# Best Practices for Applying and Interpreting the Total Operating Characteristic
## 4 Discussion  
### Interpretation of Results  
Our application demonstrates four possibilities to use exclusively the maps of the dependent variable to predict change. This is important because scientists always have the dependent variable, while the compilation and analysis of several independent variables can be burdensome. Our application considers two methods to consider the information through time. The first method considers the change between the two time points that precede the validation interval. The second method considers the single time point at the start of the validation interval. The second method requires less information but can be more helpful than the first method. The second method considers the entire presence at the start of the time interval, which is likely to be much larger than the change that the first method considers. The results show that the TOC curve for the presence pixels is above the TOC curve for the change pixels at the thresholds to the right of the correct quantity. The TOC curve for the presence patches is indistinguishable from the TOC curve for the change patches, which demonstrates that a simpler map of the presence patches can have as much predictive power as the more complicated map of the change patches. This illustration does not prove a general rule. [PERSON] et al. (2024) found that the distance to change is more helpful than the distance to presence to predict gain where loss precedes gain in a temporal pattern called alternation [30,31]. A similar pattern exists in Figure 5b, where the TOC curve for the change pixels is above the TOC curve for the presence pixels at thresholds near the origin. The TOC curves for patches are above the TOC curves for pixels. This indicates that the decision to use spatial filtering to remove isolated pixels of crop is more important than the decision to use change or presence when computing the distance maps. Spatial filtering is important for a variety of reasons. First, commission error is more likely to generate isolated pixels of crop than large patches of crop. Thus, the spatial filtering might reduce the error in the dependent variable. Second, isolated small farms account for a smaller portion of crop than large farms. These large farms tend to be near each other; thus, spatial filtering to identify patches is helpful for focusing on the large farms that account for most of the crop. Third, an isolated pixel of crop generates the smallest possible distances on all sides of the pixel; thus, a map of the distance to pixels tends to have more of the smallest possible distance than a map of distance to patches. If many of the earliest ranked pixels have a tied smallest distance, then the first segment that emanates from the TOC curve's origin is so large that it is difficult to distinguish among TOC curves. The spatial filtering reduces these weaknesses of isolated pixels. We have not seen other literature that uses spatial filters in the way our manuscript demonstrates. Other literature typically follows procedures that are much more complicated. For example, others have used numerous independent variables such as elevation, slope, soils, temperature, precipitation, protected areas, human population, administrative units, distance to transportation infrastructure, distance to central business districts, distance to particular land covers, etc. [16, 17, 18, 19, 14, 12, 16, 17, 18, 20, 21, 32, 33, 34, 35, 36, 37]. However, these independent variables are likely to be highly correlated, and thus redundant with the four distance variables that our manuscript considers. Therefore, the value added by the additional independent variables is questionable, but testable via substantial effort. If scientists use additional independent variables, then the scientists are likely to use an algorithm to calibrate a model. Many algorithms exist, such as logistic regression, neural nets, etc. Each algorithm has numerous parameters. Some algorithms require substantial expertise and computing resources.