*3.1. Calculation of Weights for Each Climate Model*

As the first step, the delta change factor is applied in this study. Then, the unequal weights are added to define weights for each climate simulations as mentioned in Table 1over the whole VG-TB. Table 2 shows the statistical indices for these climate simulations. For RMSE index, the perfect score is zero with a wide range (0 ≤ RMSE <∞). While the perfect score of Nash and ln(Nash) indices is 1 with a wide span (−∞ ≤ Nash < 1; −∞ ≤ ln(Nash) < 1). Based on these score, ranks (i.e., ensemble rank, rank sum) are calculated and presented in Table 3. It is observed that REG/IPSL gives the best score of ln(Nash), RMSE and Nash. This means that REG/IPSL simulation is well matched to observations. As opposed to this, RSM/HadGEM gives the worst score of ln(Nash), RMSE and Nash. Although REG/IPSL is the best, a multiple climate model is always highlighted in the climate simulations and projections due to reducing uncertainty [7]. The reason for this is that the uncertainty comes from lots of factors such as convective cloud parameterization, greenhouse gases scenarios or land surface parameterization. The best rank sum belongs to the model REG/IPSL, followed by REG/ICHEC. In contrary to this, the worst rank sum belongs to the model RSM/HadGEM as shown in Table 3.


**Table 2.** Statistical indices for multiple climate models.


**Table 3.** Ranks of climate models.

A reciprocal matrix between sets of models is created based on the rank sum of climate models. As shown in Table 4, it is illustrated that REG/IPSL is more important than REG/ICHEC and much more important than REG/MPI. Meanwhile, REG/ICHEC is more important than REG/MPI and much important than SNU/HadGEM. The least importance within all considered climate simulations is RSM/HadGEM, followed by REG/HadGEM.


**Table 4.** A reciprocal matrix between sets of models.

After establishing a weights matrix, weights are estimated for each climate model based on the 4th and 5th steps in Section 2.3.2. Weights are 0.344 (REG/IPSL), 0.254 (REG/ICHEC), 0.177 (REG/MPI), 0.116 (SNU/HadGEM), 0.07 (REG/HadGEM) and 0.038 (RSM/HadGEM). The results from this method are against the results from the method of equal weights. Both of them are compared with observations using the statistical indices of mean absolute error (MAE) and RMSE. The pair of equal weights and observations gives results of 99.3 for MAE and 164 for RMSE, meanwhile a pair of unequal weights and observations gives results of 95.4 and 155 for MAE and RMSE, respectively.

The perfect score of MAE and RMSE is zero. In other words, the minimum of the RMSE and MAE is obtained when simulated time series perfectly matches observed time series. As mentioned in the delta change factor approach, an assumption of the delta factor is equal to one (i.e., there is no variation) for each climate runs, and the condition of minimum of the RMSE and MAE is obtained. Climate models with convective precipitation schemes, however, often tend to produce higher precipitation [6,55] compared with the observed data. Moreover, incomplete understanding of natural and its representation is presented within the climate models. Thus, the delta factor assumed by one is unreal, especially in Vietnam where the rain regime is strongly dominated by convective processes. The calculated statistics of MAE and RMSE indicate a better implementation of unequal weights. Climate simulations are closely fit to observations with the unequal weights method as presented in Figure 3.

**Figure 3.** Scatter plot ofthe precipitation simulations with equal and unequal weights minus observations over the whole VG-TB during 1989–2008 (mm/month).
