## 3 The Proposed Recommendation System
### Collaborative Filtering
For example, we consider four users and three items (4 \(\times\) 3 matrix), as shown in Table 4. It initializes the user latent and the item latent. Then it optimizes the user latent while fixing the item latent and vice versa with Equations (6) and (7), respectively. \[u_{i}=\left(V^{T}\times V\times\lambda I\right)^{-1}\times V^{T}\times R_{i} \tag{6}\]  
\[v_{i}=\left(U^{T}\times U\times\lambda I\right)^{-1}\times U^{T}\times R_{j} \tag{7}\]  
Figure 2: User page example with interest ratings. Users’ interest rating scale is 1–5 (1 is the lowest and 5 is the highest value). \begin{table}
\begin{tabular}{c c c c} \hline  & **Item 1** & **Item 2** & **Item 3** \\ \hline User 1 & 1 & 0 & 5 \\ User 2 & 0 & 0 & 3 \\ User 3 & 3 & 2 & 4 \\ User 4 & 2 & 1 & 5 \\ \hline \end{tabular}
\end{table} 
Table 6: The item latent matrix after 10 iterations. \begin{table}
\begin{tabular}{c c c c} \hline  & **Item 1** & **Item 2** & **Item 3** \\ \hline User 1 & 1 & 0 & 5 \\ User 2 & 0 & 0 & 3 \\ User 3 & 3 & 2 & 4 \\ User 4 & 2 & 1 & 5 \\ \hline \end{tabular}
\end{table} 
Table 4: An example of rating scores for four users. \begin{table}
\begin{tabular}{c c c} \hline  & **Item 1** & **Item 2** & **Item 3** \\ \hline User 1 & 1 & 0 & 5 \\ User 2 & 0 & 0 & 3 \\ User 3 & 3 & 2 & 4 \\ User 4 & 2 & 1 & 5 \\ \hline \end{tabular}
\end{table} 
Table 5: The user latent matrix after 10 iterations. \begin{table}
\begin{tabular}{c c c} \hline  & **Item 1** & **Item 2** & **Item 3** \\ \hline User 1 & 1 & 0 & 5 \\ User 2 & 0 & 0 & 3 \\ User 3 & 3 & 2 & 4 \\ User 4 & 2 & 1 & 5 \\ \hline \end{tabular}
\end{table} 
Table 6: The item latent matrix after 10 iterations. \begin{table}
\begin{tabular}{c c c} \hline  & **Item 1** & **Item 2** & **Item 3** \\ \hline User 1 & 0.57 & 0.29 & 1.02 \\ User 2 & 0.22 & \(-\)0.16 & 0.97 \\ User 3 & 1.03 & 0.95 & 0.99 \\ User 4 & 1.02 & 0.94 & 0.99 \\ \hline \end{tabular}
\end{table} 
Table 7: The result of the dot product (item latent \(\cdot\) item latent). Figure 4 shows WCSS with the number of \(k\) increasing. The WCSS is inversely proportional to the number of \(k\). The slope is steepest between \(k\) = 1 and \(k\) = 2, and the slopes decrease as the number of \(k\) increases. In this case, the proper value of \(k\) can be 2 or 3. In addition to WCSS, we use agglomerative and divisive hierarchical clustering to figure out the properties of the synthetic data set. To visualize the hierarchical clustering, we show the dendrogram for sample data. In order to calculate the distance between two clusters, we use Equation (9). \[d(u,v)=\sqrt{\frac{|v|+|s|}{T}d(v,s)^{2}+\frac{|v|+|t|}{T}d(v,t)^{2}-\frac{|v |}{T}d(s,t)^{2}}, \tag{9}\]  
where \(u\) (the union of \(s\) and \(t\)), \(v\), \(s\), and \(t\) are clusters, and \(T\) is \(|v|+|s|+|t|\). Figure 5 shows the dendrogram for hierarchical clustering with the Euclidean distance. Note that we only show a limited number of samples for visibility (100 in Figure 5a and 10 in Figure 5b). Note that the dendrogram does not indicate whether the clustering result is good or bad. Rather, it demonstrates the process of clustering to determine proper clustering. Figure 4: WCSS with the number of clusters. Data points are marked as blue dots. Figure 3: The distribution of users’ item view history. Data points are marked as blue cross. Figure 6 shows the prototype example of our hybrid recommendation system for martial arts, dance, and play in seven Silk Road countries. The figure shows the item for Taekwondo (a Korean form of martial arts) with its title, summary, representative figure, interest rating form, and four recommended items (at the bottom of the page). Note that the first two recommended items are from content-based filtering, and the last two items are from collaborative filtering.