# Probabilistic Modeling of Dust-Induced FSO Attenuation for 5G/6G Backhaul in Arid Regions
## 3 Results and Discussions
The MAE is generally less sensitive to outliers compared to the RMSE. * _Coefficient of Determination (\(R^{2}\))_: Reflects the proportion of variance in the observed data that is predictable from the fitted model. It is mathematically defined as follows: \[R^{2}=1-\frac{\sum_{i=1}^{n}(y_{i}-\hat{y}_{i})^{2}}{\sum_{i=1}^{n}(y_{i}- \bar{y})^{2}},\] (10) where \(n\) is the number of observations, \(y_{i}\) is the actual (measured) value, \(\hat{y}_{i}\) is the predicted value, and \(\bar{y}\) is the mean of the actual values. An \(R^{2}\) value close to 1 indicates an excellent fit. Typically, values above 0.90 are considered very good in modeling physical data. * _Log Likelihood (LogL)_: Represents the logarithm of the likelihood function and quantifies how well a statistical model explains the observed data. A higher LogL value indicates a better fit of the model to the data. LogL is mathematically defined as follows: \[\log\mathcal{L}=\sum_{i=1}^{n}\log f(y_{i};\theta),\] (11) where \(f(y_{i};\theta)\) is the PDF evaluated at the observed data \(y_{i}\) given the model parameter \(\theta\), and \(n\) is the number of observations. Using all these metrics provides a more comprehensive evaluation of model performance than relying on a single measure. While the RMSE highlights the presence of large deviations, the MAE offers an intuitive measure of average error, and \(R^{2}\) indicates the proportion of variance explained by the model. LogL reflects the overall likelihood of observing the data under the fitted distribution model. Together, these four metrics ensure a balanced assessment by capturing error magnitude, explanatory power, and statistical likelihood. Figure 2 presents the fitted PDFs overlaid on histograms for each dust visibility regime: dense (first column), moderate (second column), and light (third column). In dense dust conditions, most candidate distributions fail to adequately model the sharp peak and long right tail present in the data. The Johnson SB distribution stands out, as it closely follows the measured histogram's behavior, maintaining alignment across central and extreme attenuation values. In the moderate dust regime, the Lognormal, Gamma, and Johnson SB distributions perform well visually; however, Johnson SB demonstrates a better fit in the central region and avoids the over-smoothing observed in the tails of the others. For light dust, several distributions--including the Weibull, Gamma, and Exponentiated Gamma--exhibit strong visual correspondence with the measured data. However, the Johnson SB again offers the most consistent and balanced representation. These visual comparisons highlight the superior adaptability of the Johnson SB distribution in fitting varying dust conditions, a trend that matches the quantitative goodness-of-fit metrics discussed below. \[f(x)=\frac{1}{x\sigma\sqrt{2\pi}}\exp\biggl{(}-\frac{(\ln x-\mu)^{2}}{2\sigma^{ 2}}\biggr{)},\quad x>0 \tag{3}\]0 Measured histograms with fitted PDFs for various distributions across three dust conditions: dense (left), moderate (center), and light (right). The Johnson SB distribution shows superior visual alignment with empirical data in all regimes. Subfigures represent the following distributions: Lognormal (**a**)–(**c**), Gamma (**d**)–(**f**), Weibull (**g**)–(**i**), Exponential Gamma (**j**)–(**l**), and Johnson SB (**m**)–(**o**). Table 1 presents the performance metrics--the RMSE, MAE, \(R^{2}\), and LogL--for each fitted distribution in the three dust visibility regimes. Among all models, the Johnson SB distribution consistently achieved the best results, with RMSE and MAE values approaching zero and an \(R^{2}\geq 0.95\) under all conditions, indicating excellent agreement with the empirical data. However, under dense dust conditions, the Johnson SB distribution exhibited a more negative LogL value, likely due to its heavy-tailed structure capturing extreme attenuation values, which slightly reduced its statistical likelihood compared to other models in this specific regime.