## 2 Materials and Methods
### Methodology
#### 2.3.3 Characteristic Analysis of the IST-Cube  
The year when the IST-cube ends is defined as the time when the change occurs; however, numerous IST-cubes fractured suddenly in 2020, as this was the last year of the study period. To avoid this, results were only calculated for 1987-2019. Different changes shape different IST-cubes. Therefore, through the analysis of IST-cubes at different spatiotemporal scales, this study reveals the change law of the study area and explores the corresponding relationship between the change and IST-cubes. The IST-cubes were divided into four types according to the spatiotemporal scale (Table 1): long-term-large scope IST-cube, long-term-small scope IST-cube, short-term-large scope IST-cube, and short-term-small scope IST-cube, which were abbreviated as LL, LS, SL, and SS, respectively. This study obtained the trend and slope of the number of pixels per year for the long-term IST-cube (the short-term IST-cube had no trend as their duration was short) using the Mann-Kendall test and [PERSON]'s slope estimator [67]. This study then further subdivided the long-term IST-cube into four types according to the trends in the number of pixels per year for the long-term IST-cube: monotone increasing, unchanged, alternating increasing and decreasing, and monotone decreasing trends, which are referred to as increasing, stable, fluctuating, and decreasing, respectively. Figure 5 is a schematic diagram of the long-term IST-cube classification process. First, according to the trend obtained by the Mann-Kendall test, it can be divided into the increasing type and the decreasing type. Then, for the IST-cube without trend, it can be further classified according to the slope estimated by [PERSON]'s slope estimator: if the slope is zero, it is stable type, and if the slope is not zero, it is fluctuating type. The dynamic time warping (DTW) algorithm is particularly useful in comparing the similarity of unequal time series [68; 69; 70]. As shown in Figure 6, this study compares the similarity of the time series of the short-term IST-cubes in the same category with the same location but different time points by the DTW. The DTW was calculated by the diadiistance python module [71]. The smaller the value of DTW, the more similar the two-time series are. To make the DTW value comparable between large scope IST-cube and small scope IST-cube, data were normalized by Z-normalization, which is widely recommended for time series analysis before calculating the DTW [72; 73]. However, if there is a large number of situations in that DTW equal to zero, it indicates that there are many similar IST-cubes on the timeline at this location. Further, similar cubes imply the derivation from similar influences, which means a particular change repeatedly affects the IST-cube at this place. Similar changes over and over again mean that the region is changing regularly. In the same way, if DTW is above 0, it means that the region is changing irregularly. Therefore, we referred to these changes that cause these IST-cubes as irregular and regular, respectively. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Duration \(\geq\) 5 Years** &
\begin{tabular}{c} **Average Number of Pixels** \\ **per Year \(\geq\) 300 Piixels** \\ \end{tabular} & **IST-Cube Type** & **Abbreviations** \\ \hline Yes & Yes & long-term-large scope & LL \\ Yes & No & long-term-small scope & LS \\ No & Yes & short-term-large scope & SL \\ No & No & short-term-small scope & SS \\ \hline \hline \end{tabular}
\end{table} 
Table 1: IST-cube classification and abbreviations based on spatiotemporal scale. Different changes resulted in different IST-cubes. Consequently, IST-cubes with diverse spatiotemporal characteristics revealed the regularities of change that shape their structure. This change is unavoidable, and the resulting change for the IST-cube was highly complex and may cause fragmentation of the cube or the merger. The chunk of pixels that was always present in this IST-cube maintained the LL IST-cube structure and acted as a hard core.