## 4 Numerical Experiments
### Experimental Results
#### 4.2.2 Eachmovie Dataset
The RF R-square-based score performs the best for Top-1 to Top-9, whereas the RF item modeling performs the best for Top-1 to Top-3, Top-5, Top-6, and Top-10; the RF user modeling is outperformed by the two approaches. For the two-way cooperative CF, the Pearson correlation-based score and the RF R-square-based score provide the best average of the ten Top-\(N\) accuracies. Therefore, we assume that both the Pearson correlation-based score and the RF R-square-based score work very effectively for the two-way cooperative CF than the PCA+LR two-way 1. Note that there are 150 users and 150 items in section A \(\times\) C in Figure 1a. In case 3, as shown in Table 5, the PCA+LR item modeling performs better than the other approaches for all the Top-\(N\) accuracies. The PCA+LR two-way 2 seems not to outperform the PCA+LR user modeling, although it clearly outperforms the PCA+LR two-way 1. For the user-based CF and item-based CF, the Pearson correlation-based score and the item-based CF are outperformed by the user-based CF for all the Top-\(N\) accuracies. The Pearson correlation-based score does not seem to work well. The RF R-square-based score is outperformed by the RF user modeling and the RF item modeling. \begin{table}
\begin{tabular}{l|l l l|l l l|l l l} \hline \hline \multirow{2}{*}{\(\mathbf{N}\)} & **PCA+LR** & **PCA+LR** & **PCA+LR** & **Pearson** & **Pearson** & **Pearson** & **RF User** & **RF Item** & **RF _rsq_ \\  & **User** & **Item** & **Two-Way 1** & **User** & **Item** & **Score** & **User** & **Item** & **Score** \\ \hline
1 & **0.940** & **0.940** & 0.920 & 0.880 & 0.780 & **0.920** & 0.900 & **0.920** & **0.920** \\
2 & 0.900 & 0.880 & **0.910** & **0.870** & 0.790 & **0.870** & 0.850 & **0.900** & **0.900** \\
3 & 0.867 & 0.860 & **0.860** & **0.873** & 0.727 & **0.873** & 0.867 & **0.893** & **0.893** \\
4 & **0.860** & 0.855 & 0.850 & **0.875** & 0.730 & **0.875** & 0.820 & 0.860 & 0.865 \\
5 & **0.840** & 0.828 & 0.844 & 0.832 & 0.708 & **0.840** & 0.800 & **0.836** & **0.836** \\
6 & **0.803** & 0.793 & 0.797 & 0.777 & 0.683 & **0.790** & 0.770 & **0.800** & **0.800** \\
7 & **0.766** & 0.746 & 0.754 & **0.740** & 0.660 & **0.740** & 0.734 & 0.763 & **0.777** \\
8 & **0.735** & 0.705 & 0.715 & 0.705 & 0.633 & **0.710** & 0.703 & 0.725 & **0.743** \\
9 & 0.691 & 0.678 & **0.693** & **0.689** & 0.611 & 0.687 & 0.678 & 0.696 & **0.708** \\
10 & 0.660 & 0.662 & **0.672** & **0.670** & 0.592 & 0.664 & 0.654 & **0.684** & 0.674 \\ Avg. & **0.806** & 0.795 & 0.802 & 0.791 & 0.691 & **0.797** & 0.778 & 0.808 & **0.812** \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Top-\(N\) accuracy for case 2. \begin{table}
\begin{tabular}{l|l l l|l l l|l l l} \hline \hline \multirow{2}{*}{\(\mathbf{N}\)} & **PCA+LR** & **PCA+LR** & **PCA+LR** & **Pearson** & **Pearson** & **Pearson** & **RF User** & **RF item** & **RF _rsq_ \\  & **User** & **Item** & **Two-Way 1** & **User** & **Item** & **Score** & **User** & **Item** & **Score** \\ \hline
1 & **0.926** & **0.926** & 0.917 & **0.893** & 0.843 & 0.884 & 0.926 & **0.934** & **0.934** \\
2 & 0.905 & **0.921** & 0.917 & 0.868 & 0.855 & **0.872** & 0.921 & **0.917** & 0.913 \\
3 & 0.909 & **0.917** & 0.912 & 0.862 &  
Note that there are only 10 users in section A \(\times\) C in Figure 1a. In this case, the columns of some active items in the market basket data are all zeros, which is the extreme high-dimensional cold-start problem. Then, the average of the R-square values can have a negative sign, which can lead to bad prediction performance. For instance, we randomly select a test observation where 1 denotes a purchased item and \(-1\) denotes a non-purchased item where the predicted values of the item modeling and the user modeling range from \(-1\) to \(1\). The predicted value of the item modeling is \(-0.3204994\), and the predicted value of the user modeling is \(-0.5009333\). The average of the R-square values of the item modeling is \(0.251122\) and the average of the R-square values of the user modeling is \(-0.3145476\), which has a negative sign.