The monthly average SST data were obtained from the UK Met Office Hadley Centre for the region of 30∘ S–30∘ N, 120∘ E–90∘ W. The gridded 1∘×1∘ Met Office Hadley Centre sea ice and SST data set (HadISST1; Rayner et al., 2003) includes both in situ and available satellite data. The sea areas provide important information on ocean–atmosphere coupling in the east and west Pacific Ocean and the El Niño/La Niña events. The reanalysis data, zonal winds and sea level pressures were obtained from the National Centers for Environmental Prediction (NCEP) and the National Center for Atmospheric Research (Kalnay et al., 1996). The sea surface height (SSH) field was obtained from Simple Ocean Data Assimilation (SODA) data (Carton and Giese, 2008). Outgoing longwave radiation (OLR) was obtained from the National Oceanic and Atmospheric Administration (NOAA) satellites, at a resolution of 0.5∘×0.5∘ (Liebmann and Smith, 1996). The Southern Oscillation Index (SOI) data were obtained from the Climate Prediction Center (CPC). The time series of all data were from January 1951 to December 2010 (720 months in total).

The SSTA field can be calculated from the SST field and can be deconstructed into time (coefficients)–space (structure) using the empirical orthogonal function (EOF) method. Detailed information on the EOF method can be seen in the related references (Dommenget and Latif, 2002). We have used covariance matrix, because the covariance matrix was selected to diagnose the primary patterns of covariability in the basin-wide SSTs, rather than the patterns of normalized covariance (or correlation matrix).

We used the smoothing function with MATLAB to smooth the SSTA field before the EOF deconstruction, which is moving from five points, mainly filtering out some noise points and outliers. Then, an EOF analysis of smoothed anomalies was performed, and the first two SSTA EOFs are shown in Fig. 1a and c. The principal component (PC) time series corresponding to the first and second EOFs are shown in Fig. 1b and d. The first EOF pattern, which accounted for 61.33 % of the total SSTA variance, represented the mature ENSO phase (El Niño or La Niña), and the corresponding PC time series was highly correlated (with a correlation coefficient of 0.85) with the cold tongue index (SST anomaly averaged over 4∘ S–4∘ N, 180–90∘ W) over the whole period. The second EOF, accounting for 14.52 % of the total SSTA variance, indicated the ENSO signal beginning to enhance. Compared with the first mode, these were slightly attenuated in terms of the scope and intensity. The above analysis is similar to the EOF analysis of the SSTA field in the previous studies (Johnson et al., 2000; Timmermann et al., 2001). This indicates that the front two variance contribution modes can describe the main characteristics of the SSTA field and El Niño/La Niña. Therefore, we can choose the T1, T2 time series EOF decomposition modes as the modeling objects.

Considering the complexity of computation, the amount of variables in the equations of our model cannot be too large (usually three or four variable are best). This has been explained in our previous studies (Zhang et al., 2006, 2008).  If there are more than four variables in the modeling equation, it will cause the amount of parameters, such as a1, a2, …, an, b1, b2, …, bn, …, to be too large. The huge computation makes it difficult to be precisely modeled. Thus, the total number of parameters in the model of five variables was 102, which may cause an overfitting problem. Hence, when we selected the model of five or six variables, it entailed large amounts of computation that made precision difficult, and too many parameters might cause an overfitting phenomenon. If we choose only two or even fewer variables, the forecast performance is poor too. Too few variables cause reconstructed parameters to be too small, resulting in amounts of important information missing out in the model. Thus, four variables are best for modeling dynamically and accurately. Because we have chosen two time series in Sect. 2.2 as the modeling objects, now we should select the other two ENSO intensity impact factors.

