# Optimizing Latent Factors and Collaborative Filtering for Students' Performance Prediction
## 5 Experimental Results
### Datasets
There are two significant conclusions to be drawn after analyzing the GAK plots. First, the plots of \(RMSE\) have a similar behavior. For each dataset, there is only one optimal value of \(K\). Until this value is reached, \(RMSE\) decreases when \(K\) increases; after it has been reached, the reverse occurs. Second, the optimal \(K\) values depend strongly on the considered dataset; however, they are in the range between 45 and 197. This way, we could select a value in the middle of this range to be used by default for general experiments in PSP when the optimal number of latent factors for a particular dataset is unknown, provided that it is comparable in size to the dataset that has been used here. Furthermore, we can check that \(\beta\) follows a similar trend as \(RMSE\) when \(K\) increases, while \(\lambda\) does not show a particular behavior. Tables 4 and 5 show the main results after applying the GA3 and PS3 methods, respectively. As the optimization problem has a strong non-deterministic nature, each experiment of GA and PS consisted of 31 runs. Therefore, the minimum, maximum, mean, median, and standard deviation are shown. The optimal values of the parameters \(K\), \(\beta\), and \(\lambda\) correspond to the minimum value of the \(RMSE\) found in each experiment. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**GAK** & **RMSE** & \multicolumn{3}{c}{**Values for \(RMSE\) min**} \\ \hline
**Dataset** & **min** & \(\beta\) & \(\lambda\) & \(K\) \\ \hline IC & 0.218 & 0.022 & 0.0070 & 113 \\ HS & 0.201 & 0.495 & 0.1841 & 45 \\ TO & 0.211 & 0.077 & 0.0001 & 139 \\ EN & 0.159 & 0.037 & 0.0001 & 197 \\ BE & 0.156 & 0.010 & 0.0001 & 184 \\ PR & 0.089 & 0.033 & 0.0001 & 153 \\ SR & 0.092 & 0.044 & 0.0098 & 192 \\ SJ & 0.097 & 0.052 & 0.0001 & 139 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Results for the optimization method GAK. Figure 9: Optimal value of the regularization factor \(\lambda\) for each value of \(K\). predict unknown performances and building models by using matrix factorization and the gradient descent algorithm. The prediction of the unknown performances is very useful for students and teachers from the learning point of view. On the one hand, students can predict the score of an unfinished or un-attended task, according to the scores of their completed tasks and the scores of their colleagues who completed the task in question. This prediction can help the student to identify the main strengths or weaknesses in their learning process of the corresponding subject and provide insights about how to maximize the study effort. On the other hand, the teacher can monitor the level of success for a particular task by recording the performance of all the students, including the predictions of the unknown scores. The particular nature of each virtual classroom (number of students and tasks, degree, type of learning) involves considering a different number of latent factors. We have proven that selecting a right number of latent factors, together with optimal values for the learning rate and regularization factor, improves the prediction accuracy by minimizing the error in the prediction of the test dataset. Our proposal consists of optimizing these three parameters by applying a direct search of the ideal number of latent factors at the same time that the prediction model is optimized by means of a genetic algorithm, thus obtaining a minimum prediction error. The experimental framework consisted of eight real-world datasets, filtered accordingly for a realistic processing. After applying the proposed optimization method, not only did this method outdo the prediction accuracy when compared to using simple direct search for the optimal values of \(\beta\) and \(\lambda\) with less computing time, it also obtained a range where to locate the ideal number of latent factors. The mean value of the number of latent factors can be found within this range; and it can be utilized by default for prediction purposes on other datasets, without having to perform previous optimization experiments.