## 3 Methods
### Characterization of Change in Productivity Using TSS-RESTEND
#### 3.1.2 Identification of Breakpoints  
The BFAST method was applied on the ctsVPR-residual to list potential breakpoints. Briefly, the BFAST method decomposes the timeseries into season, trend, and reminder components--an approach that allows to detect changes in the season and trend components [52; 53]. The list of potential breakpoints identified by the BFAST method [8] based on the ctsVPR-Residuals were further evaluated for their significance in the VPR--allowing to assess their impact on NPP as represented by the maximum NDVI timeseries. A Chow test was applied on the VPR-Residuals on all pixels with significant VPR (\(\alpha\) = 0.05). Based on this test, all pixels that have no significant breakpoints in the VPR-Residual (\(\alpha\) = 0.05) but have significant VPR (\(\alpha\) = 0.05) were further assessed using the standard RESTREND by developing a regression between the VPR-Residuals and time (Equation (1)) [8]. \[y_{i}=\beta_{0}+\beta_{1}\ x\ (\text{RESTREND}) \tag{1}\]  
where \(\beta_{0}\) is intercept, \(\beta_{1}\) is slope, and \(x\) is year  
Pixels that showed significant breakpoints in VPR-Residuals (based on the Chow test above) and had significant VPR were further evaluated for the significance of these breakpoints but in the VPR also using Chow test. Pixels with significant breakpoints in VPR-Residuals but not in the VPR were further assessed using the Segmented RESTREND by developing a multivariate regression between VPR-Residual, time, and a dummy variable (Equation (2))  
\[y_{i}=\beta_{0}+\beta_{1}x_{i}+\beta_{2}z_{i}+\beta_{3}x_{i}z_{i}\ (\text{Segmented RESTREND}) \tag{2}\]  
where \(\beta_{0}\) is intercept, \(x\) is year, \(z\) is value of dummy variable (0 or 1), \(\beta_{1}\) is slope, \(\beta_{2}\) is offset at the breakpoint, and \(\beta_{3}\) is the change in the slope at the breakpoint. Figure 2: Depreciation of TSS-RESTRND analysis and validation of breakpoints. In step 1, Data preparation which includes (i.e., acquisition of NPP data, precipitation, and productivity data), projection, resampling, and extraction of pixel level values. In Step II (TSS-RESTREND): method of residual fit, \(p\)-vector, trend of productivity change (i.e., decreasing, increasing, non-significance change and indeterminant), and ecosystem structure change were detected. In Step III (Evaluation), include stratified random sampling for validation, Welch’s \(t\) test on randomly selected pixels (i.e., to test the significant change of productivity before and after the break years). Pixels that showed significant breakpoints in VPR-Residuals and in VPR may indicate the presence of significant structural changes and thus the assumption of stationarity of the accumulation and offset periods used in the calculations of optimal accumulated precipitation before and after a breakpoint [5; 54]. The set if the NDVImax and precipitation timeseries before and after a breakpoint was separated to recalculate new and independent VPR on either side of the breakpoints [8]. The precipitation data were standardized to account for the differences in the accumulation and offset periods among the breakpoints (Equation (3)). \[z_{i}=\frac{(x_{i}-u)}{\delta} \tag{3}\]  
where \(z\) is the standard score, \(x_{i}\) is observed values, \(\mu\) is the mean, and \(\delta\) is the standard deviation. The NDVImax and standard score timeseries were further evaluated by fitting a multivariant regression (Equation (4)) [5; 8]. \[Yi=\beta_{0}+\beta_{1}x_{i}+\beta_{2}z_{i}+\beta_{3}x_{i}z_{i}\ (\text{Segmented VPR}) \tag{4}\]  
where \(x\) is the standardized precipitation for year \(i\), \(z\) the value of the dummy variable (0 or 1), \(\beta_{0}\) is intercept, \(\beta_{1}\) is slope, \(\beta_{2}\) is the offset at the breakpoint, and \(\beta_{3}\) the change in the slope at the breakpoint.