## 3 Methods
### Breakpoints Assessment Framework  
The TSS-RESTREND method by [8] provided only a qualitative assessment of breakpoints with other independent data in terms of significance and direction of changes. These two properties are important to properly characterize changes in productivity. A framework was developed in this study to address this gap by proposed means to quantitatively assess these properties. This framework consisted of four steps: (1) Develop random samples within the identified significant breakpoints; (2) Select and use independent productivity data; (3) Evaluate the random samples at the pixel level; and (4) Group and evaluate all random samples that fall within identified coregions--allowing to identify whether the changes in productivity at the individual pixels are reflective of consistent regional changes. Random Samples: A set of random samples in terms of size and location can be developed following [56; 57]. To estimate the size of random samples, a prior knowledge of image accuracy/variability is required [57]. The degree of variability in the RPMS data is unknown, thus it was assumed that the maximum variability would be about 50% with 95% confidence level and \(\pm\)5% precision [56]. This criterion helps to determine representative sample pixels using a stratified random sampling approach [58]. The strata were developed based on coregions, land use/land cover, and direction of change. Based on the formula from [57], the sample size was calculated as (Equation (5)). \[n_{o}=\frac{Z^{2}\ pq}{e^{2}} \tag{5}\]  
where \(n_{o}\) is the sample size, \(Z\) is the selected critical value of desired confidence level (1.96), \(p\) is the estimated proportion of an attribute that is present in the population (0.5), and \(q=1-p\) (0.5) with \(p\) and \(e\) (0.5) represent the desired level of precision. The allocation of the random sample was based on the majority of the identified breakpoints in each method (Section 3.1.2). The random samples can be broadly allocated based on the direction of change and land cover within the identified significant breakpoints in the two main categories--the Segmented VPR and Segment RESTREND. Coregions and land use/land cover (NLCD 2011) [43] maps were overlaid to identify the locations of the pixels within these regions for further assessment. Selection of independent data: The productivity data from the RPMS [45] was used to evaluate the accuracy of the identified breakpoints based on the TSS-RESTREND in terms of direction and significance of change in productivity. The data was resampled to match that of GIMMS NDVI [44; 58]. Using the coordinates of GIMMS NDVI pixels, the corresponding RPMS productivity was extracted. The rangeland productivity was converted from round per acre per year to kilogram per hectare per year. this process allowed to prepare a dataset that include RPMS productivity and with the corresponding TSS-RESTREND estimates of timing and direction of change, significance of breakpoints, and method of detection for each random sample. Pixel level assessment: Using the location of the randomly selected samples (breakpoint pixels) and their identified year of breaks, the mean annual productivity for each pixel based on the RPMS data can be calculated for before and after the break years. Over each pixel, the number of years before and after the breaks were different and thus these mean values had different sample sizes. These mean values before and after the breaks over each pixel were then statistically compared using the [PERSON]'s student-test for the significance in their difference and direction of change. The [PERSON]'s \(t\) test was conducted assuming that the variances were not equal before and after the breaks [58; 59]. The obtained results based on RPMS data using the [PERSON]'s test were compared with those from the TSS-RESTREND method. A summary of the agreement of this comparison was provided over these pixels as well as over the representative land cover classes.