# Phase-Type Distributions of Animal Trajectories with Random Walks
## 3 System Model
### Trajectory Model Using Phase-Type Distributions  
Here, the phase-type PDF approximation to the animal movement is explained in detail. First, ten to twenty real trajectories were selected, each consisting of five hundred to one thousand GPS samples, extracted and calculated from the database. The selection of the number of trajectories and the sample size was based on the behavior of the species and the techniques employed by biologists to collect samples from them. For instance, ocelots are solitary animals that live alone for most of their lives, as described in [47], while mangabey monkeys, bats from Ghana, and long-tailed ducks live and move in groups. Therefore, monitoring a single animal over a period of time is sufficient to characterize their movement, as stated in [18; 32; 46; 48]. Similar values were observed among the samples from each species. Consequently, the average of the minimum and maximum travel distance (\(d_{min}\) and \(d_{max}\), respectively), maximum rest time (\(T_{fix,max}\)), maximum movement time (\(T_{mov,max}\)), and the average speed (\(v_{avg}\)) were calculated. Using these parameters, we can calculate the time the animals spend in a static state (\(T_{fix}\)), the time they spend in motion (\(T_{mov}\)), and the trajectory angles (\(A_{traj}\)). These  
Figure 4: Flow diagram of the general procedure. parameters are represented as random variables (RVs) and they characterize the trajectories followed by each animal. In general, the parameter \(\beta_{i}\) can be obtained using Equation (1). \[\beta_{i}= \begin{cases}0,&i=0,\\ \alpha_{i}-\alpha_{i-1},&i\
eq 0,\end{cases}  \tag{1}\]  
The relative angles \(\beta_{i}\) are obtained using the alpha angles \(\alpha_{i}\) with respect to the horizontal axis, as mentioned in [50]. Here, \(i=1,2,3,\dots\) represents the number of samples from the database. An example illustrating the process of obtaining the movement characterization is presented in Figure 5. This model includes the initial static distance (\(d_{fix1}\)), fixed time (\(t_{fix1}\)), and initial location (\(x_{1}\), \(y_{1}\)). When the animal moves to the first waypoint (\(x_{2}\), \(y_{2}\)), it has an initial time (\(t_{mov1}\)), initial distance (\(d_{mov1}\)), and initial angle (\(\alpha_{1}\)). Subsequently, at the second waypoint (\(x_{2}\), \(y_{2}\)), the animal rests with a fixed distance \(d_{fix2}\) and spends a rest time \(t_{fix2}\). Finally, the animal moves towards the third waypoint (\(x_{3}\), \(y_{3}\)), with a relative angle \(\beta_{2}\), a movement distance of \(d_{mov2}\), and a movement time of \(t_{mov2}\). It is required to described the random variables used in the proposed model:  
* Resting time (\(T_{fix}\)): Time at resting state. If the animal walks less than 5 m, it is assumed to be in a state of rest. * Moving time (\(T_{mov}\)): The time that the animal spends in motion. * Angle (\(A_{traj}\)): The direction in which the animal is moving. To analyze these variables, we first obtain the histograms for each animal, as depicted in Figure 5(a)-d. From these histograms, we calculate the probability density function (PDF) for each variable, as shown in Figure 6(a)-d.  
Figure 5: Trajectories model for relative angles \(\beta\). Figure 6: _Cont._  
Figure 6: Histograms of the RVs for each species. (**a**) Mangabey monkey, (**b**) ocelot, (**c**) bat, (**d**) long-tailed duck. Figure 7: _Cont._  
Following this, the coefficient of variation (\(CoV\)) is computed for each random variable of each species using the mean and variance. This is achieved using Equation (2)  
\[CoV=\frac{\sigma_{X}}{E_{X}}, \tag{2}\]  
where \(E_{X}\) is the mean of the random variable \(X\) and \(\sigma_{X}\) is the standard deviation. Table 1 shows the calculated statistical parameters. Now, we propose to use phase-type distributions to model the RVs [51].