# Spatiotemporal Dynamics and Obstacles of the Multi-Functionality of Land Use in Xiangxi, China
## 2 Data and Methods
### Research Methods
#### 2.3.4 Measuring the Versatility of Land Use  
The first step was to standardize the data. According to the land use function evaluation index system, a sample matrix \(X\) of m evaluation indexes in n areas from 2013 to 2017 was constructed, where \(X=(X_{ij})_{n\times m}\), \(i=1,2,\dots\) N, \(j=1\), 2,...M. In order to make an evaluation of the quantitative comparison of different attributes and dimension indexes, the optimal value of each index (the maximum value of the positive index and the minimum value of the negative index) during the study period was selected as the reference value \(X_{o}\) in order to conduct a dimensionless quantization of \(X\). The calculation formula used was as follows.  
\[Y_{ij}=\left\{\begin{array}{l}X_{ij}/X_{oj}\text{ the positive index}\\ X_{oj}/X_{ij}\text{ the negative index}\end{array}\right. \tag{1}\]  
where \(Y_{ij}\) was the standardized value of the \(j\) evaluation index in area \(i\), \(X_{ij}\) was the original value of the \(j\) valuation index in area \(i\), and \(X_{0j}\) was the optimal value of the \(j\) evaluation index. Evaluation samples \(Y\) were obtained after standard treatment, \(Y=(Y_{ij})_{n\times m}\), \(Y\in\)(0 1].  
The second step was to calculate a value for the land use function. Value (\(F\)) represents the value for the level of land use with regard to functions. The larger \(F\) is, the better the level of land use is, and vice versa. According to the standardized values of the evaluation samples and the weight of factors at all levels, the land single functional value (\(F1\)), sub-functional value (\(F2\)), and total functional value (\(F3\)) were calculated. The formula for calculating the functional values is shown below.  
\[F1=\sum W_{j}Y_{ij}\,F2=\sum W_{i}F1\,F3=\sum W_{n}F2 \tag{2}\]  
where \(W_{j}\) was the weight of evaluation index, \(W_{i}\) was the weight of a single function (\(F2\)), and \(Wn\) was the weight of a sub-function (\(F3\)).  
The third step was to calculate the degree of dynamic change in land use and the degree of functional advantage of the land. The degree of dynamic change in land function (\(d\)) refers to the degree of change in land function levels within a certain period. The overall degree of dynamic change in land function was, therefore, calculated for the study period of five years. The calculation used was as follows.  
\[d=\frac{F_{t+4}-F_{t}}{F_{t}}\times 100\% \tag{3}\]  
where \(t\) represents the year, \(F_{t}\) represents the land function value of the year \(t\), \(d<0\) indicates that the land function remains unchanged, \(d>\) indicates that the land function is enhanced, and \(d<0\) indicates that the land function is degraded.  
The dynamic dominance of land function reflected the differences in land use, calculated with:  
\[s=\frac{|d|_{max}}{\sum|d|} \tag{4}\]  
where \(|d|_{max}\) was the maximum absolute value of the degree of dynamic change in land function, and \(\Sigma|d|\) was the sum of the absolute value of the dynamic changes of land function. The larger \(S\) is, the more diverse the land function change is, which means that the land use tends to be simplified. Smaller values for \(S\) point to diversification.  
The fourth step was to measure the multifunctional coordination degree of land use. The difference in the standard deviation of the functional value of land use (\(\sigma\)) reflects the coordination of the functional level of the land. The calculation formula was as follows.  
\[\sigma=\sqrt{\frac{\sum\left(F-\overline{F}\right)^{2}}{N}} \tag{5}\]  
where \(\overline{F}\) was the average value for land function and \(N\) was the function number. The value of \(\sigma\) is inversely related to the coordination degree of land function. The higher the value is, the lower the coordination degree of land function is, and vice versa.