## 2 Data and Methods
### Methods
#### 2.3.3 Impacts of Urban Expansion on Vegetation  
To assess the impact of urban expansion on vegetation, we used the findings from the previous section to extract the expansion areas of the Lanxi urban cluster and analyzed the VDI values of the pixels within these areas. The impacts of urban expansion on vegetation can be categorized into two types: positive impacts and negative impacts [65; 66]. When vegetation is unaffected by urban expansion, the VDI values fluctuate within a certain range, known as the natural fluctuation range. However, when vegetation is affected by urban expansion, the VDI values deviate from this natural fluctuation range, and the larger the deviation from the natural fluctuation range is, the greater the impact [60]. Therefore, this study assesses the impact on vegetation on the basis of changes in VDI values. Specifically, when the VDI value exceeds the natural fluctuation range, it indicates that the vegetation in that area has experienced a positive impact, whereas if it falls below the range, it indicates a negative impact. This study refers to [PERSON] et al. and employs the method of the mean standard deviation classification to determine the natural fluctuation range and classifiesthe pixels within the Lanxi urban cluster into three types on the basis of their impact [60]. The specific formula is shown in Equation (6):  
\[Impact_{i}=\left\{\begin{array}{ll}PI_{i}=0,\;\;NI_{i}=1,\;\;VDI<\mu-s\\ PI_{i}=0,\;\;NI_{i}=0,\;\;\mu-s\leq VDI\leq\mu+s\\ PI_{i}=1,\;\;NI_{i}=0,\;\;VDI>\mu+s\end{array}\right. \tag{6}\]  
where \(Impact_{i}\) represents the \(i\)th pixel affected by urban expansion, \(PI_{i}=0,\;NI_{i}=1\) indicates that the pixel has experienced adverse effects, \(PI_{i}=1,NI_{i}=0\) indicates that the pixel has experienced positive effects, and \(PI_{i}=0,\;NI_{i}=0\) represents the VDI values within the natural fluctuation range, indicating that the area has not been affected. \(\mu\) and \(s\) denote the mean and standard deviation, respectively, of all the pixels within the Lanxi urban cluster.  
[PERSON] et al., building on [PERSON] et al.'s approach, assessed the effects of urban expansion on vegetation by measuring the affected area [60; 65]. We also used this method to evaluate such impacts. Various formulas are applied to compute different types of impacts. The formula for calculating the positive impact is presented in Equation (7).  
\[TA_{P}=S\times\sum_{i=1}^{n}PI_{i}\times\left(Urban_{i}^{2021}-Urban_{i}^{2001}\right) \tag{7}\]  
In the above formula, \(TA_{P}\) represents the area of vegetation positively affected by urban expansion; \(S\) represents the pixel area, totaling 0.0625 km\({}^{2}\); \(n\) represents the total number of pixels; and \(PI_{i}\) represents the class value indicating whether the \(i\)th pixel is affected positively by vegetation. A class value of 1 for a pixel indicates that the vegetation in that pixel is positively affected, whereas a class value of 0 indicates no positive impact on the vegetation in that pixel. \(Urban_{i}^{2021}\) and \(Urban_{i}^{2001}\) indicate whether the \(i\)th pixel was urbanized in 2001 and 2021, respectively. If the pixel is within an urban area, its pixel value is 1; otherwise, it is 0.  
Similarly, the formula for calculating the negative impact is as Equation (8):  
\[TA_{N}=S\times\sum_{i=1}^{n}NI_{i}\times\left(Urban_{i}^{2021}-Urban_{i}^{2001}\right) \tag{8}\]  
where \(TA_{N}\) represents the area of vegetation positively affected by urban expansion, and \(NI_{i}\) is the class value indicating whether the \(i\)th pixel is negatively affected by vegetation. A pixel with a class value of 1 indicates that the vegetation in that pixel is negatively affected, whereas a class value of 0 indicates no negative impact on the vegetation in that pixel.