###### Abstract
Let us also assume that the RS computes user similarity using the Pearson correlation coefficient [1,2] and that, at rating prediction time, the user-item ratings database is as depicted in Table 1.  
In this setting, a prediction for grey-shaded cells cannot be computed, due to the fact that the corresponding user has no near neighbours that have rated the particular item. Considering, for instance, cell (\(U_{3}\), \(i_{4}\)), a prediction for \(r_{U3,i4}\) cannot be computed, since: (1) U\({}_{3}\) has only one NN, \(U_{2}\) (they share a positive correlation with each other, since they have both rated item i\({}_{2}\) with a high score), but \(U_{2}\) has not rated item \(i_{4}\), and (2) although user \(U_{1}\) has rated item \(i_{4}\), he does not belong to \(U_{3}\)'s NN set (no common rated items). The case for cell (\(U_{1}\), \(i_{3}\)) is similar while, notably, \(U_{4}\) has not rated any item in common with other users, and \(i_{5}\) has only been rated by \(U_{4}\), Hence, no rating prediction can be formulated for any item for \(U_{4}\), and no rating prediction can be computed for any user regarding item \(i_{5}\). However, we can also observe in Table 1 that users \(U_{1}\) and \(U_{2}\) are NNs with each other (they share a positive correlation, having both rated item \(i_{1}\) with a low score) and hence if \(U_{2}\) had asked for a prediction on item \(i_{4}\) (instead of user \(U_{3}\)) that prediction could have been formulated. According to the CFDR algorithm proposed in this paper, a preprocessing step is executed that computes all the predictions that can be formulated for all the users in the CF rating database, and the predictions having a high level of confidence regarding their accuracy, i.e., the robust ones, are converted to DRs and stored into the user-item ratings database. Table 2 depicts the state of the user-item rating \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**User/Item** & \(\mathbf{i_{1}}\) & \(\mathbf{i_{2}}\) & \(\mathbf{i_{3}}\) & \(\mathbf{i_{4}}\) & \(\mathbf{i_{5}}\) \\ \hline U\({}_{1}\) & 1 & & & 5 & \\ U\({}_{2}\) & 1 & 5 & & & \\ U\({}_{3}\) & & 5 & 1 & & \\ U\({}_{4}\) & & & & & 5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: User-item rating database excerpt. matrix from Table 1 after the execution of the preprocessing step of the CFDR algorithm (the newly inserted DRs are shown in bold). We can observe that the user-item rating matrix is now considerably denser. Its density, computed as \(\frac{\#ratings}{\#items\#users}\), has increased from 0.35 to 0.55 and, due to this increase, the prediction for the ratings \(r_{U3,i4}\) and \(r_{U3,i3}\) can now be formulated. The ability to compute rating predictions that could not be formulated on the basis of the original user-item rating matrix stems from the two following properties of the DR-enriched -item rating matrix:  
In the DR-enriched item rating matrix, the near neighbourhood of a user \(U\) may extend to accommodate a set of additional users \(\text{NN}_{\text{U}}^{\text{New}}\). The ratings of the users in the set \(\text{NN}_{\text{U}}^{\text{New}}\) can be exploited to compute rating predictions that could not have been computed on the basis of the original user-item rating matrix. Indeed, we can observe that, due to the addition of DRs \(r_{U3,i2}\) and \(r_{U3,i1}\) in the DR-enriched database depicted in Table 2, \(U_{3}\)'s near neighbourhood has extended to include \(U_{1}\), hence any rating recorded by \(U_{1}\) can now be used for computing rating predictions for \(U_{3}\), as is the case with \(r_{U3,i4}\). It is worth noting at this point that the ability to compute a prediction for \(r_{U3,i4}\) can lead to the formulation of a personalized recommendation for user \(U_{3}\): in the situation depicted in Table 1, only the value of \(r_{U3,i1}\) could be predicted. However, this value is very low (1) and hence \(\text{i}_{1}\) would not be recommended to \(U_{3}\).