## 2 Methods
### Detecting Major Trend Shifts in Vegetation Activity  
We proceeded as follows, in order to assess when and where a major trend shift occurred within a NDVI time series. First, we selected a season-trend model that adequately accounts for typical change dynamics within NDVI time series (_i.e._, seasonal cycles within years and stable trends across years). Second, we assessed the stability of this model using tests for determining if structural trend breaks occur within time series (_i.e._, _break detection_ or _testing_). Given evidence for a significant structural trend break, we then determined when it actually occurred (_i.e._, _breakpoint estimation_ or _dating_) and how it influenced the time series (_i.e._, _characterization_ or _classification_, Section 2.3). To capture gradual and phenological dynamics typically occurring within climate-driven biophysical indicators derived from satellite data (e.g., NDVI), [PERSON] _et al._[18] proposed a season-trend model. They utilized the model for detection of changes at the end of NDVI time series. Here, we use the same season-trend model to detect change--specifically trend shifts--within the time series. More specifically, for the observations \(y_{t}\) at time \(t\), a season-trend model is assumed with linear trend and harmonic season:  
\[y_{t} = \alpha_{1}+\alpha_{2}t+\sum_{j=1}^{k}\gamma_{j}\sin\left(\frac{2 \pi jt}{f}+\delta_{j}\right)\ +\ \varepsilon_{t} \tag{1}\]  
where the intercept \(\alpha_{1}\), slope \(\alpha_{2}\) (_i.e._, trend), amplitudes \(\gamma_{1},\ldots,\gamma_{k}\), and phases \(\delta_{1},\ldots,\delta_{k}\) (_i.e._, season) are the unknown parameters, \(f\) is the known frequency (e.g., \(f=12\) annual observations for a monthly time series), and \(\varepsilon_{t}\) is the unobservable error term at time \(t\) (with standard deviation \(\sigma\)). In the applications below, we employed three harmonic terms (_i.e._, \(k=3\)) to robustly detect disturbances within GIMMS NDVI3g time series, because four or more components represent variations that occur on a three-month cycle or less [23, 24]. The model (Equation (1)) can be written as a standard linear regression model (see e.g., Chapter 3.3 in [25]) whose coefficients can be estimated and tested using ordinary least squares (OLS) techniques. If there are gaps in the time series \(y_{t}\), these observations can simply be omitted prior to estimation of the model parameters. These parameters can still be identified consistently, unless _all_ observations at certain frequencies are missing. In this way, the quality flags could be used to omit unreliable observations (see example in Figure 1). The season-trend model takes the underlying trend and seasonal variation within a time series into account so that season-trend patterns are removed in the resulting residuals. This corresponds to approaches where in multiple steps a time series is \"detrended\" and \"deseasonalized\" (e.g., [5]), whereas here the adjustment was performed in a single step by OLS fitting of the season-trend model. If no trend and/or no seasonality are present in the data investigated, the corresponding elements of the regressor can also be omitted. The proposed season-trend model is analogous to the BFAST [16] approach, but differs in a way that it has been optimized for detecting the most influential trend shift in the time series, as opposed to multiple smaller shifts. Also, BFAST iteratively fits a separate seasonal and trend model and is therefore computationally more intensive. Figure 1 illustrates a season-trend model (dashed black line) fitted to an NDVI time series (gray) while accounting for the most influential trend shift. Based on the season-trend model introduced above, we assessed whether the temporal trend was a linearly developing equilibrium or included shifts, _i.e._, if the model parameters remained stable or were changing over time. In statistical and econometric literature, this approach is known as _testing_ for structural changes.