# Phase-Type Distributions of Animal Trajectories with Random Walks
## 3 System Model
### Trajectory Model Using Phase-Type Distributions
#### 3.1.1 Mangabey Monkey  
According to the real trajectories obtained from the mangabey monkey, the parameters of the phase-type distributions are shown in Table 2a, and the corresponding parameters are calculated using Equations (6)-(8).  
Specifically, we observe that \(T_{fix}\) and \(T_{mov}\) have \(CoV>1\), indicating a hyper-exponential distribution, while \(A_{traj}\) has \(CoV<1\), indicating an Erlang distribution.  
The level of significance for the \(\chi^{2}\) test is set to 5%, with \(K=50\) bins. Figure 8a shows the comparison between the proposed and real distributions for the RVs.  
A level of significance for the Kolmogorov-Smirnov test (LSK) is proposed of 1%, and the value of \(D_{table}\) is calculated from Appendix A. Figure 8b shows the comparison between the proposed and real cumulative distributions. It can be observed that in Figure 8b, parameter \(D\) is the difference between the values of the proposed and real CDFs.  
The value of \(\chi^{2}\) is compared with the \(\chi^{2}_{table}\), and the value of \(D_{max}\) is compared to the value \(D_{table}\) to validate the hypothesis distribution. The calculated values of the parameters are shown in Table 3. Hence, \(T_{fix}\) follows a hyper-exponential, \(T_{mov}\) follows a hyper-exponential, and \(A_{traj}\) follows an Erlang distribution.  
Finally, in Figure 8c, we can observe that for lower theoretical quantiles, the sample quantiles of the random variable follow a straight line; however, this alignment degrades with larger values.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{5}{c}{**(a) Mangabey Monkey**} \\ \hline
**Random Variable** & \(p\) & \(\lambda_{1}\) & \(\lambda_{2}\) & \(\lambda_{3}\) & \(\lambda_{4}\) & \(k\) \\ \hline \(t_{fix}\) & 0.01247 & \(2\times 10^{-5}\) & \(4\times 10^{-4}\) & - & - & - \\ \(t_{mov}\) & 0.021713 & \(1\times 10^{-6}\) & \(5\times 10^{-5}\) & - & - & - \\ \(A_{traj}\) & - & - & - & - & 0.0100 & 1 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameters from the random variables.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{5}{c}{**(b) Ocelot**} \\ \hline
**Random Variable** & \(\mathbf{p}\) & \(\mathbf{\lambda_{1}}\) & \(\mathbf{\lambda_{2}}\) & \(\mathbf{\lambda_{3}}\) & \(\mathbf{\lambda_{4}}\) & \(k\) \\ \hline \(t_{fix}\) & 0.2484 & \(1\times 10^{-4}\) & \(3\times 10^{-3}\) & - & - & - \\ \(t_{mov}\) & 0.1103 & \(3\times 10^{-6}\) & \(5\times 10^{-5}\) & - & - & - \\ \(A_{traj}\) & - & - & - & - & 0.01650 & 2 \\ \hline \multicolumn{5}{c}{**(c) Bat**} \\ \hline
**Random Variable** & \(\mathbf{p}\) & \(\mathbf{\lambda_{1}}\) & \(\mathbf{\lambda_{2}}\) & \(\mathbf{\lambda_{3}}\) & \(\mathbf{\lambda_{4}}\) & \(k\) \\ \hline \(t_{fix}\) & 0.03352 & \(5\times 10^{-5}\) & \(9\times 10^{-4}\) & - & - & - \\ \(t_{mov}\) & 0.02275 & \(4.7\times 10^{-5}\) & \(2\times 10^{-3}\) & - & - & - \\ \(A_{traj}\) & 0.6355 & 0.01 & 0.15 & - & - & - \\ \hline \multicolumn{5}{c}{**(d) Long-Tailed Duck**} \\ \hline
**Random Variable** & \(\mathbf{p}\) & \(\mathbf{\lambda_{1}}\) & \(\mathbf{\lambda_{2}}\) & \(\mathbf{\lambda_{3}}\) & \(\mathbf{\lambda_{4}}\) & \(k\) \\ \hline \(t_{fix}\) & 0.0137 & \(1\times 10^{-5}\) & \(3\times 10^{-4}\) & - & - & - \\ \(t_{mov}\) & 0.08980 & \(9\times 10^{-6}\) & \(5\times 10^{-4}\) & - & - & - \\ \(A_{traj}\) & - & - & - & - & 0.0280 & 4 \\ \hline \hline \end{tabular}
\end{table}
Table 2: _Cont._  
Figure 8: Hypothetical and real PDFs and CDFs and Q–Q plots for the mangabey monkey. **(a)** PDF comparison, **(b)** CDF comparison, **(c)** Red and blue lines are the theoretical and practical quantiles respectively.