## 3 Modeling the Time Effect on Music Recommendations
### Time-Aware Music Recommendation Method
#### 3.2.1 Modeling the Evolution of Preferences
Since the similarity is based on the angle between two vectors, this measure is more sensitive to the fact that two users have ratings on the same items than to the magnitude of those ratings. This is especially important in problems such as the one addressed in this paper, where there are no negative ratings, since it is assumed that users play the songs they like. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & & \multicolumn{2}{c}{**AUC**} & \multicolumn{2}{c}{**NDCG**} & \multicolumn{2}{c}{**MAP**} \\ \hline  & & **Ratings** & \begin{tabular}{c} **Decay** \\ **Ratings** \\ \end{tabular} & **Ratings** & \
\begin{tabular}{c} **Decay** \\ **Ratings** \\ \end{tabular} & **Ratings** &
\begin{tabular}{c} **Ratings** \\ \end{tabular} & **Ratings** \\ \hline \multirow{3}{*}{Day (24 h.)} & K-NN & 0.422 & 0.422 & 0.327 & 0.327 & 0.006 & 0.006 \\  & WRMF & 0.500 & **0.525** & 0.331 & **0.336** & 0.007 & **0.009** \\  & BPRMF & 0.528 & **0.535** & 0.330 & **0.332** & 0.005 & **0.006** \\ \hline \multirow{3}{*}{Morning} & K-NN & 0.413 & 0.413 & 0.315 & 0.315 & 0.005 & 0.005 \\  & WRMF & 0.516 & **0.519** & 0.316 & **0.321** & 0.006 & **0.008** \\  & BPRMF & 0.473 & **0.491** & 0.312 & **0.313** & 0.004 & 0.004 \\ \hline \multirow{3}{*}{Evening} & K-NN & 0.378 & 0.378 & 0.297 & 0.297 & 0.004 & 0.004 \\  & WRMF & 0.590 & **0.597** & 0.316 & **0.322** & 0.009 & **0.011** \\ \cline{1-1}  & BPRMF & 0.562 & **0.539** & 0.306 & **0.311** & 0.004 & **0.006** \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The area under the receiver operating characteristics (ROC) curve (AUC), normalized discounted cumulative gain (NDCG), and mean average precision (MAP) values obtained for the rating prediction when contextual information is not considered (day) and when contextual pre-filtering is performed (morning and evening). When WRMF and BPRMF were used, the values of the AUC and NDCG were higher for the decay ratings than for the ratings (except the AUC in the evening periods with BPRMF), although the improvements were not as good as those of preference predictions reported in Table 1. The MAP for the decay ratings was higher or equal than for the ratings. The behavior for prec@5, prec@10 and prec@15 was similar for both kinds of evaluations when using the K-NN. However, for WRMF and BPRMF, the decay ratings performed better than the ratings, except for prec@5 and prec@10 for the complete day and prec@5 for the evening period. We can also find an explanation for this performance of the rank-based metrics through the detailed analysis of the individual results. We have verified that some of the most relevant songs for users have very similar ratings and/or decay ratings. Hence, small fluctuations in the predicted values can cause a change in the order of the songs in the top-N list and make an item go out or into the top-5, top-10 or top-15 lists from which these metrics are computed. In some cases, this fact leads to some of the rank-based metrics being worse for the decay ratings than for the ratings, even though the predictions of the decay ratings are better than the predictions of the ratings. Additionally, the greatest differences between the lists produced with the ratings and decay ratings are also significant. Figure 2: Values of the top-N metrics given by user K-nearest neighbor (K-NN), weighted matrix factorization (WRMF) and Bayesian personalized ranking matrix factorization (BPRMF) using rating and decay ratings. ratings occur for ranks higher than 15, so these changes are not reflected in these metrics. The behaviors described above may be aggravated by the high number of songs available for recommendation, which makes it difficult to obtain a rank close to the actual rank, since little changes in the values of ratings may change the position of the songs in the list. In Figure 3, the results obtained for the complete day and the morning and evening periods are compared.