# A Technique of Recursive Reliability-Based Missing Data Imputation for Collaborative Filtering
## 3 Related Studies
The same applies to the weight of \(Sim^{\prime}_{u}(s,\ t)\): the lower value between \(|U_{s}\cap U_{t}|\) and \(\delta\) is taken as the numerator and is divided by the denominator \(\delta\) to prevent \(Sim^{\prime}_{i}(s,\ t)\) from exceeding 1. For example, in the case of the parameter value \(\gamma=10\), if the number of items rated by both users is 10 or more, then \(Sim^{\prime}_{u}(a,\ b)=Sim_{u}(a,b)\). If the number of items rated by both users is 5, then \(Sim^{\prime}_{u}(a,\ b)=\frac{1}{2}Sim_{u}(a,b)\), and half of the computed similarity value is taken. As the value of \(\gamma\) increases, the number of items rated by both users exerts higher influence on the similarity value, whereby it is always equal to or lower than the value computed with Equation (1). Similarly, as the value of \(\delta\) increases, the number of users that rated both items exerts higher influence on the similarity value, whereby it is always equal to or lower than the value computed from Equation (2). \[Sim^{\prime}_{u}(a,\ b)=\frac{Min(|I_{a}\cap I_{b}|,\ \gamma)}{\gamma}\times Sim _{u}(a,b) \tag{6}\]  
\[Sim^{\prime}_{i}(s,\ t)=\frac{Min(|U_{s}\cap U_{t}|,\ \delta)}{\delta}\times Sim _{i}(s,t) \tag{7}\]  
There were several methods that solved a data sparsity problem based on EMDP in various fields. For example, [PERSON] et al. [13] proposed to solve the data sparsity problem in movie recommendation applications by combining EMDP and Local and Global User Similarity (LU&GU) [28] methods. These approaches were improved by integrating the content information of the movies (e.g., cast, director, genre) during the item similarity calculations. The experiment results showed that the overall system performance improved when the proposed method was augmented with content information. [PERSON] and [PERSON] et al. [14] proposed a CF method for friend recommendation based on their interaction intensity and adaptive user similarity in social networks. To solve the data sparsity problem, the authors incorporated EMDP into the proposed method. As a result, the proposed method with EMDP outperformed its variants on a sparse dataset. [PERSON] et al. [15] proposed a content-aware CF for expert recommendation in social networks. The proposed method uses a new cosine similarity measure, where a weight is assigned for the local similarities among users. By using this similarity measure, the authors predicted missing values based on EMDP and formed a preference score prediction table, where experts are selected with the highest scores. The experiment results demonstrated that the proposed method outperforms the traditional content-aware CF method. [PERSON] et al. [16] proposed to combine EMDP with a method proposed by [PERSON] et al. [13] to achieve more efficient movie recommendations. Similar to [13], the proposed method was integrated with content information of the movies during the item similarity calculations. In addition,the authors used a goal programming technique to calculate the movie feature's similarity scores. The experiment results of the proposed method showed that it outperforms existing methods when it uses content information with a goal programming technique. The main advantage of EMDP is that it can avoid poor imputation (no or insufficient amount of imputed data) because it imputes all the data whose similarities with the active user and active item exceed the thresholds. However, the main disadvantage of EMDP is a relatively poor accuracy because it gives the same value to all missing data [17]. According to a study by [PERSON] [29] on the nearest neighbor rule, the nearest neighbor of a data point in an infinite training dataset shares over 50% of the information with that data point. In other words, when predicting a value, neighboring data have information values of different importance. [PERSON] et al. [18; 19] proposed the Auto-Adaptive Imputation (AutAI) method, which considers neighborhood information when imputing missing data. AutAI imputes data of high importance by focusing on data rated by both users.