# Probabilistic Modeling of Dust-Induced FSO Attenuation for 5G/6G Backhaul in Arid Regions
## 3 Results and Discussions  
The distributions selected for this study--Lognormal, Gamma, Weibull, Exponentiated Gamma, and Johnson SB--were chosen based on their widespread application in modeling positively skewed and bounded physical phenomena, including those in optical and atmospheric communication channels. These distributions provided varying levels of shape  
Figure 1: Experimental dust chamber setup for controlled FSO attenuation measurements. flexibility and tail behavior, allowing us to assess their effectiveness in modeling empirical attenuation data under different dust visibility levels [13; 21; 22]. These distributions are mathematically defined as follows:  
_Lognormal Distribution_[21; 22]:  
\[f(x)=\frac{1}{x\sigma\sqrt{2\pi}}\exp\biggl{(}-\frac{(\ln x-\mu)^{2}}{2\sigma^{ 2}}\biggr{)},\quad x>0 \tag{3}\]  
where \(\mu\) is the mean and \(\sigma\) is the standard deviation of the variable's natural logarithm. _Gamma Distribution_[23]:  
\[f(x)=\frac{\beta^{\alpha}x^{\alpha-1}e^{-\beta x}}{\Gamma(\alpha)},\quad x>0 \tag{4}\]  
where \(\alpha\) is the shape parameter and \(\beta\) is the rate parameter (inverse scale). _Weibull Distribution_[23]:  
\[f(x)=\frac{k}{\lambda}\Bigl{(}\frac{x}{\lambda}\Bigr{)}^{k-1}e^{-(x/\lambda) ^{k}},\quad x>0 \tag{5}\]  
where \(k\) is the shape parameter and \(\lambda\) is the scale parameter. _Exponentiated Gamma Distribution_[24]:  
\[f(x)=\frac{a^{b}x^{b-1}e^{-ax}[1-e^{-ax}]^{c-1}}{\Gamma(b)},\quad x>0 \tag{6}\]  
where \(a\) is the rate parameter, \(b\) is the shape parameter, and \(c\) is the exponentiation parameter controlling the tail behavior. _Johnson SB Distribution_[13; 25]:  
\[f(x)=\frac{\delta}{\lambda z(1-z)\sqrt{2\pi}}\exp\left(-\frac{1}{2}\biggl{[} \gamma+\delta\ln\biggl{(}\frac{z}{1-z}\biggr{)}\biggr{]}^{2}\right),\quad z= \frac{x-\xi}{\lambda},\ 0<z<1 \tag{7}\]  
The Johnson SB distribution, a member of the [PERSON] system of distributions, has four parameters--shape parameters \(\gamma\) and \(\delta\) and bounded range parameters \(\xi\) and \(\lambda\)--which allow the distribution to model various degrees of skewness and kurtosis flexibly. A critical parameter in histogram-based probability density function (PDF) fitting is the number of bins used for visualizing and computing the distribution fit. In this study, we selected 75 bins for the dense and light dust data and 60 bins for the moderate dust data. These numbers were empirically determined to balance resolution and noise suppression. Too few bins may smooth out important variations in the data, while too many bins may lead to overfitting or erratic fit quality. The chosen bin numbers produced stable and visually interpretable histograms, facilitating effective model fitting. Three commonly used statistical metrics were employed to evaluate the accuracy of the distribution fitting to the empirical FSO attenuation data: the RMSE, MAE, and \(R^{2}\). Additionally, the Log Likelihood (LogL) was included to assess the likelihood of the observed data, thereby providing a probabilistic evaluation of model fit. These metrics are defined as follows:  
* _Root Mean Square Error (RMSE):_ Measures the square root of the average squared differences between the observed and predicted data values. It is computed as follows: \[\text{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^{n}(y_{i}-\hat{y}_{i})^{2}},\] (8)where \(n\) is the total number of data points, \(y_{i}\) is the actual (measured) value, and \(\hat{y}_{i}\) is the predicted value from the model. A lower RMSE value indicates a better fit. Values approaching zero suggest minimal deviation from the measured data. * _Mean Absolute Error (MAE):_ Computes the average of the absolute differences between the predicted and observed values. It is given mathematically by the following: \[\text{MAE}=\frac{1}{n}\sum_{i=1}^{n}|y_{i}-\hat{y}_{i}|,\] (9) where \(n\) is the number of observations, \(y_{i}\) is the actual (measured) value, and \(\hat{y}_{i}\) is the predicted value. Similar to the RMSE, a lower MAE value indicates a better fit.