## 3 Modeling the Time Effect on Music Recommendations  
### Problem Statement  
Although the effects of time on recommender systems have been the focus of many studies, those addressing particular issues related to the music domain are very few. Most of these particularities are derived from the way users consume musical items, which is different from other products, as well as from the listening habits of users regarding the time context. In this work, both aspects are addressed by means of a recommendation approach incorporating time factors, which is specifically designed for the music area when explicit ratings are unavailable. In this context, analyzing the evolution of user preferences requires new approaches, since methods defined for other domains are not applicable for music recommendation. Let us consider a set of users \(U\) and a set of songs \(S\) where \(u_{i}\in U,\ i=1,\ldots,n\) and \(s_{j}\in S,\ j=1,\ldots,m\) represent a user and a song, respectively. When explicit ratings are available, each user \(u_{i}\) has a list of ratings that he/she has given to a subset of songs \(S_{u_{i}}\), where \(S_{u_{i}}\subseteq S\). The ratings are stored in a \(m\times n\) matrix called the rating matrix, where each element \(r_{ij}\) is the rating that a user \(u_{i}\) gives to a song \(s_{j}\). In the context of this work, explicit ratings are not available, so the methods described in Section 2.2.1 cannot be applied. As seen in that section, both recency-based models and models using time as an independent variable require the use of timestamped ratings. For certain types of items, those typically consumed only once by the user (for instance, a book), these ratings can be obtained from the time spent by the user examining an item or from the item's purchase record. This case corresponds to the one-class CF scenario where \(r_{ij}\) can only take the binary values corresponding to the positive and unknown user-item feedback, respectively. However, for music items, which are consumed many times by the user (a song is usually listened to many times by a user), it is very difficult to derive timestamped ratings in an implicit way. Using the number of plays is a common way to obtain this information, which requires a specific time frame that is wide enough to get reliable ratings. Thus, ratings are not associated with a specific time point, but rather with a longer period of time. This situation does not allow, however, the previously described approaches, such as time-SVD++ or decay functions, to be applied. In order to overcome these problems, we propose that the evolution of user preferences be modeled in the same process employed to compute implicit ratings, which are inferred from the plays of songs by users. For this purpose, the concepts of decay play and implicit decay ratings are introduced. The definition of these concepts is given below. **Definition 1**. : _(decay play). Each user \(u_{i}\in U\) has played a subset of songs where each song \(s_{j}\in S\) is played at a known given time \(t\). Each play \(play_{i,j,t}\) is assigned a weight, called decay play (dPlay), which meets the following condition: \(dPlay(play_{i,j,t})<dPlay(play_{i,j,t})\), when \(t^{\prime}>t\). This property allow us to establish a sequence of decay plays per user and song that is defined in the form: sequence \(\left[dPlay\left(u_{i},\ s_{j},\ p\right)\right]\)\(\left|\ p=1,\ \ldots,\ n_{p}\left(u_{i},\ s_{j}\right)\right.\) and \(dPlay\left(u_{i},\ s_{j},\ p\right)<dPlay\left(u_{i},\ s_{j},\ p+1\right)\), where \(n_{p}\left(u_{i},\ s_{j}\right)\) is the number of times that the user \(u_{i}\) plays the song \(s_{j}\)_  
**Definition 2**. : _(implicit decay rating). A rating inferred from the frequency of decay plays (dPlayF).