## 3 Proposed Method
### Multi-Time Contextual Profiling  
The proposed system recommends relevant live TV programs to users based on their preference for the features of broadcasted programs. In general, channels and genres are the available features of programs in standalone TVs as explained in Section 1. So, we recommend TV programs based on the channel and the genre preferences of users. Users' channel and genre preferences are inferred from the channels and genres of TV programs the user watched. We assume that users' watching logs include channels, time stamps, and genres of programs. Since there is no direct users' preference on channels or genres, we estimate the implicit preference on those using the time stamps. The channel preference is calculated based on the total watching time for the channel. A user \(u\)'s preference for channel \(c\) in time slot \(s\), notated as \(p_{u,c,s}\), is calculated as follows:  
\[p_{u,c,s}=\frac{\sum_{l_{c,s}}(end_{l_{c,s}}-begin_{l_{c,s}})}{\sum_{n\in \text{Channels}}\sum_{l_{n,s}}(end_{l_{n,s}}-begin_{l_{n,s}})}, \tag{1}\]  
where \(l_{c,s}\) is a user \(u\)'s log watching channel \(c\) in time slot \(s\). \(begin_{l_{c,s}}\) and \(end_{l_{c,s}}\) stand for the viewing start time and the viewing end time, respectively. Time slot \(s\) can be any period such as Monday, 1-2 p.m., etc. The preference is a normalized value, so its range is from 0 to 1. The value of channel preferences can be high if a user frequently watches a specific channel. For example, if a user watches channel 1 for total of 20 h and channel 2 for 30 h on all Mondays, the preference for channel 1 is 0.4 and channel 2 is 0.6 on Monday. In the same way, a user \(u\)'s preference for genre \(g\) in time slot \(s\), notated as \(p_{u,g,s}\), is calculated as follows:  
\[p_{u,g,s}=\frac{\sum_{l_{g,s}}\left(end_{l_{g,s}}-begin_{l_{g,s}}\right)}{ \sum_{n\in\text{Genes}}\sum_{l_{n,s}}\left(end_{l_{n,s}}-begin_{l_{n,s}} \right)}, \tag{2}\]  
where \(l_{g,s}\) is a user \(u\)'s log watching a program of genre \(g\) in time slot \(s\). \(begin_{l_{g,s}}\) and \(end_{l_{g,s}}\) stand for the viewing start and end time, respectively. Most TV programs are broadcasted periodically. For example, some programs are broadcast at 9 p.m. every day, and others are broadcast every Sunday. The main temporal factors of periodicity are day and hour, so a user's preference is highly associated with them. We model user preferences in four-time context: usual context, day context, hour context, and day-hour context. Usual context is a contextual factor covering all duration of a user's watching history. So, the preference of usual context refers to a user's ordinary taste. In day context, there are seven-time slots from Monday to Sunday to model user preference on each day. In hour context, there are 24-time slots from 0 to 23, which represent time slots of each hour. In day-hour context, there are 168 (7 \(\times\) 24) time slots, which represent combinations of days and hours to model user preferences at a specific time on a specific day. For example, \(p_{u,c,\text{Mon}}\) is user \(u\)'s preference on channel \(c\) on Monday time slot. Thus, the proposed system constructs four time-aware preference matrices for channels and genres as shown in Figure 1. It shows a user's four matrices of time contextual channel preferences. The rows and columns of matrices in Figure 1 denote time slots and TV channels, respectively. User \(u\)'s channel preference matrices in usual, day, hour, and day-hour contexts will be denoted as \(U^{u}\), \(D^{u}\), \(H^{u}\), and \(T^{u}\). For example, \(D^{u}_{Mon,c_{1}}\) is a user \(u\)'s preference for channel 1 on Monday in day matrix, and its value is 0.1 in Figure 1b. \(H^{u}_{2-3h,c_{2}}\) is user \(u\)'s preference for channel 2 at 2-3 a.m. in hour matrix and its value is 0.5 in Figure 1c. Matrices of time contextual preferences on the genre, \(U^{u}\), \(D^{u}\), \(H^{u}\), and \(T^{u}\), are constructed in the same way.