## 2 Materials and Methods
### Methodology
#### 2.2.2 Detecting Vegetation Change  
Vegetation changes include short-term abrupt changes and long-term trend changes. In this study, the BFAST algorithm was used to detect short-term changes in vegetation, and the BFAST01 algorithm was used to detect long-term trend changes in vegetation. 1. BFAST method for short-term abrupt change  
The BFAST algorithm utilizes piecewise linear trend and seasonal models to decompose time series into trend, seasonal, and residual components, and can detect abrupt changes in trend and seasonal components [23]. Assuming that an additive decomposition  
Figure 2: Workflow of the research method applied in this study; Google Earth Engine, GEE. model can iterate into a piecewise linear model that matches trend and seasonality, the algorithm of the model is of the form:  
\[Y_{t}=T_{t}+S_{t}+e_{t}\qquad t=1,\;2,\;\ldots n \tag{1}\]  
where \(Y_{t}\) is the observation data of \(t\), \(T_{t}\) is the trend component, \(S_{t}\) is the seasonal component, and \(e_{t}\) is the remainder component which contains the variation beyond what is explained by \(Y_{t}\) and \(S_{t}\). It is assumed that \(T_{t}\) is piecewise linear, with breakpoints \(t=1,\cdots n\) and define \(t_{0}=0\), so that  
\[T_{t}=\alpha_{j}+\beta_{j} \tag{2}\]  
where \(\alpha_{j}\) and \(\beta_{j}\) are intercept and slope of the piecewise linear, respectively. For \(t_{j-1}<t<t_{j}\) and where \(j=1,\cdots,m\). \(\alpha_{j}\) and \(\beta_{j}\) can be used to obtain the magnitude and direction of the abrupt change:  
\[Magnitudes=\left(\alpha_{j-1}-\alpha_{j}\right)+\left(\beta_{j-1}-\beta_{j} \right)t \tag{3}\]  
The seasonal component can be characterized as below:  
\[S_{t}=\sum_{j=1}^{k}y_{j}\sin\left(\frac{2\pi jt}{f}+\delta_{j}\right) \tag{4}\]  
where \(y_{j}\) is amplitudes, \(\delta_{j}\) is phases, \(f\) is observation, and \(j\) represents the order of the harmonic term. Furthermore, the seasonal breakpoints may occur at different times from the breakpoints detected in the trend component. Once the model is fitted, the breakpoint(s) is detected based on computing \(OLS-MOSUM_{t}\) of residuals at time t as a form of:  
\[OLS-MOSUM_{t}=\left\{\frac{1}{\sigma\sqrt{[S.h]}}\times\sum_{i=t-[S.h]+1}^{t} (y_{i}-\hat{y_{i}})\right\}_{t-[S.h]}^{s} \tag{5}\]  
where \(S\) represents the length of the time series, the user-defined value \(h\) is the minimal time length of a trend segment in fitting the model, \([S.h]\) is the length of the moving windows. \(y_{i}\) and \(\hat{y_{i}}\) are actual and expected observation, respectively, so \((y_{i}-\hat{y_{i}})\) gives OLSs residual, and \(\widehat{\sigma}\) is their standard deviation. If the model remains stable, the \(|OLS-MOSUM_{t}|\) should be close to zero and only fluctuate randomly. However, if a structural change occurs, the \(|OLS-MOSUM_{t}|\) will systematically deviate from zero. If \(|OLS-MOSUM_{t}|\) deviate from zero beyond a 95% significance boundary, a structural change has occurred and it is considered as a breakpoint. (2) BFAST01 method for long-term trend  
The BFAST01 algorithm is an adjusted version of the season-trend model, designed to detect the most important trend shift in the time series, calculated with either 0 or 1 breakpoint from the moving sum of OLS residuals. BFAST01 offers an effective and comprehensive change detection method for the assessment of major turning points in long-term seasonal and interannual variability as well as important disturbance events [37]. The algorithmic form of the model is as follows:  
\[Y_{t}=\alpha_{j}+\beta_{j}t+\sum_{j=1}^{k}y_{j}\sin\left(\frac{2\pi jt}{f}+ \delta_{j}\right)+e_{t} \tag{6}\]  
where the intercept \(\alpha_{j}\), slope \(\beta_{j}\) (i.e., trend), amplitudes \(y_{1}\),... \(y_{k}\), and phases \(\delta_{1}\),..., \(\delta_{k}\) (i.e., season) are the unknown parameters, \(f\) is the known frequency (e.g., \(f=12\) annual observations for a monthly time series), and \(e_{t}\) is the unobservable error term at time \(t\) (with standard deviation \(e\)).