 {{\mathrm{N3}}{\mathrm{.4}}_i = c_3{\mathrm{ACE}}^ \ast _{i-3} + g_3,{\mathrm{N3}}{\mathrm{.4}}_{i-3} \le - {\mathrm{0}}{\mathrm{.5}},{\mathrm{ACE}}({\mathrm{10}}^\circ - {\mathrm{25}}^\circ {\mathrm{N}},{\mathrm{130}}^\circ - {\mathrm{155}}^\circ {\mathrm{E}})} \end{array}} \right.,$$
(8)

where N3.4i is N3.4 at the ith month, $${\mathrm{ACE}}^ \ast _{i-3}$$ the standardized ACE anomalies at the (i−3)th month, and c1, c2, c3 and g1, g2, g3 are the regression coefficients and constants. Similarly, three ACE + N3.4 models are built, as follows:

$$\left\{ {\begin{array}{*{20}{c}} {{\mathrm{N3}}{\mathrm{.4}}_i = c_4{\mathrm{ACE}}^ \ast _{i-3} + d_1{\mathrm{N3}}{\mathrm{.4}}^ \ast _{i-3} + g_4,{\mathrm{N3}}{\mathrm{.4}}_{i-3} \ge {\mathrm{0}}{\mathrm{.5}},{\mathrm{ACE}}({\mathrm{10}}^\circ - {\mathrm{20}}^\circ {\mathrm{N}},{\mathrm{135}}^\circ - {\mathrm{170}}^\circ {\mathrm{E}})} \\ {{\mathrm{N3}}{\mathrm{.4}}_i = c_5{\mathrm{ACE}}^ \ast _{i-3} + d_2{\mathrm{N3}}{\mathrm{.4}}^ \ast _{i-3} + g_5,{\mathrm{|N3}}{\mathrm{.4}}_{i-3}| < {\mathrm{0}}{\mathrm{.5}},{\mathrm{ACE}}({\mathrm{10}}^\circ - {\mathrm{20}}^\circ {\mathrm{N}},{\mathrm{135}}^\circ - {\mathrm{155}}^\circ {\mathrm{E}})} \\ {{\mathrm{N3}}{\mathrm{.4}}_i = c_6{\mathrm{ACE}}^ \ast _{i-3} + d_3{\mathrm{N3}}{\mathrm{.4}}^ \ast _{i-3} + g_6,{\mathrm{N3}}{\mathrm{.4}}_{i-3} \le - {\mathrm{0}}{\mathrm{.5}},{\mathrm{ACE}}({\mathrm{10}}^\circ - {\mathrm{25}}^\circ {\mathrm{N}},{\mathrm{130}}^\circ - {\mathrm{155}}^\circ {\mathrm{E}})} \end{array}} \right.$$
(9)

Where $${\mathrm{N3}}{\mathrm{.4}}^ \ast _{i-3}$$ is the standardized N3.4 at the (i−3)th month, and c4, c5, c6, d1, d2, d3 and g4, g5, g6 are the regression coefficients and constants.

Correction to the ACE  +  N3.4 model: Analysis shows that the role of ACE is reduced in the ACE + N3.4 model owing to the persistence of SST, especially in the development stage of ENSO. To reduce this error, the ACE + N3.4 model is corrected using generalized cross-validation (GCV)56. In this case, leave-one-out cross-validation (LOO-CV)57,58 is employed. A brief description of the methodology is as follows.

Step 1. For a given time series with length L, one time point is chosen to be a hindcasting point, and the rest of the time series (length: L−1) is used to construct the forecast model with regression analysis.

Step 2. Repeat step 1; i.e. choose another time point to be a new hindcasting point, resample the residuals, and obtain a new forecast model. Repeat this process for each time point of the given time series. This will result in the ensemble hindcast.

