# Phase-Type Distributions of Animal Trajectories with Random Walks
## 3 System Model
### Trajectory Model Using Phase-Type Distributions
The rationale behind using these distributions, such as the Erlang and hyper-exponential distributions, is that they allow us to utilize a Markovian model, specifically Markov chains, to describe the movement of the animals. These distributions can be easily incorporated into tele-traffic analysis, which is commonly used for performance analysis in wireless sensor networks (WSNs) [7]. Figure 7: Probability density functions of the RVs for each species. (**a**) Mangabey monkey, (**b**) ocelot, (**c**) bat, (**d**) long-tailed duck. **Remark 1**. : _In future work, the use of phase-type distributions to calculate the average times and distances between sensors placed on the animals will be considered. This can help to mitigate the effect of areas without transceiver coverage caused by ground-based sensors. However, these studies are beyond the scope of the current contribution, as the main aim of this approach is to present the analytical tool for generating virtual traces of animals._  
Hence, we define that RVs with \(CoV>1\) have a hyper-exponential PDF, RVs with \(CoV=1\) have an exponential PDF, and RVs with \(CoV<1\) have an Erlang PDF. The probability density functions (PDFs) of the random variables are calculated using Equations (3)-(5). \[f_{hyper}(x)=p\lambda_{1}e^{-\lambda_{1}x}+(1-p)\lambda_{2}e^{-\lambda_{2}x}, \tag{3}\]  
\[f_{expn}(x)=\lambda_{3}e^{-\lambda_{3}x}, \tag{4}\]  
\[f_{Erlang}(x)=\frac{\lambda_{4}x^{k-1}e^{-\lambda_{4}k}}{(k-1)!} \tag{5}\]  
The parameter \(p\) is obtained with (6)  
Figure 5:0  
and \(\lambda_{1}\) and \(\lambda_{2}\) are proposed constants. The parameter \(\lambda_{3}\) is calculated using (7). Figure 5:1  
Finally, the parameter \(\lambda_{4}\) is calculated using Equation (8) for the negative exponential distribution,  
Figure 5:2  
where \(k\) is an arbitrary number used to calibrate the PDF. None of the RVs obtained in this manuscript have a \(CoV=1\) (or follows an exponential distribution). However, we include it in case a different species actually has that coefficient of variation. To validate these hypothetical distributions, two theoretical goodness of fit tests were used: the chi-squared (CS) test [52] and the Kolmogorov-Smirnov (KS) test [53]. These tests are used to verify if a random variable follows a certain PDF, as explained in Appendix A. For the CS test, we use the distribution of each random variable for each species, respectively. The significance level (\(\alpha\)) for the \(\chi^{2}\) test is set to 0.05 (or 5%) as is recommended for statistically significant tests with large sample sizes. A result probability \(p\leq 0.05\) indicates that the hypothetical PDF test is negative or should be rejected, while \(p\geq 0.05\) \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Random Variable** & **Animal Species** & \(CoV\) & \(\sigma_{X}\) & \(E_{X}\) \\ \hline \multirow{4}{*}{\(T_{fix}\)} & Mangabey monkey & 3.86 & \(1.22\times 10^{4}\) & \(6.10\times 10^{3}\) \\  & Ocelot & 3.44 & \(1.48\times 10^{4}\) & \(4.32\times 10^{3}\) \\  & Bat & 2.95 & \(5.14\times 10^{3}\) & \(1.74\times 10^{3}\) \\  & Long-tailed duck & 2.91 & \(1.36\times 10^{4}\) & \(4.66\times 10^{3}\) \\ \hline \multirow{4}{*}{\(T_{mwo}\)} & Mangabey monkey & 4.42 & \(3.26\times 10^{5}\) & \(7.39\times 10^{4}\) \\  & Ocelot & 1.93 & \(1.04\times 10^{5}\) & \(5.37\times 10^{4}\) \\  & Bat & 4.61 & \(4.48\times 10^{3}\) & \(9.72\times 10^{2}\) \\  & Long-tailed duck & 2.11 & \(2.5\times 10^{4}\) & \(1.18\times 10^{4}\) \\ \hline \multirow{4}{*}{\(A_{traj}\)} & Mangabey monkey & 0.92 & 91.79 & 99.16 \\  & Ocelot & 0.65 & 79.23 & 121 \\ \cline{1-1}  & Bat & 1.28 & 84.57 & 65.98 \\ \cline{1-1}  & Long-tailed duck & 0.53 & 76.76 & 142.60 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Statistical parameters of the RVs. means that the hypothesis is accepted.