The ENSO intensity impact factor is an important issue in ENSO prediction. Previous studies have been completed in this area, which found that teleconnection patterns, temperature, precipitation, wind and SSH may affect ENSO strength. For example, Trenberth et al. (1998) noted that Pacific North American teleconnection (PNA), SOI and OLR in the Pacific Intertropical Convergence Zone (ITCZ) are all closely related to ENSO. Webster (1999) pointed out that, after 1970, the Indian Ocean dipole (IOD) was not only affected by ENSO but also affected the strength of ENSO (Ashok et al., 2001). Yoon and Yeh (2010) reported that the Pacific Decadal Oscillation (PDO) disrupts the linkage between El Niño and the following Northeast Asian summer monsoon (NEASM) by inducing the Eurasian pattern in the mid- to high latitudes. The vast majority of studies (Tomita and Yasunari, 1996; Zhou and Wu, 2010; Kim et al., 2017) have concentrated on the impacts of ENSO on the East Asian winter monsoon (EAWM). During the EAWM season, ENSO generally reaches its mature phase and has the most prominent impact on the climate. B. and C. Wang et al. (1999) suggested that the zonal wind factors in the eastern and western equatorial Pacific play a critical role in the phase of transition of the ENSO cycle, which could excite eastward propagating Kelvin waves and affect the SSTA in the equatorial Pacific. Zhao et al. (2012) analyzed the characteristics of the tropical Pacific SSH field and its impact on ENSO events.

Based on the above analysis, we have selected nine factors, which may be closely related with the ENSO index (Niño3.4).

The zonal wind in the eastern equatorial Pacific factor (u1) was calculated as the grid-point average of zonal wind in the area of 5∘ S–5∘ N, 150–90∘ W.

Table 1 shows that SOI and EAWMI have the stronger correlation with the front two time series (T1, T2) than the other seven factors. The results are also consistent with previous research (Clarke and Van Gorder, 2003; Drosdowsky, 2006; Zhang et al., 1996; Wang et al., 2008; Yang and Lu, 2014). Therefore, the first time series (T1), the second time series (T2), SOI and EAWMI will be selected as prediction model factors.

The training sample for the model was from January 1951 to April 2008. Here, from Eq. (3), the forecast results using T1, T2, SOI and EAWMI factors can be calculated; this is called a step-by-step forecast.

When the retrospective order p is confirmed, step-by-step forecasts can be carried out. For example, when the T1, T2, SOI and EAWMI values of May 2008 were forecast, yi was obtained from the previous p+1 time of T1, T2, the SOI and the EAWMI data, and Fi(x1i,x2i,x3i,x4i) was obtained from the previous p times of T1, T2, the SOI and the EAWMI data. All four equations were integrated simultaneously. Taking these in Eq. (3), we can get the T1, T2, SOI and EAWMI values of May 2008, which these can be taken as the initial values for the next prediction step. Then, the T1, T2, SOI and EAWMI values from June 2008 and so on can be generated.

As in Sect. 3, the model's skill should be further assessed by cross-validated retroactive hindcasts of the time series. Because our step-by-step forecasts need the earlier data of seven observations (p+1=7), we can obtain cross-validated retroactive hindcast results of T1 and T2 from August 1951 to December 2010, as shown in Fig. 5.

From Fig. 5, the forecast performance of T1 and T2 was good. The CCs of T1 and T2 were 0.7124 and 0.7036, respectively. The MAPEs of T1 and T2 were small at only 19.57 and 19.79 %, respectively. The peaks and valleys of T1 and T2 were also forecasted accurately. The forecast results indicated that the cross-validated retroactive hindcast results of T1 and T2 were close to the observed values. Compared to Fig. 3, the improved model had better forecast abilities than the original model.

Many researchers (Zhang et al., 2003b; Smith, 2004) have used the Oceanic Niño Index (ONI) which is used by the US NOAA Climate Prediction Center to determine the El Niño and La Niña years. It was defined that when the ONIs of 5 consecutive months in winter are all more than 0.5 (less than -0.5), it is an El Niño (La Niña) year. Based on the above criterion, we can divide the total 60 years (1951–2010) into three categories. It includes the 18 examples of El Niño years (such as 1958, 1964, 1966, etc.), 22 examples of La Niña years (such as 1951, 1955, 1956, etc.) and the remaining 20 experiments of neutral years. Since the details in Fig. 5 are not clear, we list the forecast results of 60 experiments (including 18 El Niño examples, 22 La Niña examples and 20 neutral examples) in Table 3.

