model's reliability and practical utility in realworld scenarios.
The second part of the prediction results (Proposed B) indicates the likelihood of a robust El Niño phenomenon occurring in the latter half of 2023, albeit with a gradual weakening trend.This observation is consistent with the conclusion of the International Research Institute for Climate and Society (IRI) in May [60].

IV. CONCLUSION
In this paper, we propose a TimesNet-based El Niño/La Niña event prediction model, which combines the (CEEMDAN) with a conventional ARIMA model and obtains better results than ConvLSTM while using only the Niño3.4index.In addition, we have compared the combined forecasting results with those of the dynamic and statistical models, and the results show that the combined model has better forecasting results, and some of the key findings are summarized below:  [56]).Compared to the more costly dynamic models, statistical models can be used as an easily implemented forecast reference and are expected to be a viable alternative forecast method in short-term forecasting.5) In the selected years (2013-2021), the hybrid model outperforms the dynamic model with the statistical model regarding forecasting effectiveness, providing a more straightforward means of forecasting.In addition, our model can better handle the need for forecasting in ations with insufficient data granularity than a deep learning with multi-factor inputs, demonstrating low data-dependent forecasting capabilities.

V. FURTHER DISCUSSION
We tried to use the more popular long-order prediction means (such as Informer and Autoformer) to predict the Niño3.4index for 24 periods or even longer during the preliminary experiments.However, we did not find its significant prediction advantage on the dataset used in this paper, so we did not show the comparison in the article (We show the specific details in Appendix B&C).TimesNet, as an alternative implementation of the current long-short order prediction, distinguished from the self-attentive mechanism, has achieved SOTA on the M4 public dataset up to now.
In the future, we can continue to pay attention to the brilliant performance of TimesBlock, which realizes multi-feature extraction technology through dimensionality enhancement in long-term prediction and further practical application in El Niño/La Niña event prediction.
In addition, a number of improvement schemes based on CEMMDAN as well as VMD have been derived for time series feature extraction, and decomposition methods such as SVMD [57] and ICEEMDAN [58] have been applied in practice in many fields, providing more ideas for our future research.
We will continue to focus on the prediction benefits of long-term prediction techniques such as TimesNet in common meteorological data and further try to improve the prediction ability of El Niño/La Niña events using multidimensional data, especially the complex problem that intense El Niño/La Niña events are challenging to be accurately predicted in statistical models.Considering other factors affecting La Niña/El Niño events (e.g., climate change, rainfall, cloud thickness, etc.) may also improve the prediction accuracy.However, it also implies more consumption, and how to find the optimal balance of input factors deserves further in-depth research.Furthermore, the method proposed in this paper can be further applied to data in other fields, where prediction can be attempted for data with complex cycles, non-stationary or symmetric nature.
Currently, we only use TimesNet-ARIMA models for single-order forecasting.The advantage of this approach is that this simple, non-reliance on extensive historical data can be a good reference when there is insufficient granularity in the data set or when the study area is macroscopic.At the same time, TimesNet can handle two-dimensional features, which also means that spatio-temporal data forecasting can be directly extended to TimesNet, but how to apply two-dimensional data to CEEMDAN and ARIMA requires further research and discussion in the future.

APPENDIX A
In this section we will show the algorithmic flow of EMD and CEEMDAN, and since the EMD algorithm is included in CEEMDAN, we will show the two separately.

APPENDIX B
We used Autofromer for the equivalent dataset in our actual tests, and the results can be viewed in Table 12 and Table 13.Although AutoFormer is very effective in long-order prediction [48], it did not achieve satisfactory results in the short  term in the dataset we tested and was closer to TimesNet and ConvLSTM in terms of RMSE metrics, but the MAE metrics differed significantly from the optimal values.

