## 3 Modeling the Time Effect on Music Recommendations
### Time-Aware Music Recommendation Method
#### 3.2.1 Modeling the Evolution of Preferences
To perform this validation, records for all except the five selected users were removed from the test set. The objective was to prove that the use of the decay ratings with the recommendation methods is better or at least equal to the use of the ratings for new users. The study has been carried out only for a 24-h period because when doing the contextual pre-filtering for short periods with only five users, the test set would have been a very small size. The results are shown in Table 2. The best values for each method in the comparison of ratings and decay ratings and the percentages of improvement are highlighted in bold. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**MAE**} & \multicolumn{2}{c}{**RMSE**} & \multicolumn{2}{c}{**NMAE**} \\ \hline  & **Ratings** & \begin{tabular}{c} **Decay** \\ **Ratings** \\ \end{tabular} & **Improv.** & **Ratings** & \
\begin{tabular}{c} **Decay** \\ **Ratings** \\ \end{tabular} & **Improv.** & **Ratings** &
\begin{tabular}{c} **Decay** \\ **Ratings** \\ \end{tabular} \\ \hline K-NN Pearson & 0.898 & **0.890** & 0.89\% & 1.162 & **1.153** & 0.77\% & 0.225 & **0.222** \\ K-NN Cosine & 0.985 & **0.948** & 3.76\% & 1.228 & **1.212** & 1.30\% & 0.246 & **0.237** \\ MF & 0.788 & **0.777** & 1.40\% & 0.999 & **0.998** & 0.10\% & 0.197 & **0.194** \\ BMF & 0.785 & **0.775** & 1.27\% & 0.977 & **0.982** & 0.51\% & 0.196 & **0.194** \\ FWMF & 0.803 & **0.792** & 1.37\% & 0.995 & **0.993** & 0.20\% & 0.201 & **0.198** \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The RMSE, MAE and NMAE values obtained for the rating prediction in the cold-start scenario. Figure 1: The MAE and RMSE values given by some recommendation methods using ratings and decay ratings for different time periods. We can observe that the results obtained with decay ratings are better than those obtained with ratings for all recommendation methods, although the percentage of improvement was lower than when using the complete data set. These results were predictable since the evolution of preferences of users who have been registered in the system for a shorter period of time, is reflected to a lesser extent. Therefore, decay ratings and ratings have similar values for these users. #### 4.3.2 Top-N Recommendation

After analyzing the rating prediction results, the reliability of the top-N recommendations was studied. This kind of recommendation consists of a ranked list containing the N songs with the highest values of predicted ratings. The evaluation of these lists was usually performed by means of rank-based metrics. We used N=100 for obtaining the results. The results of the tests performed using ratings and decay ratings are presented in Table 3 and Figure 2. The table shows the values of the AUC, NDCG and MAP for the experiments corresponding to the complete day, morning and evening (the best results are shown in bold). The figure illustrates the comparison of the two types of evaluation (ratings and decay ratings) according to these metrics and others such as prec@5, prec@10, prec@15. When the K-NN was applied, the values of the AUC, NDCG and MAP were the same for the decay ratings and ratings in spite of the prediction that the preferences would be better when using decay ratings against ratings, as shown in Table 1. After a detailed examination of the predictions and the top-N lists generated for some users, we verified that the predictions of decay ratings were closer to the actual values than the prediction of ratings, but the ordered list of songs provided by the algorithm is very similar. This indicates that, despite obtaining better results in the prediction when using decay ratings, these do not translate into a great change in the order of the songs in terms of their relevance. Thus, all rank-based metrics (those used for top-N recommendations) give the same results for K-NN. This behavior may be produced by the characteristics of the cosine similarity metric involved in this method.