From Table 3, the average CC of both T1 and T2 of 60 experiments within 6 months was more than 0.84, and MAPE was less than 8 %. The average CC within 12 months was more than 0.74, and MAPE was less than 12 %. According to the literature (Tofallis, 2015), when MAPE was less than 15 %, it meant the error was not great and the forecast results were good. Obviously, the forecast results of the El Niño/La Niña experiments were a little worse than those of neutral examples, which means the forecast ability of our model for the abnormal situation was a little worse than that for the normal situation. However, even for El Niño/La Niña experiments, the average CC was still more than 0.7 and MAPE was less than 15 %, which means the error was not too large and was still within an acceptable range.

When we obtained the forecast results of the time coefficient series T1 and T2, we submitted them into the following equation to reconstruct the forecast SSTA field: x^t=∑n=12En⋅Tnt,t=1,2,…,12, where En, Tnt are the EOF space fields and forecast time coefficients, respectively, and x^ij is the forecast SSTA field reconstructed by EOF.

After reconstruction of the space mode (treated as constant) and time coefficient series (model prediction), the forecast of the SSTA fields was obtained based on the forecast results of T1 and T2 in Sect. 5.2. For economy of space, we cannot draw all of the forecasted SSTA fields, so we selected a strong El Niño event (December 1997), a strong La Niña event (December 1999) and a neutral event (November 2002) as examples.

Figure 6 shows the forecast SSTA field during a strong El Niño event. From the actual SSTA field in December 1997 (Fig. 6a), an obvious warm tongue structure occurred in the area of 10∘ S–5∘ N, 90–150∘ W in the eastern equatorial Pacific, and a warm anomalous distribution arose in the west Pacific, which indicated a weak El Niño event. The forecasted SSTA field of December 1997 is shown in Fig. 6b. Although the range of warm tongue was a litter bigger than the actual situation, the forecast shape was similar to the actual field and also the contour lines were similar. The average MAPE between the forecast field and the actual field is 8.56 %, which was controlled within 10 %. The forecast results of the improved model event were quite good for the El Niño event.

Figure 7 shows the forecasted SSTA field of a strong La Niña event. From the actual SSTA field in December 1999 (Fig. 7a), an obvious cold pool occurred in the area of 10∘ S ∼ 10∘ N, 120∘ W ∼ 180∘ W in the equatorial Pacific, which covered the Niño3.4 area. This SSTA field presented a strong strength La Niña event. The forecast SSTA field from December 1999 is shown as Fig. 7b. Although the strength of the cold pool was weaker than the actual situation, the forecast shape was similar to that of the actual field. The average MAPE between the forecast field and the actual field was 9.69 %. The errors were larger than those of the El Niño event, but they can be controlled within 10 %, which is acceptable.

Figure 8 shows the forecasted SSTA field of a neutral event. From the actual SSTA field in November 2002 (Fig. 8a), a warm pool occurred in the area of 10∘ S–10∘ N, 120–180∘ W in the equatorial Pacific, which covered the Niño3.4 area. However, the warm pool was small and weak, which represented a neutral event. The forecasted SSTA field from November 2002 is shown in Fig. 8b. Comparing Figs. 6–8, we can see that the forecasted SSTA field of a neutral event was a little worse than that of the El Niño and La Niña events. The forecasted shape of the SSTA field basically described the actual situation, but the warm pool in the Niño3.4 area was stronger and bigger than that of the actual situation, which indicated a borderline El Niño event. The average MAPE between the forecasted field and the actual field was 14.50 %, which was big but can be accepted.

