# An Efficient Approach to Manage Natural Noises in Recommender Systems
## 3 The Proposed Approach of Managing Natural Noises
### Detection of Natural Noise
#### 3.1.2 The Subcategories of Users and Items  
For a given user \(u\), the percentages of low, medium, and high ratings among all historical ratings can be calculated as follows:  
\[x_{u}=\frac{|UR_{low}|}{|UR|},y_{u}=\frac{|UR_{middle}|}{|UR|},z_{u}=\frac{|UR _{high}|}{|UR|}, \tag{3}\]  
where \(UR\) denotes the set of all ratings provided by user \(u\), and \(UR_{low}\), \(UR_{middle}\), and \(UR_{high}\) are the subsets of \(UR\), which consist of low ratings, medium ratings, and high ratings as given by Equations (1) and (2). The function \(|\cdot|\) returns the number of elements in a set. Based on this, we use the triple \(\omega_{u}=(x_{u},y_{u},z_{u})\) to describe the trend of how user \(u\) rates items. Similarly, the triple \(\omega_{i}=(x_{i},y_{i},z_{i})\) is used to describe the tendency of how item \(i\) obtains ratings from users. Here, \(x_{i}\), \(y_{i}\), and \(z_{i}\) are the corresponding percentages of low ratings, medium ratings, and high ratings of all the ratings of item \(i\), which are calculated as  
\[x_{i}=\frac{|IR_{low}|}{|IR|},y_{i}=\frac{|IR_{middle}|}{|IR|},z_{i}=\frac{| IR_{high}|}{|IR|}, \tag{4}\]where \(IR\) denotes the set of all ratings on item \(i\), and \(IR_{low}\), \(IR_{middle}\), and \(IR_{high}\) are the subsets of \(IR\) that consist of low ratings, medium ratings, and high ratings, respectively. According to Equation (3), we define three kinds of special virtual users, i.e., Negative-Extreme user, Average-Extreme user, and Positive-Extreme user. A Negative-Extreme user is a virtual user who only gives low ratings for all items, i.e., \(x_{u}=1,y_{u}=0,z_{u}=0\). Similarly, an Average-Extreme user is a user whose \(x_{u}=0,y_{u}=1,z_{u}=0\), and a Positive-Extreme user is a user whose \(x_{u}=0,y_{u}=0,z_{u}=1\). For items, according to Equation (4), we define three kinds of special virtual items, i.e., No-Preferred-Extreme item (\(x_{i}=1\), \(y_{i}=0,z_{i}=0\)), Av-Preferred-Extreme item (\(x_{i}=0,y_{i}=1,z_{i}=0\)), and Preferred-Extreme item (\(x_{i}=0,y_{i}=0,z_{i}=1\)). Table 2 lists these special virtual users and items. Here, the virtual users and items in Table 2 are used as criteria for classifying users and items into subcategories. For a given user \(u\), we determine which subcategory user \(u\) should belong to by calculating the distances from \(u\) to the Negative-Extreme user, the Average-Extreme user, and the Positive-Extreme user. The distance between two triples \(\omega_{1}=\left(x_{1},y_{1},z_{1}\right)\) and \(\omega_{2}=\left(x_{2},y_{2},z_{2}\right)\) is defined by  
\[d(\omega_{1},\omega_{2})=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1} \right)^{2}+\left(z_{2}-z_{1}\right)^{2}}. \tag{5}\]  
With Equation (5), we can compute the distances between user \(u\) and the three kinds of users. For a given item \(i\), the distances between \(i\) and the three kinds of items can also be calculated with Equation (5). Figure 0(a) illustrates the distances from user \(u\) to the three kinds of users given in Table 2. According to the three distances in Figure 0(a), we divide users into three subcategories: Negative, Average, and Positive. The rules of classifying users are as follows:  
* If \(d(\omega_{u},\omega_{Negative-Extreme})\) is the minimum value among \(d(\omega_{u},\omega_{Negative-Extreme})\), \(d(\omega_{u},\omega_{Average-Extreme})\) and \(d(\omega_{u},\omega_{Positive-Extreme})\), the user \(u\) is regarded as a Negative user;
* If \(d(\omega_{u},\omega_{Average-Extreme})\) is the minimum value among \(d(\omega_{u},\omega_{Negative-Extreme})\), \(d(\omega_{u},\omega_{Average-Extreme})\) and \(d(\omega_{u},\omega_{Positive-Extreme})\), the user \(u\) is regarded as an Average;
* If \(d(\omega_{u},\omega_{Positive-Extreme})\) is the minimum value among \(d(\omega_{u},\omega_{Negative-Extreme})\), \(d(\omega_{u},\omega_{Average-Extreme})\) and \(d(\omega_{u},\omega_{Positive-Extreme})\), the user \(u\) is regarded as a Positive user. Here, Negative users are fastidious and tend to rate items with low ratings.