## 6 Applying HMM to Student Data  
In applying the HMM model to our problem, we begin by identifying the set of hidden states corresponding to three different enrollment strategies: full-time enrollment strategy (FES), part-time enrollment strategy (PES), and mixed enrollment strategy (MES). The probability of students changing their enrollment strategy from one semester to the next \begin{table}
\begin{tabular}{l c c c} \hline \hline  & **First-Time-in-College** & **Transfer** \\ \hline Percentage & 39.5\% & 60.5\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Admission type distribution at UCF over 10 years. \begin{table}
\begin{tabular}{l c c} \hline \hline  & **Female** & **Male** \\ \hline Percentage & 56.2\% & 43.8\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Gender distribution at UCF over 10 years. \begin{table}
\begin{tabular}{l c c} \hline \hline  & **Final-Time-in-College** & **Transfer** \\ \hline Percentage & 39.5\% & 60.5\% \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Enrollment distribution by semesters at UCF over 10 years. is represented using the probability transition matrix \(A\). The probability of observing a particular enrollment status when a student uses a specific enrollment strategy is given by the emission matrix \(B\). Finally, \(\pi_{0}\) is the initial enrollment strategy distribution by which students start their education. Beginning with an initial guess for \(A\), \(B\), and \(\pi_{0}\), the Baum-Welch algorithm is applied to estimate the optimal model parameter set, \(\lambda^{*}\). Converging after 20 iterations, the following values for \(A\), \(B\), and \(\pi_{o}\) are numerically calculated. Additionally \(\pi\), the steady-state distribution, is provided:  
\[A= \begin{bmatrix}0.898&0.05&0.052\\ 0.168&0.74&0.092\\ 0.007&0.12&0.873\end{bmatrix} ,B= \begin{bmatrix}0.974&0.026\\ 0.611&0.389\\ 0.061&0.939\end{bmatrix} \]  
\[\pi_{0}= \begin{bmatrix}0.718&0.113&0.169\end{bmatrix} \]  
\[\pi= \begin{bmatrix}0.417&0.239&0.344\end{bmatrix} \]  
Between any two subsequent semesters at time-steps \(t\) and \(t+1\), the rows in the transition matrix \(A\) correspond to states FES, MES, and PES at semester \(t\), while the columns correspond to states FES, MES, and PES at semester \(t+1\). Each element of \(A\) thereby describes the probability of switching from one enrollment strategy to another between semesters. Based on the estimated transition matrix \(A\), most students maintain their enrollment strategy with high probability from one semester to the next. Reading the matrix's diagonal, with 0.898 probability, a student employing a FES will continue employing a FES. Similarly, with probabilities 0.74 for PES and 0.873 for MES, students maintain their strategy. These findings indicate that most students persist in their enrollment strategy regardless of which strategy they deploy. For emission matrix \(B\), each row corresponds to the probability of full-time and part-time enrollment status in a semester for a given enrollment strategy. The results indicate that students employing FES register full-time with a probability of 0.974, or, part-time with a probability of 0.026. Students employing a PES register full-time with a probability of 0.061 and part-time at a probability of 0.939. Most interesting are students with MES, as the probability of their full-time or part-time enrollment is 0.611 and 0.389, respectively. The initial probabilities vector \(\pi_{0}\) indicates that most undergraduate students start their academic career with a full-time enrollment strategy (with a probability of 0.718). The initial probabilities of using PES or MES are 0.169 and 0.113, respectively. Furthermore, the steady-state probabilities vector \(\pi\) shows that 0.417 of students employ a full-time enrollment strategy at any given time. Moreover, the probability of using PES and MES is 0.239 and 0.344, respectively.