2.3.2. Unequal Weights

In this study, a method is suggested to estimate the unequal weights based on the rank sum from each climate simulation run for the analytical hierarchy process. It is called unequal weights. The rank sum is calculated using the statistical indices. The statistical indices are Nash–Sutcliffe efficiency with logarithmic value (ln(Nash)), root-mean-square error (RMSE) and Nash–Sutcliffe efficiency (Nash) for the studied domain. They are basically quantified by measuring the difference between the observed data and the outputs of regional climate models forced by lateral and surface boundary conditions from the European Centre for Medium-Range Weather Forecasts at the monthly scale. The Nash–Sutcliffe efficiency with logarithmic values ln(Nash) is selected because it can be added to expect a better quantification of the performance in different conditions (maximum and minimum values) [41]. The other remaining value is widely applied in hydrometeorological fields. The formulations used to compute the goodness-of-fit statistical indicators for each climate simulation run are presented as follows:

$$\text{RMSE} = \sqrt{\frac{1}{n} \sum\_{i=1}^{n} (\mathbf{F}\_i - \mathbf{O}\_i)^2} \tag{2}$$

$$\ln\left(\text{Nash}\right) = 1 - \frac{\sum\_{1}^{n} \left(\ln\text{O}\_{\text{i}} - \ln\text{F}\_{\text{i}}\right)^{2}}{\sum\_{1}^{n} \left(\ln\text{O}\_{\text{i}} - \ln\overline{\text{O}}\right)^{2}}\tag{3}$$

$$\text{Nash} = 1 - \frac{\sum\_{\mathbf{i}}^{n} (\mathbf{O\_{i}} - \mathbf{F\_{i}})^{2}}{\sum\_{\mathbf{i}}^{n} (\mathbf{O\_{i}} - \overline{\mathbf{O}})^{2}} \tag{4}$$

where n denotes number of months, Fi and Oi represent simulated and observed monthly data, respectively. The method of unequal weights is briefly expressed with the following steps with an assumption of N climate simulations:

(1) Calculation of the statistical indices on the basic of the historical observations and climate simulations from regional climate models forced by the reanalysis data of the European Centre for Medium-Range Weather Forecasts during 1989–2008. Each climate simulation receives a rank from 1 to N depending on the levels of perfect score for each statistical index, starting with the best as 1 and the worst is N. As an example, if the RMSE index of the ith climate model has the best score (the perfect score of RMSE is zero), the received rank is 1. Then, an ensemble rank order (r) as an integer number is calculated from the average of the ranks they span for each climate simulation.


2.3.3. Standardized Precipitation Index (SPI)

In this study, the SPI is constructed for multiple timescales ranging from 1 month to 24 months. The calculation of the SPI is separately applied for each month on the basis of the Gamma distribution. The reason for this is that the Gamma distribution is the most frequently used and fits well with daily precipitation in different studies across Vietnam [42]. The alpha and beta parameters of the Gamma probability density function are estimated for each station and for the required multiple timescales. They are used to calculate the cumulative probability distribution of accumulated precipitation. The maximum likelihood solutions [43] and Thom's study [44] are applied to optimally estimate alpha (α) and beta (β). It is especially noteworthy that the reference period adopted to compute the best-fit parameters for the gamma distribution spans 30 years (1986–2015).

A probability transformation is then applied to transform monthly precipitation to a standard normal distribution with a zero mean and standard deviation of one to yield SPI values by preserving probabilities [45]. Figure 2 shows the fitness of the SPI data [46].

**Figure 2.** The probability transformation from fitted gamma distribution to the standard normal distribution.
