## 2 Background and Related Work
### Time-Aware Recommender Systems
#### 2.2.1 Evolution of User Preferences  
Proposals for addressing preference evolution commonly require explicit ratings and information about the time when they were obtained from users. Timestamped ratings are necessary to adjust their weight [37], using either decay functions or time windows. In the group of recency-based models, decay-based models usually involve exponential functions of time. Let us consider a target time \(t_{\text{f}}\) representing a future time when recommendations will be given, and \(t_{\text{u,j}}\) the time when the user \(u\) rates the item \(j\). The importance of every rating at the recommendation time is given by a weight \(w_{\text{u,j}}\) computed from a decay function \(f(t)\) that decreases the weight of the ratings as their timestamp moves away from the target time [18]. [PERSON] and [PERSON] [38] proposed the following decay function to reduce monotonically rating importance with time in the range \((0,1)\). \[w_{\text{u,j}}(t_{\text{f}})=f(t)=e^{-\lambda\ (t)} \tag{4}\]  
where \(\lambda\) is the decay rate and \(t=t_{\text{f}}-t_{\text{u,j}}\). The decay rate \(\lambda\) is defined as the inverse of the half-life parameter \(T_{0}\) (Equations (5) and (6)). \[\lambda=\frac{1}{T_{0}} \tag{5}\]  
The parameter \(T_{0}\) is defined by Equation (6), which indicates a reduction in the weight by half in \(T_{0}\) days after the initial time \(t=0\). \[F(T_{0})=\frac{1}{2}\;f(0) \tag{6}\]  
In order to predict the rating \(pr_{aj}\) for the active user \(u_{a}\) and the item \(j\) at the target time \(t_{\text{f}}\) in user-based CF, Equation (3) is modified by introducing the weights \(w_{u,i}(t_{\text{f}})\) of the ratings given by the \(k\) neighbors \(u_{i}\) of the active user who have rated the item \(j\):  
\[pr_{aj}=\bar{r}_{a}+\frac{\sum_{i=1}^{k}w_{u\text{j}}(t_{\text{f}})\cdot sim(u _{a},u_{i})\big{(}r_{ij}-\bar{r}_{i}\big{)}}{\sum_{i=1}^{k}\left|sim(u_{a},u_{i })\right|} \tag{7}\]  
This approach can also be used in item-based CF by incorporating the weight in a similar way to the corresponding formula. In window-based models, corresponding to the second subdivision of the category of recency-based models, a time interval preceding the target time is established. Then, ratings with a timestamp outside the time window are discarded [39; 40]. The size of the window depends on the application domain. An approach combining time windows and decay-based models is presented in [41], where time is partitioned in intervals and the users' interest distribution in these intervals is analyzed in order to model its evolution. As a result, the following decay function, Equation (8), for an elapsed time \(t\) was proposed:  
\[f(t)=0.5+0.5\times e^{-\lambda\;t} \tag{8}\]  
The decay rate \(\lambda\) is obtained from Equation (9), where \(N_{c}\) is the number of preferences of catalog c, and \(n\) is the total number of ratings. Moreover, \(\alpha\) is a constant whose value lies between 0 and 1, and \(\sum_{i=1}^{k}d(i)\) is the number of rated items during one period. \[\lambda=\alpha\;\sum_{i=1}^{k}d(i)\times N_{c}/n \tag{9}\]  
In [42], the window and decay-based approaches are also used together in a bipartite network-based recommendation method. They define the short- and long-term windows from the timestamp information associated to explicit ratings, then an exponential attenuation function is applied to the ratings in order to decrease their importance at a different speed in each long-term window. The method is validated with two MovieLens datasets of different sizes, showing a better performance than other network-based algorithms. Using time as an independent variable allows us to model preference trends with respect to both users and items. Although time-series methods can be applied, the most extended approaches are temporal factor models. They have the advantage over the previously described techniques in that they allow noise and temporary trends to be differentiated from long-term trends, but with more complexity [18].