## 2 Methods
### Detecting Major Trend Shifts in Vegetation Activity
Structural, in this context, refers to the statistical property of the time series and not to characteristics of a vegetation canopy. A broad range of statistical tests for structural change detection have been suggested, using linear regression models in particular [26, 27, 28]. Because the season-trend model can be formulated as an OLS linear regression, all of these methods can be leveraged directly. Below, we focus on structural change tests based on moving sums (MOSUMs) of OLS residuals [29] that have higher power for detecting changes in the trend (_i.e._, the parameters \(\alpha_{1}\) and \(\alpha_{2}\)) while being less sensitive to changes in cyclical patterns [30], _i.e._, to phenological changes. Given detection of a significant structural change, the associated breakpoint was estimated [31]. This is referred to as _dating_ of the breakpoint. Both testing and dating were carried out using the implementation of [32, 33] in the R software for statistical computing [34]. The mentioned MOSUM test for break detection is based on moving sums with bandwidth \(h\) of the OLS residuals \(\hat{\varepsilon}_{t}\) (Equation (2)). The bandwidth \(h\) operates as a smoothing parameter, although only for the break detection on OLS residuals and not for the subsequent breakpoint estimation on the NDVI3g time series. \[M_{t} = \frac{1}{\hat{\sigma}\sqrt{n}}\sum_{s=t-h+1}^{t}\hat{\varepsilon} _{s} \tag{2}\]  
where \(h=40\) for our application (see below for more details). If the model remains stable, the MOSUM process \(M_{t}\) should be close to zero and fluctuate only randomly. However, if a structural change occurs,  
Figure 1: Detecting one major trend shift within an NDVI time series. A season-trend model (dashed black line) was fitted to the NDVI time series (gray). The vertical dashed line depicts the detected trend shift, together with the confidence interval of its timing (red). This particular shift was detected in 2000, yielding separate trend segments (dark red) before and after. Data gaps due to non-zero quality flags (e.g., in 1988) do not influence the season-trend model parameter estimation. \(M_{t}\) will deviate systematically from zero. A structural break was declared if the maximum absolute value \(\max_{t}|M_{t}|\) exceeded its 5% asymptotic critical value [29]. If the MOSUM test signaled a significant instability in the season-trend model (at 5% level), we captured this change by estimating one breakpoint and fitting separate season-trend models to the two segments before and after the break. The breakpoint was estimated (or _dated_) using OLS, _i.e._, by minimizing the combined sum of squared residuals from the two segments. Confidence intervals for the breakpoint estimates can also be calculated [31]. To ensure that the 8 regression parameters (2 trend and 6 season parameters) of the season-trend model can be sensibly estimated in both segments, a minimal segment size of 40 observations (_i.e._, 3.5 years) was enforced and the MOSUM bandwidth was chosen correspondingly. Although such a two-segment season-trend model may still contain further structural changes, its breakpoint consistently identifies the largest shift (in terms of the residual sum of squares), see [35]. In summary, this modeling approach either identifies no break at all (if the MOSUM test results is non-significant) or a single \"most important\" break that can occur between 1985 and 2008. Typically, if such a break is found, it can be interpreted as a long-term change. However, breakpoints occurring very early or late in the sampling period need to be carefully interpreted with respect to their long-term significance, since one of the two segments might reflect a short-lived process.