## 3 Results
### Kullback-Leibler Divergence  
A fundamental information theoretical concept is the Kullback-Leibler (KL) divergence, providing a method to measure the conventionally defined \"distance\" among two distributions [21]. Let \(p\) and \(q\) be two distinct distributions defined on the same probability space, where \(p\) refers to a theoretical probability distribution and is measured against an experimental or modeled distribution, \(q\). The KL divergence, also called the relative entropy, measures the inefficiency of assuming a distribution to be \(q\) when in reality it is \(p\). The relative entropy or KL divergence between two distributions is given as [22],  
\[D_{KL}(p||q)=\sum_{i}p_{i}\log_{2}\frac{p_{i}}{q_{i}} \tag{11}\]  
The KL divergence is non-negative and asymmetric in \(p\) and \(q\), i.e.,  
\[D_{KL}(p||q)\
eq D_{KL}(q||p) \tag{12}\]  
The lower the value of the \(D_{KL}\), the less the distance between \(p\) and \(q\), which ultimately becomes zero if the two distributions are identical. Additionally, if \(p_{i}>0\) and \(q_{i}=0\) for a given \(i\), the KL divergence goes to infinity. [PERSON] and [PERSON] give a technical interpretation of the KL divergence, as the \"coding penalty\" associated with a distribution \(q\) selected to compare with a distribution \(p\)[22].  
Five theoretical distribution models where utilized to apply the Kullback-Leibler divergence Equation (11) in order to evaluate and compare their fit performance to the real-valued RSSI data. The theoretical PDFs used were:  
\[\text{Weibull}:\ f(x)=\frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1}e^{ -\left(x/\lambda\right)^{k}} \tag{13}\]  
\[\text{Gamma}:\ f(x)=\frac{\beta^{q}x^{a-1}e^{-\beta x}}{\Gamma(a)},\ \text{for}\ x>\ 0\ \text{and}\ \alpha,\ \beta>\ 0 \tag{14}\]  
\[\text{Log}-\text{Normal}\ (\text{LN}):\ f(x)=\frac{1}{x\sigma\sqrt{2\pi}}\exp \left(-\frac{\left(\ln(x)+\sigma^{2}/2\right)^{2}}{2\sigma^{2}}\right) \tag{15}\]  
\[\text{Burr}:\ f(x)=ck\frac{x^{c-1}}{\left(1+x^{c}\right)^{k+1}} \tag{16}\]  
\[\text{Extreme}-\text{Value}\ (\text{EV}):\ f(x)=\frac{1}{\beta}e^{-\left(x+e^{-x }\right)} \tag{17}\]  
The Kullback-Leibler divergence has been calculated for these five PDFs and the best fit to the observed RSSI data (30 November 2019-27 October 2020) was deduced. Initially, the frequency occurrence of each RSSI observation is calculated. The probability density of the RSSI data was then evaluated by dividing the frequency of each observation by the total number of observations. We then utilized the _distiff_ MATLAB application to obtain the parameters of the theoretical distributions within the observed RSSI values range (187-517). By employing Equation (11) for each model, the KL divergence has been calculated (see Figure 2). The Burr distribution yields the best fit to experimental data and, as expected, it exhibited the lowest KL divergence value, \(2.77\times 10^{-2}\), followed by the Weibull distribution yielding a value of \(3.07\times 10^{-2}\).  
Figure 4: Probability plots for (**a**) Burr and (**b**) Extreme-Value distribution against RSSI data.  
Figure 5: Probability plots for (**a**) Gamma and (**b**) Log-Normal distribution against RSSI data.  
It is obvious graphically from the probability plots that the Burr distribution best fits the RSSI data. In particular, an excellent agreement with the experimental data is observed, with the exception of the far upper tail of the distribution. In contrast, the rest four of the tested distributions exhibit a noteworthy disagreement at both tails (Figures 4-6).