# Phase-Type Distributions of Animal Trajectories with Random Walks
## 3 System Model  
In this section, we provide a detailed description of the animal trajectory model along with the assumptions made. We consider a realistic scenario where animals move with random trajectories that are characterized and approximated using phase-type distributions. Multiple virtual trajectories will be generated, possessing the same (or similar) statistical properties as the real animal paths. These virtual traces allow for the evaluation of the monitoring system performance without the need to collect large amounts of scarce and difficult-to-obtain data. To begin, we identify the region where the animal of interest naturally moves, denoted as \(A_{obs}\). Real traces of the four animals considered in this study are obtained from available studies in the Movebank database [18; 32; 46; 47; 48]. These traces are depicted in Figure 2.  
Using the real traces, we compare the statistical characteristics of the virtual traces generated by the proposed random walk model. If the virtual traces demonstrate similar characteristics, they can serve as alternatives to the real traces. Based on the available data, we analyze different observation areas, \(A_{obs}\), depending on the animals' movement patterns. Specifically, an area ranging from 20,000 \(\times\) 20,000 m to 200,000 \(\times\) 200,000 m is selected for the monkey, ocelot, and bat. For the long-tailed duck, an area ranging from 1,000,000 \(\times\) 1,000,000 m to 9,000,000 \(\times\) 9,000,000 m is chosen, as this species does not venture beyond this range in its natural habitat [49]. Figure 2: Real trajectories in Google Maps. (**a**) Mangabey monkey, (**b**) ocelot, (**c**) bat, (**d**) long-tailed duck. Individual virtual trajectories of each animal within the aforementioned areas are illustrated in Figure 3a-d. The black line represents the virtual path, while the blue and red dots represent the trajectory's start and end points, respectively. The procedure is depicted in the flow diagram shown in Figure 4, starting with the selection of a species' real trajectory available in the database. Random variables (RVs) and coefficient of variation (\(CoV\)) are calculated based on the real trajectories of each animal. Suitable phase distributions are selected for each RV according to their \(CoV\). Goodness of fit tests are performed to determine if the RVs follow the proposed phase distributions. If the tests yield positive results, the phase distributions for each RV are accepted and can be used to simulate and validate the movement patterns for each animal. If none of the proposed phase distributions are validated, an alternative species from the database must be selected. Utilizing a model selection method introduces the challenge that a phase-like distribution may not be chosen, rendering the advantages of using Markov chains unsuitable. Furthermore, it becomes impossible to measure or detect if the distribution was poorly selected since our aim is to approximate the random walk behavior through the \(CoV\). It is possible that the random walk follows a distribution other than phase-type that provides a better approximation, but there is no inherently incorrect selection. Our objective is not to identify the optimal distribution through distribution selection methods, but rather to approximate the observed data using Markov chains with phase-type distributions. Figure 3: Random walk examples inside the \(A_{\mathrm{obs}}\) (snapshots of the simulation trajectories animation in real time captured while they are still in progress and have not yet concluded). (**a**) Monkey virtual trajectory, (**b**) ocelot virtual trajectory, (**c**) bat virtual trajectory, (**d**) long-tailed duck virtual trajectory. For an exponential PDF, even if the experimental results do not have a coefficient of variation (\(CoV\)) of exactly 1, the exponential distribution can still be employed for \(CoVs\) that are close to 1. This can lead to a good approximation, which can be verified through goodness of fit tests.