This method will provide a new ensemble hindcast of N3.4 using the predictor variables $$A{\mathrm{CE}}^ \ast$$ and $${\mathrm{N3}}{\mathrm{.4}}^ \ast$$ The new ensemble hindcast of N3.4 produced by LOO-CV is mainly affected by the persistence of SST. Hence, by comparing this ensemble hindcast with the observations during the training period, we can obtain the correction ε1 that will be used to correct the ACE + N3.4 models. And in this study, ε1 depends on the base period. Because the effect of TCs on N3.4 varies with ENSO development, the whole N3.4 series is divided into four cases to obtain a more exact forecast: Case A, $${\mathrm{N3}}{\mathrm{.4}}_{i-3} \ge {\mathrm{N3}}{\mathrm{.4}}_{i-4}$$ and $${\mathrm{N3}}{\mathrm{.4}}_{i-3} \ge 0$$; Case B, $${\mathrm{N3}}{\mathrm{.4}}_{i-3} \ge {\mathrm{N3}}{\mathrm{.4}}_{i-4}$$ and $${\mathrm{N3}}{\mathrm{.4}}_{i-3} \, < \, 0$$; Case C, $${\mathrm{N3}}{\mathrm{.4}}_{i-3} \, < \, {\mathrm{N3}}{\mathrm{.4}}_{i-4}$$ and $${\mathrm{N3}}{\mathrm{.4}}_{i-3} \ge 0$$; Case D, $${\mathrm{N3}}{\mathrm{.4}}_{i-3} \, < \, {\mathrm{N3}}{\mathrm{.4}}_{i-4}$$ and $${\mathrm{N3}}{\mathrm{.4}}_{i-3} \ge 0$$. Then, four different corrected values (ε2) are obtained for the four cases during the training period. Thus, the ACE+N3.4 model is built as follows:

$$\left\{ {\begin{array}{*{20}{c}} {{\mathrm{N3}}{\mathrm{.4}}_i = c_4{\mathrm{ACE}}^ \ast _{i-3} + d_1{\mathrm{N3}}{\mathrm{.4}}^ \ast _{i-3} + g_4 + \varepsilon,{\mathrm{N3}}{\mathrm{.4}}_{i-3} \ge {\mathrm{0}}{\mathrm{.5}},{\mathrm{ACE}}({\mathrm{10}}^\circ - {\mathrm{20}}^\circ {\mathrm{N}},{\mathrm{135}}^\circ - {\mathrm{170}}^\circ {\mathrm{E}})} \\ {{\mathrm{N3}}{\mathrm{.4}}_i = c_5{\mathrm{ACE}}^ \ast _{i-3} + d_2{\mathrm{N3}}{\mathrm{.4}}^ \ast _{i-3} + g_5 + \varepsilon,{\mathrm{|N3}}{\mathrm{.4}}_{i-3}| < {\mathrm{0}}{\mathrm{.5}},{\mathrm{ACE}}({\mathrm{10}}^\circ - {\mathrm{20}}^\circ {\mathrm{N}},{\mathrm{135}}^\circ - {\mathrm{155}}^\circ {\mathrm{E}})} \\ {{\mathrm{N3}}{\mathrm{.4}}_i = c_6{\mathrm{ACE}}^ \ast _{i-3} + d_3{\mathrm{N3}}{\mathrm{.4}}^ \ast _{i-3} + g_6 + \varepsilon,{\mathrm{N3}}{\mathrm{.4}}_{i-3} \le - {\mathrm{0}}{\mathrm{.5}},{\mathrm{ACE}}({\mathrm{10}}^\circ - {\mathrm{25}}^\circ {\mathrm{N}},{\mathrm{130}}^\circ - {\mathrm{155}}^\circ {\mathrm{E}})} \end{array}} \right.$$
(10)

where ε depends on the case, ε = ε1 + ε2.

Running holdout method: This method only differs from the holdout method in its use of a variable-length training period that changes with the hindcasting time point. That is, the required set of regression coefficients and constants is based on the time series before each hindcasting target point. The base period is the shortest training period. Other processes are as described for the holdout method.

Leave-p-out Cross-Validation (LPO-CV): Leave-p-out is similar to LOO-CV, but the single time point is replaced by a time interval. A brief description of the methodology is as follows.

Step 1. A given time series with length L is divided into time intervals of length m. If L is not exactly divisible by m, we suppose that the whole time series can be divided into n parts and some (<m) remainder time points that cannot be used as the hindcasting object.

Step 2. One part is chosen to be a hindcasting object, and the rest of the time series (length: Lm) is used to construct a forecast model according to regression analysis.

Step 3. Repeat step 2, using another part as a new hindcasting object, resample the residuals, and obtain a new forecast model. Repeat this process n times. This will result in the ensemble hindcast without remainders.

Step 4. Compare the ensemble hindcast with the corresponding observations, and obtain correlation coefficients and root mean square errors. Then, change the value of m, repeat the above steps, and obtain a new ensemble hindcast.

To facilitate modelling, the start of the first time interval is 1970.04.

In the model comparison, the lead