# An Efficient Approach to Manage Natural Noises in Recommender Systems
## 3 The Proposed Approach of Managing Natural Noises
### Correction of the Natural Noise  
After detecting the natural noise in ratings, we design a method to correct it according to the characteristics of the user. As we know, positive users are most likely to provide high ratings to Preferred items, but there still exists the possibility that they provide items with low or medium ratings. Thus, we correct the natural noise with threshold values weighted by probabilities. The details of our scheme are as follows. Firstly, we calculate the probabilities that user \(u\) belongs to the three subcategories of users, i.e., \(p(u\in\textit{Negative})\), \(p(u\in\textit{Average})\), and \(p(u\in\textit{Positive})\). Since the probabilities are correlated to the distance given by Equation (5), they are defined as  
\[p(u\in\textit{Negative})=\frac{\varphi(u,\textit{Negative})}{\varphi(u, \textit{Negative})+\varphi(u,\textit{Average})+\varphi(u,\textit{Positive})}, \tag{6}\]  
\[p(u\in\textit{Average})=\frac{\varphi(u,\textit{Average})}{\varphi(u,\textit{ Negative})+\varphi(u,\textit{Average})+\varphi(u,\textit{Positive})}, \tag{7}\]  
\[p(u\in\textit{Positive})=\frac{\varphi(u,\textit{Positive})}{\varphi(u, \textit{Negative})+\varphi(u,\textit{Average})+\varphi(u,\textit{Positive})}, \tag{8}\]  
where \(\varphi(u,\textit{Negative})=\frac{1}{d\left(\omega_{u},\omega_{\textit{ Negative-Extreme}}\right)}\), \(\varphi(u,\textit{Average})=\frac{1}{d\left(\omega_{u},\omega_{\textit{ Average-Extreme}}\right)}\) and \(\varphi(u,\textit{Positive})=\frac{1}{d\left(\omega_{u},\omega_{\textit{Positive-Extreme}}\right)}\). Secondly, we calculate the probabilities that item \(i\) belongs to the three subcategories of items. The corresponding probabilities are denoted as \(p(i\in\textit{No-Preferred})\), \(p(i\in\textit{Av-Preferred})\), and \(p(i\in\textit{Preferred})\), which are obtained by  
\[p(i\in\textit{No-Preferred})=\frac{\varphi(i,\textit{No-Preferred})}{\varphi( i,\textit{No-Preferred})+\varphi(i,\textit{Av-Preferred})+\varphi(i,\textit{Preferred})}, \tag{9}\]  
\[p(i\in\textit{Av-Preferred})=\frac{\varphi(i,\textit{Av-Preferred})}{\varphi( i,\textit{No-Preferred})+\varphi(i,\textit{Av-Preferred})+\varphi(i,\textit{Preferred})}, \tag{10}\]  
\[p(i\in\textit{Preferred})=\frac{\varphi(i,\textit{Preferred})}{\varphi(i, \textit{No-Preferred})+\varphi(i,\textit{Av-Preferred})+\varphi(i,\textit{ Preferred})}. \tag{11}\]  
where \(\varphi(i,\textit{No-Preferred})=\frac{1}{d\left(\omega_{i},\omega_{\textit{ No-Preferred-Extreme}}\right)}\), \(\varphi(i,\textit{Av-Preferred})=\frac{1}{d\left(\omega_{i},\omega_{\textit{ No-Preferred-Extreme}}\right)}\) and \(\varphi(i,\textit{Preferred})=\frac{1}{d\left(\omega_{i},\omega_{\textit{No-Preferred-Extreme}}\right)}\). Thirdly, based on the probabilities of users and items, we calculate the probabilities that user \(u\) gives low ratings, medium ratings, and high ratings to item \(i\) as  
\[p(r_{ui}\leq k)=\frac{\pi(r_{ui}\leq k)}{\pi(r_{ui}\leq k)+\pi(k<r_{ui}<v)+\pi(r _{ui}\geq v)}, \tag{12}\]  
\[p(k<r_{ui}<v)=\frac{\pi(k<r_{ui}<v)}{\pi(r_{ui}\leq k)+\pi(k<r_{ui}<v)+\pi(r_{ ui}\geq v)}, \tag{13}\]  
\[p(r_{ui}\geq v)=\frac{\pi(r_{ui}\geq v)}{\pi(r_{ui}\leq k)+\pi(k<r_{ui}<v)+\pi(r _{ui}\geq v)}. \tag{14}\]  
where \(r_{ui}\) is the value rated by user \(u\) on item \(i\), \(\pi(r_{ui}\leq k)=p(u\in\textit{Negative})\times p(i\in\textit{No-Preferred})\), \(\pi(k<r_{ui}<v)=p(u\in\textit{Average})\times p(i\in\textit{Av-Preferred})\) and \(\pi(r_{ui}\geq v)=p(u\in\textit{positive})\times p(i\in\textit{Preferred})\). Finally, the proposed rules to correct the natural noise in ratings are as follows:  
1.