## 4 Methodology  
In this study, we generate and apply a Hidden Markov Model (HMM) to identify students' enrollment strategy and characterize the impact of enrollment strategy on student outcomes. The use of HMM is not new to educational data mining and modeling. Previously it has been used to investigate students' sequential behaviors, decision-making, and performance [35; 36; 37; 38; 39; 40]. As an example, ref. [41] classified students engagement (in termsof checking their course progress) in a massive open online course using Hidden Markov models and then predicted students retention for each class. Results show that students who never or scarcely check their progress are more likely to drop the course than students who actively check their progress. Other papers have used HMMs to model sequential student behavior. Ref. [37] modeled high school students' actions and behaviors using HMMs. By estimating HMM parameters with the Baum-Welch algorithm for each student, the authors clustered the students based on the individual transition matrices to assess differences in behavior and achievement between clusters. This study aimed to investigate the relationship between students' educational outcomes and learning tutorial modes. The results of this study identify a significant correlation between tutorial dialogue structure and student learning outcomes.  
As depicted in Figure 1, similar to ordinary Markov Models, an HMM represents the dynamics of a system as it moves between operating states or modes (e.g., Modes 1, 2, and 3 in the figure). When operating within a state or mode, the system generates state-related output \(O_{i}\) at each time-step. Unlike Markov Models, in the case of the standard HMM problem, the states are not always directly observable, and as such, they can only be inferred by observing a sequence of outputs. For the problem under consideration here, the hidden state corresponds to the enrollment strategy of a student (i.e., full-time enrollment strategy, mixed enrollment strategy, and part-time enrollment strategy), while the observations refer to the actualized enrollment in any given semester as reported in the student's academic history.  
To give a formal definition of Hidden Markov models, we begin with the following notations: \(Q=\{q_{1},q_{2},\ldots,q_{N}\}\) represents the set of \(N\) possible states in the system; \(A=[a_{ij}]\in\mathcal{R}^{NxN}\) is a transition matrix, where each \(a_{ij}\) denotes the probability of transitioning from state \(i\) to \(j\) at any given time-step; \(O=o_{1},o_{2},\ldots,o_{T}\) represents a sequence of observations of length \(T\), each drawn from the set of \(M\) possible observations \(V=\{v_{1},v_{2},\ldots,v_{M}\}\); and \(\pi_{o}\) represents the initial distribution of the system. When a system is operating in a specific state \(q_{i}\), the output \(o_{t}\) at time \(t\) is generated according to a unique probability distribution denoted as \(B=b_{i}(o_{t})\), the emission probability.  
To generate an HMM to represent student enrollment strategies, we must learn the optimal model parameters \(\lambda^{*}=(A,B,\pi_{0})\) that reproduce known observations. The process of learning \(\lambda^{*}\) is based on the Baum-Welch algorithm, an iterative process that requires calculating the likelihood of any sequence of observations given \(\lambda^{*}\), and decoding  
Figure 1: Representation of a simple Hidden Markov Model.  
relationships between observations and hidden variables. As the model is iteratively updated, the likelihood calculations and the decoding is updated.