## 4 The Proposed Framework
### The Methodological Framework
#### 4.1.5 Data Classification and Hotspot Detection  
A plot of the clusters' centroids is made after each pixel has been identified with the cluster to which it belongs. This allows for the identification of two thresholds, called hot and cold thresholds, which, respectively, are used to identify the hot and cold spots in urban zones, which are defined by high LST and low NDVI and low LST and high NDVI.  
In the graph, the abscissa shows the LST, and the ordinate represents the NDVI.  
The hot threshold identifies a point on the curve beyond which there is no significant change in NDVI as the LST increases.  
The cold threshold identifies a point on the curve before which there is no significant change in the NDVI as the LST increases.  
The plot in Figure 2 is an example of a trend in the NDVI with respect to the LST. Initially, the NDVI does not vary significantly with increasing LST; beyond the cold threshold point, the NDVI decreases exponentially with increasing LST, up to the hot threshold point, beyond which the curve shows a plateau, and the NDVI does not vary significantly with increasing LST.  
The two lines in Figure 2, labeled hot threshold line and cold threshold line, are the two horizontal lines intersecting the curve in the hot and cold spot points, respectively.  
Once the centroids are plotted, a slope analysis is performed between adjacent centroids in order to fix the two thresholds.  
Slope is a measure of the change in one value relative to another. In the context of centroid analysis, slope is calculated between adjacent centroids and represents the change in LST and NDVI values. A steep slope indicates a rapid and significant change in values, while a gentler slope suggests a more gradual change.  
An area with a steep slope could indicate a transition from an area with high vegetation and low temperatures to an area with low vegetation and high temperatures. These transitions can be indicative of phenomena such as UHIs, where temperatures increase dramatically due to the lack of vegetation and the presence of artificial surfaces.  
The two thresholds are fixed based on the value of the slope. The cold threshold is set as the point on the curve beyond which the slope increases to more than 10%. Likewise, the hot threshold is set as the point on the curve beyond which the slope definitively decreases to below 10%.  
To set the threshold values, a preprocessing activity was performed in which, by varying these values, the corresponding hot and cold spots were detected, and their average reliability was measured. The threshold values of hot and cold spots equal to the points on the curve beyond which (below which) the slope decreases (increases) by 10% are those that provided the highest average reliability.  
Furthermore, the resulting hot and cold spots were compared with a high-resolution digital orthophoto. This comparison was performed by observing that the detected hotspots corresponded on the digital orthophoto to predominantly impermeable areas with high building density and/or no vegetation and that the cold spots corresponded to areas predominantly with vegetal cover.  
Figure 2: Example of a centroid Cartesian plot: LST vs. NDVI.  
After selecting the two thresholds in the plot, the urban area hotspots are generated by aggregating neighboring pixels belonging to clusters having centroids beyond the hot threshold. Similarly, cold spots are generated by aggregating neighboring pixels belonging to clusters having centroids before the cold threshold.