# Vegetation Dynamics and Recovery Potential in Arid and Semi-Arid Northwest China
## 4 Materials and Methods
### Model and Methodology
#### 4.3.4 Mann-Kendall Trend Test  
The Mann-Kendall test, as a non-parametric method, has the advantage of being able to test for both linear and non-linear trends and has been widely used in the analysis of time series in hydrology, drought studies, and other fields. Research has found that the MK test trend exhibits autocorrelation, particularly when dealing with long-term time series data, which can affect the scientific accuracy of the results. To correctly eliminate the influence of sequence autocorrelation on the calculation results, the MK test has been improved and modified, resulting in the modified Mann-Kendall (MMK) test. Based on this, this paper will use the MMK trend test to perform a significance test on the spatial changes in vegetation in the northwest region during different periods. The calculation steps for the statistical value S and the standardized test statistic Z are as follows:  
For a given time series \(X=\left\{x_{1},x_{2},\cdots,x_{n}\right\}\),  
\[S=\sum_{i<j}a_{ij}=\sum_{i<j}sgn\big{(}x_{j}-x_{i}\big{)} \tag{6}\]  
\[var(S)=\eta\times\frac{n(n-1)(2n+5)}{18} \tag{7}\]  
\[\eta=1+\frac{1}{n(n-1)(n-2)}\times\sum_{i=1}^{n-1}(n-i)(n-i-1)(n-i-2)r_{i} \tag{8}\]  
In the formula, \(S\) is the test statistic, sgn is the sign function, \(X\) is the time series, and \(r_{i}\) is the autocorrelation coefficient. Given a significance level of \(\alpha\), if the autocorrelation is significant, the variance, \(var(S)\), is calculated using the modified formula. Finally, the standardized test statistic \(Z\) in the MMK trend test can be obtained using the following formula:  
\[Z=\begin{cases}\frac{S-1}{\sqrt{var(S)}}&S>0\\ 0&S=0\\ \frac{S+1}{\sqrt{var(S)}}&S<0\end{cases} \tag{9}\]  
In the formula, \(Z\) represents the trend of changes in the time series data. If \(Z\) is positive, it indicates an increasing trend in vegetation over time in the study area; conversely, if \(Z\) is negative, it indicates a decreasing trend over time. According to the standard normal distribution table, if the absolute value of \(Z\) is greater than or equal to 1.64, 1.96, and 2.58, it indicates that the time series has passed the significance tests at the 90%, 95%, and 99% confidence levels, respectively. This corresponds to a slightly significant change, a significant change, and a highly significant change in vegetation. For an overall increasing or decreasing trend, this study selects a significance level of \(\alpha=5\%\), corresponding to a \(Z\) value of 1.96.  
The NDVI, the LPJ dynamic global vegetation model, and the Mann-Kendall test method demonstrate significant advantages in this study. The NDVI is a commonly used remote sensing index for assessing vegetation coverage and health, providing a direct and intuitive measure of vegetation growth. The NDVI data, derived from remote sensing imagery, offer large-scale spatial and temporal distribution information, making them particularly suitable for studying northwest China, a region characterized by vast areas and diverse ecosystems. Additionally, its long time series supports the analysis of vegetation dynamics and restoration potential. The LPJ model, as a dynamic global vegetation model, simulates interactions between vegetation and climate, including carbon, water, and energy cycles. It compensates for gaps in remote sensing data across time and space, enabling precise predictions of vegetation dynamics under various climatic conditions. This is particularly beneficial for assessing the impacts of climate warming and humidification on vegetation restoration potential. The Mann-Kendall test, as a non-parametric statistical method, is highly robust and insensitive to outliers, making it well suited for analyzing vegetation coverage trends over time. It effectively identifies significant trends in NDVI data, revealing spatiotemporal patterns of vegetation change. Together, these three methods complement each other, providing a solid technical foundation for a comprehensive investigation of vegetation dynamics and restoration potential in northwest China.