## 3 Methodology
### Drought Spatial Distribution Pattern Identification  
Using drought characteristics, extracted from meteorological station recordings to explore drought spatial distribution patterns, is important for drought hazard management, for which several cluster methods are generally used. Given that the unsupervised K-means method is relatively simple to implement and easily adapts to new examples, it was utilized in this study to explore the drought spatial distribution pattern. The drought characteristics extracted in Section 3.1.2 and the longitude and latitude of the corresponding meteorological stations were used as input variables for the K-means cluster model. These variables have different ranges and units, so they were normalized before being inputted into the K-means model to eliminate negative effects [5]. In general, the results of K-means depend largely on the initialization of centroids, and the method is less efficient in certain cases and easily falls to a local extreme value [45; 46]. To eliminate these limitations, the revised method proposed by [PERSON] et al. [47] was used in this study. In this revised method, an optimized selection of initial centroids is applied, and the information entropy theory is introduced when the unsupervised clustering algorithm is implemented. Compared with the traditional K-means algorithm, the revised method has a higher detection rate and a lower false alarm rate.  
The K-means method is unsupervised, and the number of optimal \(k\) partitions or the optimal number of clusters needs to be determined. In this study, we utilized several quantitative measures to evaluate the cluster results and determine the optimal number of clusters. These quantitative measures include two R packages (Nbclust and Mclust) and two criteria (Elbow and Calinsky). Nbclust and Mclust provide 26 and 14 different indicators, respectively [48; 49]. The number of indicators corresponding to the optimal number of clusters is greater. The Elbow standard calculates the sum of squared errors (SSE) to determine the number of clusters. As the number of clusters increases, SSE decreases accordingly; when the decline slows down, an inflection point arises, and the cluster value at the inflection point is used as the optimal number of clusters [50]. The Calinsky criterion evaluates the local optimal clustering number by calculating the SSE between groups and the SSE within a group. The smaller the SSE within a group is, the larger the SSE between groups is, and this result is reflected in the variance ratio criterion (VRC) value. A large VRC value indicates an optimal number of clusters [51].  
Another task that should be performed when analyzing cluster results is to determine whether the drought characteristics from meteorological stations within a given partition are homogeneous or heterogeneous. The heterogeneity test [52] that compares the sample L-moment ratio with kappa distribution parameters can effectively deal with this matter. The heterogeneity statistic, \(H_{i}\), can be calculated as follows [25]:  
\[H_{i}=\frac{(V_{i}-\mu_{gi})}{\sigma_{zi}}\;i=1,\;2,\;3, \tag{4}\]  
\[V_{i}=\sqrt{\left\{\frac{\sum_{j=1}^{N}n_{i}(a-b)^{2}}{\sum_{j=1}^{N}n_{j}} \right\}}\;i=1,\;2,3, \tag{5}\]  
where \(\mu_{vi}\) is the means of simulated \(V_{i}(i=1,2,3)\) values, \(\sigma_{vi}\) is the standard deviations of simulated \(V_{i}(i=1,2,3)\) values, \(a\) is the sample L-moment at each site, \(b\) is the average sample L-moments at a regional scale, and \(n_{j}\) is the length of the time series at site \(j\). If \(H_{i}<1(i=1,2,3)\), then the region can be regarded as acceptably homogeneous. If \(1<H_{i}<2\), then the region can be regarded as possibly homogeneous. If \(H_{i}>2\), then the region can be viewed as heterogeneous [4; 37].