APPENDIX C
This table is a further addition to Table 4 (adding Autoformer).Based on the data in the table, Autoformer does not exploit its strengths in short-term forecasting metrics.
steps: a) Running CEEMDAN on the original Niño3.4index sequence and getting several ''Intrinsic Mode Functions'' (IMFs).b) Normalizing the obtained IMFi components and the residual r. c) Inputting the IMF1-IMFk components into the TimesNet model for forecasting and the remaining IMFk+1 -IMFi and the residual r into the ARIMA model for prediction.d) The i+1 predicted values are obtained, and the inverse normalization is performed and summed to obtain the prediction series of the Niño3.4index.

FIGURE 2 .
FIGURE 2. The hybrid model's main architecture: comprises three components, namely CEEMDAN, TimesNet, and ARIMA.TimesNet and ARIMA generate predictions for different IMFs, and the combination of all predictions results in the final prediction sequence.

FIGURE 6 .
FIGURE 6. Prediction effects of TimesNet and ARIMA for different IMFs: (a) For the first three IMFs, TimesNet outperforms ARIMA.(b) For IMF4, ARIMA can already fit the true value well, while TimesNet fluctuates under the true value.(c) The prediction effect of residuals, ARIMA can perfectly fit the real data, while TimesNet does not work well.(d) Using the proposed model for forecasting in 2012, the results can outperform the TimesNet and ARIMA models used alone.
2229 • C, RMSE = 0.3140 • C).The MAE of the model led the rest of the models in 2013, 2016, 2019, and 2020.The RMSE of the model leads the rest of the models in 2013, 2016, 2019, 2020, and 2022.

FIGURE 8 .
FIGURE 8. Comparison of the prediction effect of the proposed approach on 2013-2022 with other models.

Niño3. 4
index change is stable in 2016 and 2020, the proposed model achieves the optimal prediction in both years (MAE = 0.134 • C, MAE = 0.2298 • C), indicating that the proposed model is more capable of predicting event shifts (Especially the shift from El Niño to La Niña event).

FIGURE 9 .
FIGURE 9. Comparison of prediction effects of several models in 2019-2020: The results of the comparison model are marked in the figure with curves of different colors, and the two dashed lines in the figure are marked with the 0.5 and −0.5 bounds.

FIGURE 10 .
FIGURE 10.Correlation analysis of several models predicting lag order and RMSE.

FIGURE 11 .
FIGURE 11.Relationship between model training time/RMSE and length of training set.

FIGURE 12 .
FIGURE 12. Results of the proposed model in the Niño3.4forecast for 2023.

TABLE 1 .
Statistical properties of the data in the training and validation sets.

TABLE 2 .
Main statistical characteristics of each IMFs and residual.

TABLE 3 .
The ability of each model in predicting the Niño 3.4 index (2013-2022): the LSTM in the table refers to the ConvLSTM, and the input values for the predictions are obtained from CEEMDAN processing; the bolded values in red indicate that the model has the smallest prediction error indicator in a single year, and the data underlined in blue indicate the second smallest error indicator in that year.

TABLE 4 .
Cross-correlation coefficients of each model in predicting the Niño 3.4 index.(In this table, the value bolded in red indicates the maximum value for that row, and the value marked by the underlined number in blue is the second larges t.The p-values are indicated by * , where * * * indicates p<0.01 and * * indicates p<0.05.)

TABLE 5 .
Comparison of predictive accuracy metrics of proposed models for El Niño/La Niña events.

TABLE 6 .
MAE of dynamic models, statistical models and Deep learning model approaches in predicting El Niño/La Niña events.

TABLE 7 .
RMSE of dynamic models, statistical models and Deep learning model approaches in predicting El Niño/La Niña events.

TABLE 8 .
List of TimesNet parameter settings.

TABLE 9 .
Comparison of TimesNet predictions with different losses.

TABLE 12 .
MAE of each model in predicting the Niño 3.4 index (Including Autoformer).Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

TABLE 13 .
RMSE of each model in predicting the Niño 3.4 index (Including Autoformer).

TABLE 14 .
Cross-correlation coefficients of each model in predicting the Niño 3.4 index (Including Autoformer).(In this table, the value bolded in red indicates the maximum value for that row, and the value marked by the underlined number in blue is the second largest.The p-values are indicated by * , where * * * indicates p<0.01 and * * indicates p<0.05.)