# Enhancing Recommendation Accuracy of Item-Based Collaborative Filtering via Item-Variance Weighting
## 2 Related Work  
### Overview of the Traditional IBCF  
First, we explain the notations and terms related to RSs. Given an RS, \(U\) and \(I\) denote finite sets of users and items, respectively, \(R\cup\{\star\}\) denotes the set of possible item rating scores and \(RM\) is the user-item rating matrix. \(T\cup\{\circ\}\) represents the set of possible item rating time and \(TM\) is the user-item time matrix. The rating score of user \(u\) for item \(x\) is denoted \(r_{u,x}\in R\cup\{\star\}\), where \((\star)\) represents a missing rating and \(t_{u,x}\in T\cup\{\circ\}\) is the time at which item \(x\) was rated by user \(u\), where \((\circ)\) indicates a missing rating time. The average of the valid ratings of item \(x\) is denoted \(\ r_{x}\).  
The IBCF approach was first proposed by [PERSON] [16] and it is widely used in RSs. IBCF has good scalability and can be applied to extremely large numbers of items and users that are typical of modern RSs. In contrast to user-based CF, IBCF can perform well even for RSs that have many items but comparatively few ratings. Furthermore, the IBCF approach can compute item-item similarity offline, which reduces online time and results in more effective recommendations. The IBCF procedure is described in the following.  
Step 1: Item-item similarity computation. On the basis of the user-item rating matrix \(RM\), IBCF computes the similarity between all items. Note that several measures can be used to compute similarity; however, we used the Pearson correlation coefficient, which was widely used in IBCF. Similarity computed using the Pearson correlation coefficient is expressed as follows.  
\[sim(x,y)=\frac{\Sigma_{u\in U_{x}\cap U_{y}}(r_{u,x}-\overline{r_{x}})*\left( r_{u,y}-\overline{r_{y}}\right)}{\sqrt{\Sigma_{u\in U_{x}\cap U_{y}}(r_{u,x}- \overline{r_{x}})^{2}}\ \sqrt{\Sigma_{u\in U_{x}\cap U_{y}}(r_{u,y}-\overline{r_{y}})^{2}}} \tag{1}\]  
Figure 1: The flow chart of proposed approach. Our approach comprises the following main phases: 1 combine user-item rating matrix and user-item time matrix; 2 apply the proposed item-related correlation degree function into similarity measure to compute item-item similarity; 3 further select the neighborhood for each similar item of a target item; 4 insert proposed covering degree function into rating prediction; 5 recommend top \(N\) items with highest predicted rating scores.  
Here, \(U_{x}=\{u\in U\big{|}\tau_{u,x}\
eq\star\}\) is a set of users who have rated item \(x\) and \(\overline{r_{y}}\) denotes the average rating of item \(y\).  
Step 2: Neighborhood selection. After computing item-item similarity, for each target item \(ti\) that has not been rated by target user \(tu\), items with high similarity with target item \(ti\) will be selected as its neighborhood. Generally, in traditional IBCF, after all items are sorted in descending order of similarity values, the \(k\) most similar (nearest) items will comprise the neighborhood for target item \(ti\).  
Step 3: Rating prediction. Note that ratings are normalized and according to the rating information of the target item's neighborhood for target user \(tu\), a rating score is predicted for each target item \(ti\). Note that the weighted sum is a useful measure often used in the IBCF approach:  
\[p_{tu,ti}=\frac{\Sigma_{q\in N_{ti}(k)\cap I_{tu}}sim(t_{i},q)\star r_{tu,q}}{ \Sigma_{q\in N_{tu}(k)\cap I_{tu}}\big{|}sim(t_{i},q)\big{|}} \tag{2}\]  
Here, \(I_{tu}=\{x\in I\big{|}\tau_{tu,x}\
eq\star\}\) is the set of items rated by target user \(tu\), \(N_{ti}(k)\) is the neighborhood of target item \(ti\) and \(P_{tu,ti}\) is the prediction made about target item \(ti\) for target user \(tu\).