# Mapping Abandoned Cultivated Land in China: Implications for Grain Yield Improvement
## 2 Materials and Methods
### Methodology
#### 2.2.2 Characterization of the Spatial Pattern of Abandoned Cultivated Land  
1. Trend surface analysis  
Trend surface analysis is a mathematical method that uses a mathematical curved surface to simulate the spatial distribution and change trend of geographical system elements. This approach has found extensive application in simulating the spatial patterns of resources, environmental factors, population distribution, and economic elements [45]. In this paper, the trend surface analysis method was used to show the spatial distribution pattern and change trend of the \"abandoned area\" and \"abandoned rate\" in China according to the definition of the trend surface as follows:  
\[Z_{i}(X_{i},Y_{i})=T_{i}(X_{i},Y_{i})+\epsilon_{i} \tag{1}\]  
where \(X\) and \(Y\) are plane coordinates; \(Z_{i}(X_{i},\,Y_{i})\) is the trend function, which represents the object value of the whole region; and \(\epsilon_{i}\) stands for the autocorrelation random error, showing the difference between the actual value and the estimated trend value of the deserted area or the abandoned rate of the \(i\)-th prefecture-level city. In this paper, the second-order polynomial was used to calculate the trend value of the regional abandoned area and the abandoned rate. The trend function can be expressed as follows:  
\[T_{i}(X_{i},Y_{i})=\beta_{0}+\beta_{1}x+\beta_{2}y+\beta_{3}x^{2}+\beta_{4}y^{ 2}+\beta_{5}xy \tag{2}\]
2. Global spatial autocorrelation  
Global spatial autocorrelation was mainly used to describe the overall distribution of abandoned cultivated land across the country and to judge whether the phenomenon had spatial aggregation characteristics [46; 47]. The Moran's \(I\) coefficient enables the evaluation of both the overall spatial correlation and the spatial disparity among regions. The specific formula is as follows:  
\[\text{Moran's }I=\frac{n\sum_{i=1}^{n}\sum_{j=1}^{n}w_{ij}(x_{i}-\overline{x}) \big{(}x_{j}-\overline{x}\big{)}}{\Big{(}\sum_{i=1}^{n}\sum_{j=1}^{n}w_{ij} \Big{)}\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}} \tag{3}\]  
where \(\overline{x}=\frac{1}{n}\sum_{i=1}^{n}x_{i}\); \(n\) is the number of spatial units of data; and \(x_{i}\) and \(x_{j}\) represent the attribute values of spatial units \(i\) and \(j\), respectively, which make up the spatial weight matrix. The value of the global [PERSON]'s \(I\) index ranges from \(-1\) to \(1\). If the value is above \(0\), it indicates a positive spatial autocorrelation in the spatial distribution of the research unit, with the observation attributes exhibiting a clustered spatial pattern. In this case, as the value approaches \(1\), the positive correlation becomes more pronounced, indicating a closer relationship and stronger connection between the units. A value under \(0\) suggests a negative spatial autocorrelation in the study area, where the observed characteristics show a clear spatial layout Moreover, as the value approaches \(1\), the spatial autocorrelation of the observed attributes decreases, suggesting a random distribution across space. 3. Local spatial autocorrelation  
The global spatial autocorrelation index can only determine if there is clustering at the scale of abandoned cultivated land, but it cannot provide information regarding the specific location or characteristics of such clusters. To obtain insights into the attribute value of a particular spatial unit and its neighboring units in the local space, a local spatial autocorrelation analysis is required [48; 49; 50; 51]. Therefore, this research used local spatial autocorrelation to measure the extent of local spatial connection and highlight the spatial differences between a region and its neighboring regions. The findings were then visualizedusing a LISA cluster map, effectively showing the spatial pattern of local variations.