# A Technique of Recursive Reliability-Based Missing Data Imputation for Collaborative Filtering
## 4 k-Recursive Reliability-Based Imputation (k-RRI)
### k-RRI  
In this paper, we propose a k-RRI, which is a recursive imputation method for CF based on reliability. We first define the related key terms used in this paper as follows:  
* Active user: the user of the data targeted for prediction. * Active item: the item of the data targeted for prediction. * Key neighbors: a set of data necessary for predicting the active user and the active item. Each imputation algorithm will define the key neighbors and impute the missing data, i.e., unrated data, among them. * High-reliability data: a set of data whose similarity with the active user and active item exceeds the given threshold. According to the prerequisite for neighborhood-based CF defined by [17], the similar nearest neighbor has the most information regarding the data to be predicted. Accordingly, k-RRI is implemented according to the criteria defined below. * Definition 1: Missing data are selectively imputed in a stepwise and decreasing order of reliability. * Definition 2: Data imputed in the preceding step can be used for imputation in the succeeding step. * Definition 3: The threshold cutoff value decreases as the algorithm progresses. That is, the reliability-based threshold cutoff criteria are strict in earlier recursive steps and become increasingly relaxed toward the end. Similarity is determined by the [PERSON]'s modified similarity computation equation [27] presented in Section 2, whereby the number of data unrated by both users is included in the weight. In other words, in the case of the identical [PERSON]'s Correlation Coefficient, the larger the size of the unrated data subset, the higher the similarity. In k-RRI, the user-user similarity is defined by Equation (8), which is a modification of Equation (1) by incorporating a weight factor. The purpose of adding weight to the similarity measure in k-RRI is to lower the similarity value obtained in cases where there are only a small number of data rated or unrated by both users. In Equation (8), \(\left|I_{a}^{c}\cap I_{b}^{c}\right|\) is the number of items that have not been rated by both users. In the same manner, the item-item similarity is defined by Equation (9), which is a modification of Equation (2) by adding a weight factor. In Equation (9), \(\left|I_{\delta}^{c}\cap U_{I}^{c}\right|\) is the number of the users who have not rated both items. Parameters \(\gamma_{I}\) and \(\gamma_{I^{c}}\) are required as input values when computing user similarity with the k-RRI algorithm, whereby a user similarity value is considered reliable only if the number of the data rated by both users is higher than or equal to \(\gamma_{I}\) and the number of the data unrated by both users is higher than or equal to \(\gamma_{I^{c}}\). Thus, when assigning weight to \(Sim_{u}^{\prime}(a,\ b)\), the lower values between \(\left|I_{a}\cap I_{b}\right|\) and \(\gamma_{I}\) and between \(\left|I_{a}^{c}\cap I_{b}^{c}\right|\) and \(\gamma_{I^{c}}\) are taken as the numerator and divided by the product of \(\gamma_{I}\) and \(\gamma_{I^{c}}\) to prevent \(Sim_{u}^{\prime}(a,\ b)\) from exceeding 1, as per the basic definition of similarity. Similarly, a user similarity value is considered reliable only if the number of data rated by both users is higher than or equal to \(\delta_{U}\) and the number of the data unrated by both users is higher than or equal to \(\delta_{U^{c}}\). Accordingly, when assigning weight to \(Sim_{u}^{\prime}(s,t)\), the lower values between \(\left|U_{s}\cap U_{I}\right|\) and \(\delta_{U}\) and between \(\left|U_{s}^{c}\cap U_{I}^{c}\right|\) and \(\delta_{U^{c}}\) are taken as the numerator and divided by the product of \(\delta_{U}\) and \(\delta_{U^{c}}\) to prevent \(Sim_{u}^{\prime}(s,t)\) from exceeding 1. For example, in the case of the parameter values \(\gamma_{I}=10\) and \(\gamma_{I^{c}}=20\), if the number of items rated by both users is 10 or more and the number of the items unrated by both users is 20 or more, then \(Sim_{u}^{\prime}(a,\ b)=Sim_{u}(a,b)\).