## 4 Results
### Recommendation Process on Coursera
#### 4.3.3 Calculating Learner Similarity  
Using both the course attribute preferences and multidimensional capability to cluster learners (into similar learning groups), better clustering performance was shown. The same weight was given to the course attribute preferences and multidimensional capabilities in Equation (8), that is, \(\beta=\gamma\). Consequently, according to Step 3, shown in Figure 1, an example of the learner similarity matrix is illustrated in Table 9 below. The diagonal elements represent the similarity of each learner to himself/herself, thus taking the value of 1. For example, the similarity between Learner 8064 and Learner 10,269 was 0.21. Moreover, to calculate the comprehensive similarity of learners so that the top \(k\) similar learners in clusters could be found, the optimal values of the course rating similarity weight coefficient \(\alpha\) in Equation (8) and the top \(k\) nearest neighbors were determined through sensitivity analysis. In this study, the mean absolute error (MAE) between predicted and actual course ratings was obtained when the value of the weight coefficient \(\alpha\) changed by iterating the nearest top \(k\) neighbors. Usually, MAE changes smoothly with the nearest top \(k\) neighbors. The step size \(k\) is generally set to 5 or 10 [26; 32]. This study started with \(k=10\), and a change in MAE was observed, as shown in Figure 2. \begin{table}
\begin{tabular}{c c c c} \hline
**Cluster** & **Number** & **Percentage** & **Characteristics** \\ \hline
1 & 329 & 17.79\% & High price, short class hours, and advanced level \\
2 & 392 & 21.20\% & Low price, unpopular course, and beginner level \\
3 & 288 & 15.58\% & Popular and high rating \\
4 & 256 & 13.85\% & Few materials and low rating \\
5 & 584 & 31.58\% & Few materials, short class hours, and niche \\ \hline \end{tabular}
\end{table} 
Table 8: Characteristics of clusters. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline ta & 8164 & 10269 & 14654 & 15598 & 26307 & 28185 & 47728 & 60377 & 88789 & 90744 \\ \hline
8164 & 1.00 & 0.21 & 0.03 & 0.01 & 0.02 & \(-\)0.24 & 0.42 & 0.33 & \(-\)0.17 & \(-\)0.02 \\ \hline
10269 & 0.21 & 1.00 & \(-\)0.05 & 0.01 & 0.07 & \(-\)0.07 & 0.20 & 0.06 & \(-\)0.11 & \(-\)0.12 \\ \hline
14654 & 0.03 & \(-\)0.05 & 1.00 & \(-\)0.20 & \(-\)0.14 & \(-\)0.05 & 0.15 & \(-\)0.08 & 0.01 & 0.06 \\ \hline
15598 & 0.01 & 0.01 & \(-\)0.20 & 1.00 & 0.10 & 0.16 & \(-\)0.10 & \(-\)0.01 & 0.22 & 0.35 \\ \hline
26307 & 0.02 & 0.07 & \(-\)0.14 & 0.10 & 1.00 & 0.19 & 0.15 & 0.03 & \(-\)0.11 & 0.05 \\ \hline
28185 & \(-\)0.24 & \(-\)0.07 & \(-\)0.05 & 0.16 & 0.19 & 1.00 & \(-\)0.15 & \(-\)0.27 & 0.15 & 0.46 \\ \hline
47728 & 0.42 & 0.20 & 0.15 & \(-\)0.10 & 0.15 & \(-\)0.15 & 1.00 & 0.40 & \(-\)0.20 & \(-\)0.09 \\ \hline
60377 & 0.33 & 0.06 & \(-\)0.08 & \(-\)0.01 & 0.03 & \(-\)0.27 & 0.40 & 1.00 & \(-\)0.29 & \(-\)0.11 \\ \hline
88789 & \(-\)0.17 & \(-\)0.11 & 0.01 & 0.22 & \(-\)0.11 & 0.15 & \(-\)0.20 & \(-\)0.29 & 1.00 & 0.12 \\ \hline
90744 & \(-\)0.02 & \(-\)0.12 & 0.06 & 0.35 & 0.05 & 0.46 & \(-\)0.09 & \(-\)0.11 & 0.12 & 1.00 \\ \hline \end{tabular}
\end{table} 
Table 9: A subset of data on learners similarity. As indicated in Figure 2, when the number of the nearest neighbors \(k\) is fixed and the course rating similarity weight coefficient \(\alpha\) increases from 0 to 1, MAE fluctuates; it decreases at first and then it increases after \(\alpha=0.8\). Finally, when the course rating similarity weight coefficient \(\alpha\) is 0.8 and the number of the nearest neighbor \(k\) is 20, the model obtains the smallest MAE. Therefore, the course rating similarity weight coefficient \(\alpha\) and the top \(k\) nearest neighbors were determined to be 0.8 and 20, respectively, through sensitivity analysis. Accordingly, based on Equation (8), both the course attribute similarity weight coefficient \(\beta\) and multidimensional capability similarity weight coefficient \(\gamma\) took the value of 0.1.