# Analyzing Student Behavioral Patterns in MOOCs Using Hidden Markov Models in Distance Education
## 2 Materials and Methods
Its values are denoted by the variable \(a\). The Backward algorithm is the ID sequence reverse of the Forward algorithm. We use the Backward Algorithm to find the probability that the machine will be in a particular hidden state \(S_{i}\) for the ID \(t\) and will generate the remaining part of the sequence of the visible symbols (grades) \(O\). The output of the Backward algorithm is denoted by the variable \(\beta\). A special case of the Expectation Maximization algorithm is the Baum-Welch algorithm. We will estimate the latent variables \(\gamma\), \(\xi\) with this algorithm (Figure 2; Table 2), the values of \(\gamma\), and the sum of the columns of \(\xi\). (Table 2). The learning problem is a constrained optimization problem. To this aim, we use Lagrange Multipliers, as we mention in another section below. We have calculated the variables \(\alpha\), \(\beta\) in the Forward-Backward step. We use the \(\alpha\) and \(\beta\) values obtained from the Forward-Backward process to calculate the variable \(\gamma\). Specifically:  
\[\gamma_{t}(i)=\frac{\alpha_{t}(i)*\beta_{t}(i)}{\sum_{i=1}^{N}\beta_{t}(i)* \alpha_{t}(i)}\]  
where \(\gamma\) represents the probability of being in a given state \(i\) for student ID \(t\) for the entire students' range. In addition, we denote as \(\xi_{t}\). the probability for a student ID \(t\) to be in state \(S_{i}\) and for a student ID \(t+1\) being in state \(S_{j}\). The variables \(\gamma\) and \(\xi\) are related to the equation  
\[\gamma_{t}(i)=\sum_{j=1}^{N}\xi_{t}(i,j)\]In our methodology with \(\gamma\), we express the probability of transitioning from hidden to the given observed states. Hence, the expected number of transitions from state \(S_{i}\) to state \(S_{j}\) is the summation of \(\xi_{t}\). The difference between the Forward-Backward and Viterbi algorithms lies in the induction step: the Forward-Backward algorithm uses summation, while the Viterbi algorithm uses minimization. In the Viterbi algorithm, we use \(\delta\) (it is notated as omega in the source code in R) to determine the probability of ending the most probable path in a particular state at a student ID. Formally, \(\delta_{t}(i)\) represents the probability of the most probable path, which takes into account the first \(t\) observations and terminates in state \(i\):  
Table 3.0  
* As \(t\) observations, we consider the final grades of \(t\) corresponding students. The magnitude \(\delta_{t-1}(i)\) is the maximum probability of any path that ends to state \(i\) for student ID \(t-1\). * \(a_{ij}\) is the transition probability from hidden state \(i\) to hidden state \(j\). * \(b_{j(o_{t})}\) is the emission probability of observing \(o_{t}\) from hidden state \(j\). We have employed the omega factor to estimate the relationship between the last state and previous states. The resulting sequence appears in Table 3.1  
Next, we apply the Forward-Backward algorithm to calculate the factors \(\alpha\) and \(\beta\), which are the outputs of the algorithm. The variable \(\alpha\) corresponds to the output of the Forward and the \(\beta\) to the output of the Backward part of the algorithm. Using the Forward-Backward algorithm allows us to calculate \(P[O\,|\,\lambda]\) with significantly fewer computations, where \(\lambda\) represents the model (HMM), and \(O\) is the observation sequence. The Forward-Backward algorithm provides the probability \(P[O|\lambda]=\sum_{t=1}^{N}\beta_{t}(i)a_{t}(i)\), where \(t\) represents the user ID for each student in the dataset. We assign the IDs in ascending order, from \(t=1\) to \(t=4037\). The above probability represents the solution to the evaluation problem, i.e., the likelihood that the model generated the given observation sequence. The dataset has a length of \(T=4037\) for both hidden and observed states, meaning that 4037 students participate in the course, each one with a unique user ID. We observe that \(\alpha\) and \(\beta\) values converge to zero exponentially.