## 3 Methods
### Bayesian Estimator of Abrupt Change, Seasonal, and Trend (BEAST)  
#### 3.2.1 Decomposition Model of BEAST  
In this research, the BEAST approach was utilized to decompose the reconstructed NDVI time series to obtain abrupt changes as well as seasonal and trend signals for monitoring the dynamic disturbances of wetland vegetation [36]. By the BEAST algorithm, the data space (\(y\)) is transformed to the ecological or biophysical space with interspersed abrupt changes over time (\(t\)) and remainder (\(\varepsilon\)) by the decomposition model \(\hat{y}\)\((t_{i})=f(t_{i;}\ \Theta)\) (Equation (1), f (\(\cdot\)) donates the function of the decomposition model).  
\[\hat{y}\)\((t_{i})=f(t_{i};\Theta)=S\)\((t_{i};\Theta_{S})+T\)\((t_{i};\Theta_{T})+\varepsilon_{i},\ i=1,\ldots,\ n \tag{1}\]  
where \(i\) is the index of time dimension, \(\Theta_{s}\) and \(\Theta_{T}\) represent the parameters related to seasonality (\(S\)) and trend (\(T\)), respectively, and \(\varepsilon\) is the residual.  
In the BEAST algorithm, the definition of abrupt changes, trend and seasonality is broader than that in other decomposition algorithms. Seasonality (\(S\)\((t)\)) is considered as a seasonal signal embedded with disturbance that indicates the phenological change, and it is fitted by a piecewise harmonic model (Equation (2)).  
\[S\)\((t)=\sum_{l=1}^{L_{k}}\)\(\left[a_{k,\ l}\cdot\sin\left(\frac{2\pi lt}{P}\right)+b_{k,\ l}\cdot\cos\left(\frac{2\pi lt}{P} \right)\right],\ \ \xi_{k}<t<\xi_{k+1}\)\((k=0,\ldots,\ P)\) (2)  
where \(p\) is the time-series period and the \(p\) separates time scale into \(k\) segments; \([\xi_{k},\ \xi_{k+1}]\) is the interval of the (\(p+1\))-th segment; \(t\) represents the time; \(L_{k}\) is the harmonic order at segment \(k\), \(a_{k,\ l}\) and \(b_{k,\ l}\) specify the parameters for sins and cosines, respectively.  
The changepoints, including all turning points, are deviated from previous regular signals. Trend (\(T\)\((t)\)) is defined as a complex nonlinear curve consisting of numerous piecewise lines (the time span was divided by knots (\(\tau_{j},\ j=0,\ \cdots m\)) into (m + 1) intervals \([\tau_{j},\ \tau_{j+1}]\); \(a\) and \(b\) are the coefficients of the piecewise linear function) (Equation (3)).  
\[T\ (t)=a_{j}+b_{j}t,\ (\tau_{j}<t<\tau_{j+1},\ j=0,\ldots,\ m) \tag{3}\]  
\[\Theta_{s}=\{P\}\ \cup\ \{\xi_{k}\}_{k=1,\ \cdots,P}\ \cup\ \{L_{k}\}_{k=0,\ \cdots,P}\ \cup\ \{a_{k,l},\ b_{k,\ l}\}_{k=0,\ \cdots,P;l=1,\ \cdots,L_{k}} \tag{4}\]  
\[\Theta_{T}=\{m\}\ \cup\ \left\{\tau_{j}\right\}_{j=1,\ \cdots,m}\ \cup\ \left\{a,\ b_{j}\right\}_{j=0,\ \cdots,m} \tag{5}\]When parameters \(\Theta_{T}\) and \(\Theta_{s}\) are assumed as: \(\{\Theta_{T},\Theta_{s}\}=\{\text{M},\ \beta_{\text{M}}\}\), M is the model structure to be estimated (Equation (6)), including the total number and occurrence time of changepoints, harmonic orders of seasonal and trend signals; \(\beta_{\text{M}}\) represents the coefficients in M (Equation (7)). Thus, Equation (1) can be rewritten as Equation (8):  
\[\text{M}=\{m\}\ \cup\ \left\{\tau_{j}\right\}_{j=1,\ \cdots,m}\ \cup\ \{P\}\ \cup\ \{\xi_{k}\}_{k=1,\ \cdots,P}\ \cup\ \{L_{k}\}_{k=0,\ \cdots,P} \tag{6}\]  
\[\beta_{\text{M}}=\left\{a,b_{j}\right\}_{j=1,\ \cdots,m}\ \cup\ \left\{a_{k,l},\ b_{k,l}\right\}_{k=0,\ \cdots,P=1,\ \cdots,L_{k}} \tag{7}\]  
\[\hat{y}\ (t_{i})=\text{M}\ (t_{i})\cdot\ \beta_{\text{M}}+T\ (t_{i};\Theta_{T}),\ i=1, \ldots,\ n \tag{8}\]