## 4 Methodology
### Validation  
For validation, the damage identified by our approach was compared with the official map by the US Geological Survey (USGS), which it was obtained from their portal. The FEMA (Federal Emergency Management Agency) HAZUS loss model was used for earthquake damage validation. We used fundamental Equation (2) to create a simulated earthquake official damage map. \[\text{Loss}=\text{Hazard}\times\text{Vulnerability}\times\text{ Exposure} \tag{2}\]  
In Equation (2), Hazard represents earthquake intensity by USGS. The intensity of the earthquake takes into account the magnitude of ground shaking at a distance from the epicenter and offers a specific estimate of the probable damage. The HAZUS building grid was applied for exposure and vulnerability variables, which included data on the compiled building category and building costs [5]. The HAZUS map consists of two fields, AEL and APEL. AEL has estimated annualized earthquake losses for a specific spatial unit. APEL is annualized earthquake loss ratio (AELR), which is computed as the AEL proportion of a particular spatial unit compared to that unit's complete building value. AELR is multiplied by 100 to enable analysis and conversation and is called an annualized earthquake loss (APEL) percent. Therefore, the APEL field information, along with the PGA data layer, was used to evaluate the results of the damage map with the HAZUS map [38]. To validate the produced damage map by each algorithm at a 10 \(\times\) 10 km grid spatial unit, confusion matrix, Spearman's Rho, Kendall's tau, and Pearson correlation were applied. To validate the produced damage map at the city and county spatial units, Spearman's Rho, Kendall's tau, and Pearson correlation were applied. The 10 \(\times\) 10 km grid spatial unit produced a damage map based on our approach, and the official map was classified into 4 classes (weak, strong, very strong, and severe). Then, the statistical indices, including accuracy, precision, recall, and F-measure, were computed based on the confusion matrix according to Equations (3)-(6):  
\[\text{Overall Accuracy}=\text{TP+TN}/(\text{TP + FP+TN+ FN}) \tag{3}\]\[\text{Precision = TP/(TP + FP)} \tag{4}\] \[\text{Recall = TP/(TP + FN)}\] (5) \[\text{F-score = 2 \times ((Precision \times Recall)/(Precision + Recall))} \tag{6}\]  
where:  
* True Positives (TP): These are items in which predicted damage was also damage in reality. * True Negatives (TN): Predicted non-damage was, in reality, non-damage. * False Positives (FP): Predicted damage was, in reality, non-damage. * False Negatives (FN): Predicted non-damage was, in reality, damage. The Pearson correlation is a commonly utilized index of correlation. It is a metric of the linear correlation of two, \(i\) and \(j\), parameters. Where \(i\) is the amount of damage estimated by our approach at three spatial units and j is the value of official loss model at three scales (spatial units). The range of values are among +1 to \(-1\), in which 1 is a complete positive linear correlation, 0 is no linear correlation, and \(-1\) is a complete negative linear correlation [44]. The Spearman's Rho among two parameters is identical to the Pearson among these two parameters' rank values. Though [PERSON]'s evaluates linear relations, [PERSON]'s evaluates monotonous (linear or non-linear) relations. [PERSON]'s Rho is computed as Equation (7):  
\[\rho=1-\frac{6\sum_{i}F_{i}^{2}}{N(N^{2}-1)} \tag{7}\]  
where \(N\) is the number of spatial units (city, county, and grids) and \(F_{i}=r_{i}-s_{i}\) is the difference between the factual ranking (damage ranking based on our approach at each of the three spatial units) and the expected ranking (damage ranking based on official loss model data at each of the three spatial units). It could be any value between \(-1\) to \(1\), and the nearer the measure's absolute value to \(1\), the greater the relationship and \(-1\) indicates a strong negative correlation. In addition, the value of 0 indicates a lack of correlation.