# A Technique of Recursive Reliability-Based Missing Data Imputation for Collaborative Filtering
## 4 k-Recursive Reliability-Based Imputation (k-RRI)
### k-RRI
However, if the number of items unrated by both users is 5 but the number of the items rated by both users is 10, then \(Sim_{i}^{\prime}(s,\ t)=\frac{1}{4}Sim_{i}(s,t)\), where one-quarter of the computed similarity value is taken. As the value of \(\gamma_{I}\) increases, the number of items rated by both users exerts a higher influence on the similarity value. In addition, as the value of \(\gamma_{I^{c}}\) increases, the number of items unrated by both users exerts a higher influence on the similarity value, whereby it is always equal to or lower than the value computed from Equation (1). Similarly, as the value of \(\delta_{U}\) increases, the number of users that rated both items exerts higher influence on the similarity value, and as the value of \(\delta_{U^{c}}\) increases, the number of users that have not rated both items exerts a higher influence on the similarity value, whereby it is always equal to or lower than the value computed from Equation (2). \[Sim_{u}^{\prime}(a,\ b)=\frac{Min(\left|I_{a}\cap I_{b}\right|,\ \gamma_{I})}{\gamma_{I}}\times\frac{Min(\left|I_{a}^{c}\cap I_{b}^{c} \right|,\ \gamma_{I^{c}})}{\gamma_{I^{c}}}\times Sim_{u}(a,b)\] (8) \[Sim_{i}^{\prime}(s,\ t)=\frac{Min(\left|U_{s}\cap U_{I}\right|,\ \delta_{U})}{\delta_{U}}\times\frac{Min(\left|U_{s}\cap U_{I}^{c}\right|,\ \delta_{U^{c}})}{\delta_{U^{c}}}\times Sim_{i}(s,t)\] (9)  
The basic steps of the k-RRI are described in Algorithm 1. The input values for Algorithm 1 are the user-item rating matrix, user-item similarity set, active user and active item for prediction, number of recursive steps, and final user and item similarity thresholds. The output is the imputed matrix for the active user and the active item. In Algorithm 1, the final user similarity threshold \(\eta_{k}\) is used when selecting a high-reliability user set. The user similarity threshold at any given step \(\left[\eta_{r}\right]\) is a value between 1 (i.e., maximum similarity value) and \(\eta_{k}\). Similarly, the final item similarity threshold \(\theta_{k}\) is used when selecting a high-reliability item set, and the item similarity threshold at any given step \(\left[\theta_{r}\right]\) is a value between 1 (i.e., maximum similarity value) and \(\theta_{k}\). \(\eta_{k}\) and \(\theta_{k}\) indicate any real number in the closed interval \([0,1]\), which is the similarity range. In the first line, the imputed matrix is initialized with the user-item rating matrix. Lines 2-13 represent the process of imputing missing data according to the threshold cutoff value, which is reset at each iteration while performing the \(k\) recursive steps. As the algorithm progresses, the threshold cutoff values are decreased. In the third line, the current user and item similarity thresholds (\(\eta_{r}\) and \(\theta_{r}\), respectively) are set. The reliability-based threshold reset process for determining \(\eta_{r}\) and \(\theta_{r}\) is explained in detail in Section 4.2. The sets of users \(\mathcal{U}_{a}\) and items \(\mathcal{I}_{s}\) satisfying the current threshold values are generated in lines 4 and 5, respectively. In line 6, the key neighbor set \(\mathcal{N}_{a,s}\) is selected, combining the elements of \(\mathcal{U}_{a}\) and \(\mathcal{I}_{s}\). That is, \(\mathcal{N}_{a,s}\) is the set of data containing information necessary for predicting user \(u_{a}\)'s rating of item \(i_{s}\). In lines 7-12, the imputed matrix is filled by imputing the missing values among the elements of \(\mathcal{N}_{a,s}\). \(\hat{\eta}_{r^{\prime}s^{\prime}}\) in the ninth line is calculated using the rating prediction method described in Section 2.2. ```
Input: the user-item rating matrix \(\mathcal{R}\). the similarity set \(sim^{\prime}\). the active user \(u_{a}\). the active item \(i_{s}\). the number of steps \(k\). the user's final similarity threshold \(\eta_{k}\). the item's final similarity threshold \(\theta_{k}\). Output: the imputed matrix \(\mathcal{R}^{\prime}\).