## 3 Modeling the Time Effect on Music Recommendations
### Time-Aware Music Recommendation Method
#### 3.2.1 Modeling the Evolution of Preferences  
The recency models described in Section 2.2 assume that user preferences change over time in such a way that the more recent the rating is, the more important it is. Usually, an exponential function of time is used to reduce monotonically ratings' importance, but, as previously stated, it requires ratings to be stamped with a date. In the context of this work, explicit ratings are not available; therefore, implicit ratings must be computed from the frequency of plays. Given that the information about the date and time of the plays is available, it can be used to calculate the degree of importance of the implicit ratings obtained from them. Therefore, our proposal can be considered a recency approach, as it involves changes in preferences over time. However, it differs from the models described in Section 2.2 in the fact that it does not require timestamped explicit ratings since the evolution of preferences is encompassed in the process for inducing implicit ratings. In this process, unlike other recency methods, the time function is applied to plays instead of ratings. The first step of the proposal is the calculation of the decay plays defined in the previous section from the timestamped plays. In the same way that the weight of ratings decreases as their age increases, the importance of plays will also decrease. Thus, ratings can be calculated from the frequencies of weighted plays. We propose the use of an exponential function to weight the plays according to the moment in which they occur. As stated before, a decay play (\(dPlay\)) is defined as the weight that a play of a song \(s_{j}\) by a user \(u_{i}\) has according to its timestamp \(t_{i,j,p}\), as seen in Equation (12), with 1 being the maximum weight, which is assigned to the most recent play. \[dPlay\Big{(}u_{i},\ s_{j},\ p\Big{)}=e^{-\lambda(t_{max}-t_{i,j,p})/T_{u}}. \tag{12}\]The most recent time is designed as \(t_{max}\). The parameter \(\lambda\) is used to adjust the decay rate. \(T_{u}\) is used to convert the units in which \(t_{i,j,p}\) and \(t_{max}\) are measured to the target units, which depend on the application domain. We have used years as the target unit since changes in musical trends are not usually very drastic. Other functions can be used; however, several studies in which the evolution of ratings has been analyzed, have proved that exponential decay is more suitable. This occurs because the gradient of the curve reflects the change of their importance over time in a more realistic way. Although this model is indicated for users who have been registered in the system for some time, it can also be applied in the cold-start scenario referring to new users who only have recent plays. In these cases, the value of the decay plays will be 1 or close to 1, so they will receive recommendations very similar to those received without considering the evolution of preferences. The frequency of weighted plays will be used to obtain implicit ratings that incorporate time effects. The frequency of decay plays for a user \(u_{i}\) and a song \(s_{j}\) within a given time window is computed as indicated in Equation (11). Once the decay play frequencies are computed, the next step is to obtain the implicit ratings from them. To do this, we recommend following the procedure by [PERSON] [74], since this method is appropriate for power-low distributions, like in this work, where a higher number of plays is concentrated on a few songs. A rating \(r_{i,j}\) for the user \(u_{i}\) and the song \(s_{i}\) is obtained as a linear function of the frequency percentile by using Equation (13), where \(dPlayF_{k}\) denotes the decay play frequency of a song with the rank \(k\) in the ranked list of the user \(u_{i}\), as introduced in Definition 2. Simple frequency functions [75] can also be used to transform plays into ratings but are less suitable for these types of situations.