## 10 Comparative Analysis of Propagation Models  
In this section, building on the previously presented results, we evaluate the performance of the individual propagation models across the examined frequency bands using comparative metrics and statistical analysis. Particular attention is given to exploring the relationship between the standard deviation of the predicted field strength levels and the standard deviation of terrain elevation. The comparative assessment is carried out separately for each frequency, allowing for a detailed and differentiated evaluation of model performance. This article examines four statistical metrics [62]. In the following expressions, \(x_{i}\) denotes the \(i\)-th reference value determined by the deterministic Parabolic Equation Model (PEM), while \(\hat{x}_{i}\) represents the corresponding sample value, i.e., the signal strength level estimated by a non-deterministic model at the same location, and \(n\) is the number of calculated points. The first metric is the Root Mean Square Error (RMSE), which quantifies the magnitude of the average deviation on a squared scale:  
\[\text{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^{n}(x_{i}-\hat{x}_{i})^{2}}. \tag{49}\]  
The RMSE is sensitive to large errors; therefore, a high RMSE value indicates a substantial, frequent, or severe deviation from the reference value. The Mean Absolute Error (MAE) represents the average magnitude of error in absolute terms, and is less sensitive to outliers than the RMSE. Its interpretation is straightforward: on average, this is how much the model deviates from the reference. The definition is:  
\[\text{MAE}=\frac{1}{n}\sum_{i=1}^{n}|x_{i}-\hat{x}_{i}|. \tag{50}\]  
The mean deviation direction (Bias) indicates whether the model in question tends to overestimate or underestimate the values calculated by the reference model. A positive bias suggests that the model generally overestimates, while a negative bias indicates underestimation:  
\[\text{Bias}=\frac{1}{n}\sum_{i=1}^{n}(\hat{x}_{i}-x_{i}). \tag{51}\]  
Last but not least, the Relative Error (RelErr) is an intuitive metric that expresses the magnitude of deviations in relation to the reference model, expressed as a percentage:  
\[\text{RelErr}=\frac{1}{n}\sum_{i=1}^{n}\left|\frac{\hat{x}_{i}-x_{i}}{x_{i}} \right|\cdot 100. \tag{52}\]  
Further insights into the behavior of each wave propagation model can be gained by examining the standard deviation of the predicted field strength levels and the corresponding terrain elevations. For this purpose, we apply the classical formula for standard deviation:  
\[\sigma_{x}=\sqrt{\frac{1}{n}\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}}, \tag{53}\]  
where \(x_{i}\) denotes the \(i\)-th element of the sample of size \(n\), and \(\bar{x}\) represents the sample mean. We also examine the Pearson correlation coefficient, which provides information about the strength and direction of the linear relationship between two variables--in this case, terrain elevation and field strength level:  
\[r=\frac{\sum_{i=1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sqrt{\sum_{i=1}^{n}(x_ {i}-\bar{x})^{2}}\cdot\sqrt{\sum_{i=1}^{m}(y_{i}-\bar{y})^{2}}}, \tag{54}\]  
where \(y_{i}\) denotes the \(i\)-th element of the other sample of size \(n\), and \(\bar{y}\) represents its mean. The correlation coefficient \(r\) takes values between \(+1\) and \(-1\). A value of \(+1\) indicates a strong positive correlation--in this case, an increase in the standard deviation of terrain elevation is associated with an increase in the standard deviation of the predicted field strength levels, which reflects a strong terrain sensitivity of the given wave propagation model. A value of \(-1\) suggests that increasing variation in terrain elevation corresponds to decreasing variation in field strength, while a value of \(0\) implies no linear relationship between the two variables. The comparative analyses are presented separately for each frequency band, namely 3.6 GHz and 6 GHz.