## 3 The Proposed Algorithm
In Figure 10, we can notice that the CF\({}_{\text{DR}}\) algorithm, presented in this paper, achieves an average prediction RMSE reduction equal to 2.75% (in absolute terms 1.229 versus 1.264, the plain CF achieves), exceeding the performance of the CF\({}_{\text{VNN}}\) algorithm [27] (which is measured at 2.09%) by 0.66% in absolute figures, or 31.5% as a relative improvement. In comparison to the CF\({}_{\text{FOAF}}\) algorithm [29], the CF\({}_{\text{DR}}\) algorithm surpasses CF\({}_{\text{FOAF}}\) (measured at 1.46%) by 88%. Similarly to the case of the MAE, listed above, the considerable differences observed for the values of the MAE metric between the MovieLens dataset on the one hand and the Amazon datasets on the other is that in the MovieLens dataset the ratings range from 0 to 9, whereas in the Amazon datasets ratings range from 1 to 5. On the individual dataset level, the performance edge of the proposed algorithm against the CF\({}_{\text{VNN}}\) algorithm, which achieves a higher RMSE reduction than the CF\({}_{\text{FOAF}}\) algorithm, ranges from 17.5% (for the Amazon \"CDs and Vinyl\" dataset) to 88.6% (for the Amazon \"Office Supplies\" dataset). Finally, Figure 11 illustrates the results obtained regarding the prediction accuracy increase, in terms of the RMSE reduction, when the CS metric is used to quantify similarity between users. In Figure 11, we notice that the CF\({}_{\text{DR}}\) algorithm, presented in this paper, achieves an average prediction RMSE reduction equal to 2.09% (in absolute terms 1.151 versus 1.176, the plain CF achieves), exceeding the performance of the CF\({}_{\text{VNN}}\) algorithm [27] (which is measured at 1.48%) by 0.61% in absolute figures or by 41.5% as a relative improvement. The CF\({}_{\text{DR}}\) algorithm also outperforms the CF\({}_{\text{FOAF}}\) algorithm [29], whose performance is measured at 1.47%, by 42.2%. On the individual dataset level, the performance edge of the proposed algorithm against the CF\({}_{\text{VNN}}\) algorithm ranges from

\[\text{HM(s)}=2*\frac{normCovInc(s)\times normAccInc(s)}{normCovInc(s)+normAccInc (s)} \tag{6}\]6 Rating prediction RMSE for the datasets summarized in Table 4, when the PCC user similarity metric is used. 8.95% (for the Amazon \"Digital Music\" dataset) to 87.5% (for the Amazon \"Grocery and Gourmet Food\" dataset). ## 5 Discussion of the Results and Comparison with Other Works

From the experimental evaluation, we can conclude that the CF\({}_{\text{DR}}\) algorithm, presented in this paper, can successfully increase the density of the CF datasets, regardless of their initial density, since it was successfully applied to both sparse and dense datasets. At the same time, the CF\({}_{\text{DR}}\) algorithm can achieve both a prediction coverage improvement and a significant prediction error reduction, for both the error metrics tested. In regard to the DR formulation, which is performed within the CF\({}_{\text{DR}}\) preprocessing step, the optimal setting for considering a prediction as robust (and hence create a DR) is when at least three NNs of a user take part in the prediction formulation. Furthermore, the optimal significance of the DRs was found to be exactly the same with the real (explicit) user ratings. In the following subsections, we compare the performance of the CF\({}_{\text{DR}}\) algorithms against other algorithms published in the literature and target the increase of coverage of CF-based algorithms. To provide a fair comparison basis, the algorithms included in the comparison are limited to those which utilize only the user-item rating matrix, without necessitating the availability of additional data, such as social relationships [24; 36] or trust [57; 58]. In Table 5, we can observe that the CF\({}_{\text{DR}}\) algorithm achieves higher coverage increase than all other state-of-the-art algorithms, both for the cases of sparse and dense datasets. This is also true for the rating prediction accuracy improvement.