# Phase-Type Distributions of Animal Trajectories with Random Walks
## 5 Results  
In this section, we present the results of the virtual trajectory generation procedure based on phase-type distributions proposed. To validate our results, we compare the residence time of the different animals in a given region, considering both the virtual and real trajectories. According to the goodness of fit analysis, we can conclude that there are sufficient samples to approximate the random variables involved in the animal's movement (movement time, pause time, and direction) using phase-type distributions. The results of this analysis for each selected animal are provided in Table 7.  
As described in Section 3, the probabilities of finding the animal inside the region of interest (\(R_{cov}\)) (\(P_{in}\)) and outside the region (\(P_{out}\)) are calculated for both the virtual and real trajectories, considering a constant restricted area where the animal can freely move (\(A_{obs}\)). This calculation is performed in Algorithm 1 for each animal. As an example, Figure 13 presents a screen capture depicting the animated running simulation of the mangabey monkey virtual walk. This visualization offers insights into the virtual random walk, demonstrating instances where it traverses both within and outside the expanding coverage radius, symbolized by the green ring denoted as \(R_{cov}\). Figure 14a presents the probabilities associated with the mangabey monkey's presence inside and outside a specified region for \(A_{obs}\) = 80,000\(\sim\)\(m\)\(\times\) 80,000\(\sim\)\(m\) across varying values of \(R_{cov}\). These probabilities are calculated employing the statistical parameters derived from Section 3. The red line in Figure 14a represents the probability that the actual trajectory of the mangabey monkey falls within or outside the radio coverage area. In contrast, the blue line depicts the \(P_{in}\) and \(P_{out}\) values corresponding to the virtual generated trajectory, with the average speed of the animal's movement. Additionally, magenta and green lines represent the \(P_{in}\) and \(P_{out}\) for the animal when its average movement speed is increased. This illustrates the insight that an increased movement speed corresponds to a reduced probability of remaining within the radio coverage area and the probability of remaining outside is increased. Likewise, Figure 14b provides a visualization of probabilities pertaining to the ocelot's behavior. The real traces follow the trend of the red line, while the calculated and augmented ocelot's average velocities are denoted by the blue, magenta, and green lines, respectively. Notably, it becomes evident that despite the increase in the ocelot's speed, it remains within its natural habitat. Moving on to Figure 14c, we present the outcomes for bats in terms of both \(P_{in}\) and \(P_{out}\) probabilities. The simulation traces align with the real traces. The red line corresponds to the actual path, while the blue, magenta, and green lines denote the \(P_{in}\) and \(P_{out}\) probabilities at varying \(v_{avg}\). Notably, for a calculated average speed of 6 m/s, the probability of bats being inside or outside their coverage area closely resembles the real path. However, with increased velocity, it is evident that they deviate from their natural environment. It should be noted that the inside and outside probabilities for the real trajectory of the bat follow a very stable line, as the bat's trajectory has limited variations. Finally, Figure 14d shows the probabilities of the long-tailed duck moving inside and outside the area coverage and different values of \(R_{cov}\). For the duck, an average velocity of 5 m/s was calculated, hence, it is observed that the virtual trace, depicted with a blue line, is similar to the real trace (red line). If the ducks increase their speed, they remain in the coverage area, the outside probability is decreased. Figure 13: Simulation example of the mangabey monkey random walk inside and outside a radio coverage region \(R_{cov}\). Green and red dots are the start and end points respectively of the random walk.