## 2 Materials and Methods
### Standardized Precipitation Evaptranspiration Index (SPEI)  
[PERSON] et al. [18] proposed a new drought index called the SPEI, which combined the benefits of the SPI [17] being able to respond to multiple time-scales of drought conditions and the Palmer Drought Severity Index (PDSI) [16] accounting for the temperature factor. This proposed index is straightforward to calculate and is based on the original SPI calculation procedure, which uses the difference between the monthly precipitation and monthly potential evapotranspiration (PET). The Thornthwaite method [49], a simple climate water balance method, was used to obtain the SPEI. The calculation procedure for SPEI is as follows:  
1. Calculation of the PET; the monthly PET (mm) is obtained by; \[\text{PET}=16\text{K}(\frac{10T}{I})^{m}\] (5)  
where \(T\) is the monthly mean temperature (\({}^{\circ}\)C), and \(I\) is a heat index, which is calculated as the sum of 12 monthly index values \(I\); \(i\) is calculated from the mean monthly temperature using the formula:  
\[i=(\frac{T}{5})^{1.514} \tag{6}\]  
The letter \(m\) represents a coefficient that depends on \(I\); \(\text{m}=6.75\times 10^{-7}I^{3}-7.71\times 10^{-5}I^{2}+1.79\times 10^{-2}I+ 0.492\); and \(k\) is a correlation coefficient based on the function as follows:  
\[k=(\frac{N}{12})\bigg{(}\frac{NDM}{30}\bigg{)} \tag{7}\]  
where \(NDM\) is the number of days of the month, and \(N\) is the maximum number of sun hours, which is calculated as follows:  
\[N=(\frac{24}{\pi})\omega_{s} \tag{8}\]  
where \(\omega_{s}\) is the hourly angle of the sun rising, which is calculated as follows:  
\[\omega_{s}=\arccos(-\tan\varphi\tan\delta) \tag{9}\]  
where \(\varphi\) is the latitude in radians and \(\delta\) is the solar declination in radians, which is calculated using the formula:  
\[\delta=0.4093\text{sen}(\frac{2\pi I}{365}-1.405) \tag{10}\]where \(J\) is the average Julian day of the month. (b) (Calculation of the water balance \(D_{i}\), which is the difference between the monthly total precipitation and monthly \(PET\): \[D_{i}=P_{i}-PET_{i}\] (11) (c) Calculation of the cumulative difference \(D_{m,n}^{s}(t)\) in the \(m\)th month (January, February, and December) in \(n\) years is calculated: \[D_{m}^{s}(t)=\sum_{i=k-s+1}^{k}D_{i},k=12(t-1)+m\] (12) For the first month, where the order of the \(D_{i}\) series \(i\) is smaller than the time-scale \(s\), \(D_{m}^{s}(t)=\sum_{i=1}^{k}D_{i}\). Note that \(t\) = 1, 2, 3,..., \(n\), represents the yearly index. (d) Fitting of the cumulative D series using a probability density function. The general optimal distribution of the SPEI has not been specified [50]. To fit the GAMLSS model in R, a two-parameter distribution in the GAMLSS family in R, called logistic distribution, was used in this study (\(D_{m}^{s}\sim Logistic(\alpha,\beta)\)). \[\mathrm{f}(D_{m}^{s}|\alpha,\beta)=\frac{1}{\beta}\exp\left(\frac{D_{m}^{s}- \alpha}{\beta}\right)\frac{1}{\left(1+\exp\left(\frac{D_{m}^{s}-\alpha}{ \beta}\right)\right)^{2}},-\infty<D_{m}^{s}<\infty. \tag{13}\]  
where \(\alpha\) and \(\beta\) are the location and scale parameters, respectively. (e) Fitting of the cumulative probability of the D series under a particular time-scale is fitted. By transforming the cumulative probability density into a standard normal distribution with a mean of zero and variance of one, the SPEI can be obtained. A time-scale of six months was chosen in this study. Since the calculation of the SPEI index was based on that of the original SPI [18], the definition of the drought intensity of the SPEI was consistent with that of the SPI. The threshold values of the SPEI index and their corresponding drought levels are shown in Table 1.  
A drought event for each time-scale is defined as a period wherein the SPEI is continuously negative, reaching a value of \(-1.0\) or less [17]. The definition of a complete drought event should include the start time, ending time, duration, and intensity at each time point.