We obtained the average values of MAPE of 18 El Niño events, 22 La Niña events and 20 neutral events, which were 9.52, 9.88 and 14.67 %, respectively, representing a good SSTA field forecasting ability of our model.

The ENSO index can be represented as the SSTA in the Niño3.4 region (5∘ N–5∘ S, 120–170∘ W) and the ENSO index forecast was the 3-month forecast (Barnston et al., 2012). So we also can pick up the ENSO index from the above-forecasted SSTA field. The forecast results of the ENSO index within 20 months can also be obtained. The definition of lead time can be seen in the reference (Barnston et al., 2012). Therefore, similar to the forecast experiment in Sect. 5.1, a succession of running 3-month mean SST anomalies with respect to the climatological means for the respective prediction periods, averaged over the Niño3.4 region, can be obtained, as demonstrated in Fig. 9.

The evaluation criterion of the ENSO index is the temporal correlation (TC); its definition and specific calculation steps can be seen in the literature (Kathrin et al., 2016; Nicosia et al., 2013). The TC is often used to measure the prediction effect of the ENSO index. For example, Barnston et al. in 2012 also used the TC to compare the forecast skill of 21 real-time seasonal ENSO models.

The forecast results within lead times of 18 months are shown in Fig. 9, which demonstrate that the forecast results of the ENSO index are good. Within the lead time of 12 months, the TC was 0.8985 and the MAPE value was small at only 8.91 %. In addition, the borderline La Niña event in 2008–2009 was predicted well. After lead times of 12 months, forecasts began to diverge and the errors started to increase. Although the TC remained approximately 0.61, MAPE reached 18.58 %. Therefore, a moderate-strength El Niño event that occurred in 2009/2010 was not predicted.

We should give more examples to test the ENSO prediction ability of our model. As in Sect. 5.3, we can divide 60 examples into three types, which are examples of the El Niño year, La Niña year and neutral year. Finally, we can obtain the forecast results of different types of examples in different lead times, as shown in Table 4.

From Table 4, the average TC of 60 experiments was 0.712, and the average MAPE was 7.62 % within 12 months for all seasons of lead time, which indicates that the overall ENSO forecast ability of our model was good. The forecast results of the El Niño examples were significantly worse than those of La Niña examples, while the forecast results of La Niña examples were significantly worse than those of neutral examples, which show the model forecast ability of the abnormal state was worse than the normal state of the ENSO index. Even for the forecast results of El Niño examples, the average TC was still above 0.6 and the average MAPE can be controlled below 10 %, which means the forecast results were still in the acceptable range. Our model not only accurately predicted the stronger El Niño and La Niña phases but also the neutral states.

The ENSO forecast often had a spring predictability barrier (Webster, 1999), which was most prominent during decades of relatively poor predictability (Balmaseda et al., 1995). To test our model, the skill should be computed over the entire time series and separately for seasonal subsets of the time series. From Table 4, we can see that although the forecast results of the present model in the spring were worse than in the autumn, the margin was not high, which means the model can overcome the “spring predictability barrier” to some extent.

Barnston et al. (2012) compared many ENSO forecast models. Based on his research, we selected four high-quality dynamical models, including ECMWF, JMA, the National Aeronautics and Space Administration Global Modeling and Assimilation Office (NASA GMAO) and the National Centers for Environmental Prediction Climate Forecast System (NCEP CFS; version 1). Two high-quality statistical models also were selected, including the University of California, Los Angeles Theoretical Climate Dynamics (UCLA-TCD) multilevel regression model and the NOAA/NCEP/CPC constructed analogue (CA) model. The details of the above models can be found in these references (Reynolds et al., 2002; Luo et al., 2005; Barnston et al., 2012).

