# Phase-Type Distributions of Animal Trajectories with Random Walks
## 3 System Model
### Trajectory Model Using Other Distributions  
In the preceding section (Section 3.1), we introduced the phase-type distribution as a proposed model for this manuscript, primarily due to its memory properties and its applicability in Markov chain models. However, it is imperative to validate this proposal by subjecting our random walk models to a comparative analysis against other similar distributions. For the purpose of comparison, we have chosen to evaluate our proposed phase-type distribution against widely recognized distributions, namely, the normal, log-normal, and Pareto distributions. These selections were made based on their ability to exhibit different characteristics of the coefficient of variation, where \(CoV\) values approximate to 0, \(CoV\) values approximate to 1, and \(CoV\) values exceed 1, respectively. To illustrate this comparison, we have specifically considered three random variables sourced from the animals under study, taking into account their associated \(CoV\) values as presented in Table 1. For instance, for the ocelot, where \(A_{traj}\) possesses a \(CoV\) of 0.65, we have conducted a comparison against the normal distribution. Similarly, in the case of the mangabey monkey, \(A_{traj}\), with a \(CoV\) of 0.92, has been contrasted with the log-normal distribution, while \(T_{move}\), with a \(CoV\) of 4.42, has been matched with the Pareto distribution. The mathematical formulations employed for calculating the normal, log-normal, and Pareto distributions are elucidated in Appendix C. To facilitate a comprehensive understanding, Table 7 presents the \(CoV\) values of the three animal random variables, accompanied by their respective hypothetical distribution counterparts. In summary, the comparison undertaken against well-established distributions contributes to the thorough evaluation and validation of our proposed phase-type distribution within the context of our random walk models. Utilizing the methodology outlined in Section 3.1, we employed a consistent process to compute the PDFs and CDFs for each selected random variable. Following this, based \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Random Variable** & \(\chi^{2}\) & \(\chi^{2}_{table}\) & \(D_{max}\) & \(D_{table}\) \\ \hline \(T_{fix}\) & 5.7931 & 39.0882 & 0.1245 & 0.3651 \\ \(T_{move}\) & 3.7237 & 39.0882 & 0.1425 & 0.3651 \\ \(A_{traj}\) & 14.7590 & 62.9489 & 0.1337 & 0.2828 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Goodness of fit parameters for long-tailed duck. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Animal Species** & \(t_{fix}\) & \(t_{move}\) & \(A_{traj}\) \\ \hline Mangabey monkey from Uganda & Hyper-exponential & Hyper-exponential & Erlang \\ Ocelot from Barro Colorado & Hyper-exponential & Hyper-exponential & Erlang \\ Bat from Ghana & Hyper-exponential & Hyper-exponential & Hyper-exponential \\ Arctic long-tailed duck & Hyper-exponential & Hyper-exponential & Erlang \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Results of the goodness of fit tests. on their respective coefficient of variation values, we established the corresponding hypothetical distribution for each variable. Subsequently, we compared these hypothetical distributions with the computed distributions of the actual animal trajectories through rigorous goodness of fit tests. Presented in Figure 12a are comparisons between the PDFs of the actual animal trajectories and the PDFs derived from the corresponding normal, log-normal, and Pareto hypothetical distributions. Additionally, Figure 12b offer a visual exploration, showcasing the cumulative distribution functions. These figures provide side-by-side representations, contrasting the hypothetical distribution CDFs with the CDFs of the actual trajectories for each of the random variables associated with the species. This systematic approach allows us to assess the concordance between the proposed hypothetical distributions and the empirical data.