## 4 The Proposed Framework
### The Case Study  
The experiment was conducted in the urban context of the city of Naples, Italy, as shown in Figure 3, due to its characteristics of high population density and building intensification.  
Naples is characterized by a relatively small territorial surface, equal to 11,727 hectares, and a high population density, with approximately 8566 inhabitants per square kilometer. The metropolitan area of the city has a population of 295,700 inhabitants; of these inhabitants, those residing in the municipality of Naples amount to 1,004,500 inhabitants.  
The city's climate is subtropical, slightly continental, and humid-subhumid with the near absence of intermediate seasons, featuring increasingly intense rain showers interspersed with long periods of drought.  
In recent years the city has seen an increase in the frequency of heat waves lasting more than a few days throughout the summer months.  
In many areas of the historic center and in recently developed areas, the expansion of concrete and asphalt surfaces has exacerbated the risk of thermal accumulation phenomena during the summer months, a phenomenon that has intensified in recent years due to the increasing frequency and duration of heat waves. The city center is characterized by a particularly high concentration of building and population density, with a significant prevalence of impermeable surfaces, to the detriment of the presence of urban green areas. On the contrary, the western and northwestern areas of the city are characterized by a greater diffusion of forest-type green areas, compared to the rest of the urban fabric. The eastern peripheral districts are predominantly intended for industrial uses, featuring a high building density along the coastal strip and a predominance of impermeable surfaces.  
The input data necessary for UHI hotspot detection consist of LST and NDVI raster datasets, both derived from remotely sensed imagery captured by the Landsat 8 satellite. These datasets were acquired on 30 July 2022, coinciding with a recorded heatwave event lasting 28 consecutive days, from 15 July to 11 August 2022. This ensures that the analysis is representative of extreme urban warming conditions. The imagery is publicly accessible through the USGS Earth Explorer platform.  
The LST raster, derived from the Thermal Infrared Sensor (TIRS) bands (bands 10 and 11), which measure surface thermal radiation, exhibits temperature values ranging from a minimum of 28 \({}^{\circ}\)C to a maximum of 56 \({}^{\circ}\)C; it is shown in Figure 4.  
Figure 4: LST of the study area.  
Figure 3: Study area of Naples, Italy.  
The NDVI raster, derived from the spectral bands of the Landsat Operational Land Imager (OLI) with a domain between \(-1\) and +1, records values ranging from a minimum of \(-0.16\) to a maximum of +1; it is shown in Figure 5. The NDVI raster was processed to match the spatial resolution of the LST dataset, ensuring accurate data integration for subsequent analysis.  
The two raster datasets represent the input data for the execution of the FCM clustering algorithm. The results and thematic maps of hotspot detection are shown in the following subsection.  
Following the calculation of the validity indices, it emerged that the optimal number of clusters is equal to 9. The plot graph of the trend of NDVI with respect to LST is shown in Figure 6, where the two threshold lines are plotted; the hotspots are obtained by dissolving adjoining pixels belonging to clusters 3, 4, and 6. The cold spots are obtained by dissolving adjoining pixels belonging to clusters 8, 1, and 9.  
Figure 5: NDVI of the study area.  
Figure 6: Centroid Cartesian plot: LST vs. NDVI.  
The model was implemented on a GIS platform built using the ESRI ArcGIS Pro 3.4 Suite. Preprocessing, clustering, and fuzzy reliability calculations were performed using ESRI ArcPy Python libraries for ArcGIS Pro 3.4, in which the FCM algorithm was implemented.