We then compared the forecast ability of the above six models with that of our model. All of the experiments of our model and six other models were conducted under the same conditions using the same historical data for modeling and the same initial values to forecast. On the CPC website, there are detailed explanations of the six models' training samples and the initial values. So we do not need to install all these models on their own machines and run them for forecasting. We just made training samples, and initial values of our model were the same as those of the six selected models. At an 8-month lead time, the TC of our model for all seasons combined was 0.613 (Fig. 10). In brief, the forecast ability of the ECMWF model was slightly better than that of our model but the ability of the other five models was worse than that of our model. However, in regard to the forecast length, the TC within 12 months of our model is greater than 0.6, which was superior to the ECMWF model. In addition, the forecast results of the UCLA-TCD model and the CPC CA model reduced quickly after 5-month lead times, so the forecast ability of our model was more stable than theirs.

The root mean square error (RMSE) was also examined to assess the performance of discrimination and calibration. Barnston et al. (2012) believed that all seasonal RMSE values contributed equally to a seasonally combined RMSE. So we drew Fig. 11 to show seasonally combined RMSE.

From Figs. 10 and 11, we can see the highest correlation tends to have lower RMSE. So the RMSE of our model was slightly higher than that of the ECMWF model, but it was much lower than those of the other five models. Figures 10 and 11 show the average TC and RMSE of the 240 experiments compared with six mature models and cover a variety of different types of ENSO and different lead times. So those samples should be really representative.

A new forecasting model of the SSTA field was proposed based on a dynamical system reconstruction idea and the principle of self-memorization. The approach of the present paper consisted of the following steps.

The SST field can be time (coefficients)–space (structure) deconstructed using the EOF method. Take T1, T2, SOI and EAWMI and consider them as trajectories of a set of four coupled quadratic differential equations based on the dynamical system reconstruction idea. The parameters of this dynamical model were estimated using a GA.

The forecast results of the dynamical model can be improved by the self-memorization principle. The memory coefficients in the improved self-memorization model were obtained using the GA method.

The long-term step-by-step forecast results and cross-validated retroactive hindcast results of time series T1 and T2 are all found to be good, with a CC of approximately 0.80 and a MAPE of less than 15 %.

The improved model was used to forecast the SSTA field. The forecasted SSTA fields of three types of events are accurate. Not only is the forecast shape similar to the actual field but also the contour lines are similar.

The improved model was also used to forecast the ENSO index. The average TC of 60 examples within 12 months is 0.712, and the MAPE value is small at only 7.62 %, which proves that the improved model has better forecasting results of the ENSO index. Although the forecast results of the model in the summer were worse than in the winter, the margin was not high, which means that the model can overcome the spring predictability barrier to some extent. Finally, compared with the six mature models, the new dynamical–statistical forecasting model has a scientific significance and practical value for the SST in the eastern equatorial Pacific and El Niño/La Niña event predictions.

L'Heureux et al. (2013) reported that using different data sets and time periods, the second EOF is not stable, entirely due to the strong trend. So we need to do more experiments to prove that we choose the second mode of EOF to be appropriate and whether different time periods will make our forecast unstable or not. Our original data are the monthly average SST data from January 1951 to December 2010 (60 years). We will increase the length of the data for 20 years (January 1931–December 2010) and for 10 years (January 1941–December 2010), and decrease the length of the data for 10 years (January 1961–December 2010) and for 20 years (January 1971–December 2010). Then, we will use the same method to reconstruct a model and forecast the ENSO index as in Sect. 5.4. The results show that, in the 60 experiments, the difference among forecast results of both TC and MAPE of five different sample data is lower, and no abnormal changes that are suddenly worse or better appear. All this indicates that using different data sets and time periods may have a certain impact on the pattern of the second EOF, but the impact on our forecast is not great and it will not make our forecast unstable.

Actually, the amount of variables and which variables are used in our model become key issues to be resolved. We have developed a complex coupled model of four-factor differential equations, so we are more concerned with the correlations between each of them. The correlation must be considered as an important criterion to select the factors, but in order to further verify the correctness of the selection criterion, we have carried out the prediction experiments (the 60 cross-validated retroactive hindcast experiments of the ENSO index for all seasons combined at lead times of 8 months) of different variables.

We can see that for all the forecast results of the models of different variables, the prediction results of T1, T2 and SOI are the best among the three factors, and the prediction results of T1, T2, SOI and EAWMI are the best among the four factors. However, the prediction results of T1, T2, SOI and EAWMI are the best among all the factors, which proves that our selection factors are correct. In our previous study (Hong et al., 2015), the model of the western Pacific subtropical high was established by using the correlations as a criterion to select factors, and their forecast results are also good. Now, we use the correlations as a criterion to select factors also in line with our previous research.

Based on the definition of overfitting and the previous studies (Golbraikh et al., 2003; Everitt and Skrondal, 2010), there is no evidence that more parameters will result in overfitting. We can judge whether a model is overfitting or not by the accuracy of prediction results of independent samples (Golbraikh and Tropsha, 2002; Qin and Li, 2006).

In the sample training, our model does not purposely pursue the high degree of the training sample fitting and improve the effectiveness of the independent generalization. In fact, in our paper, the forecast results of the cross-validated retroactive hindcasts (Sect. 5.2) and the independent sample validation (Tables 3 and 4) are both good. Especially in the independent sample validation of the ENSO index (Table 4), we have carried out the 240 independent sample validation predictions of four seasons of different ENSO events, and the coverage of independent samples test is very wide. Moreover, compared with six mature prediction models, the forecast results of our model are also good, which proves that the overfitting problem does not exist in our model. So according to the definition of overfitting, we can say the overfitting phenomenon does not exist in our model.

Compared with the original model, the reasons why the improved model has good forecast results and can overcome the spring predictability barrier to some extent are as follows. Recently, many studies have pointed out that spring is the most unstable season of the air–sea interaction and the error is likely to develop or grow in the spring, resulting in the spring predictability barrier (Zhang et al., 2012; Philander et al., 1992). When the original model uses the indexes in summer as the initial values to predict, the SOI factor representing the air–sea interaction is most unstable in the spring and the EMWMI factor does not have much influence on ENSO in summer, so the forecast results using the indexes in summer as the initial values are certainly much worse than those using the indexes in the winter as the initial values. That is why our original model does not overcome the spring predictability barrier.

However, the introduction of the self-memorization dynamics principle can help our model overcome the spring predictability barrier to some extent. Although the lead time is still summer (such as JJA), the information of the initial value actually contains the previous p+1 month (in this case, p=6), which contains the information of the previous 7 months, including the information of the T1, T2, SOI and EMWMI factors in winter (January, February), spring (March, April, May) and summer (June and July). From the dynamical analysis, in this situation, the information and interaction relationship of four factors has been accumulating for a long period (from winter to summer), containing many air–sea interaction processes, and the winter monsoon contains abnormal information, so the forecast results of our improved model will be much better than the original model which simply uses only one initial value. That is why the improved model overcomes the spring predictability barrier to some extent.

The forecast results of our model are good, but it still has some problems:

The inclusion of these terms and the physical processes these terms represent in Eq. () are important, especially for the discussion of dynamical characteristics of the dynamical model. However, now it is difficult to give a clear meaning. Now, the main work of our paper is the prediction experiments of the model. For the reasons of time and length, this paper mainly discusses the prediction results of the model. The physical processes do these terms represent and the discussion of the dynamical characteristics of the model will be the focus of our next work. Before this, we have also used Takens' delay embedding theorem to reconstruct the dynamical model of the western Pacific subtropical high (WPSH). Based on the reconstructed dynamical model, dynamical characteristics of WPSH are analyzed and an aberrance mechanism is developed, in which the external forcings resulting in the WPSH anomalies are explored, which have been published (Hong et al., 2016). We also study the bifurcation and catastrophe of the west Pacific subtropical high ridge index of a nonlinear model (Hong et al., 2017). Based on our previous method and work, our next work is to analyze the physical processes and the dynamical characteristics of the SST field.