## 3 Problem Statement  
In the baseline estimation model in (2), related deviations \(b_{u}\) and \(b_{i}\) are both based on the system's global rating range, and it has inherent limitations in controlling the predictive value from overflowing the system's rating range. This global deviation may result in irrational predictive results, and the prediction may even overflow the rating range. For example, assume that the entire recommender system's average rating is \(\mu=4.0\), and the system's rating range is from 0.5 to 5.0. For an unromantic movie M, it obtains an average rating \(\overline{r}_{i}=2.0\), so the related global deviation of movie M would be \(b_{i}=2.0-4.0=-2.0\). [PERSON] is a critic and has a very strict rating criterion on movies, and her average rating on movies is \(\overline{r}_{u}=1.5\), so the related global deviation of Alice would be \(b_{u}=1.5-4.0=-2.5\). The possible baseline estimate on movie M by [PERSON] would be \(\hat{r}_{ui}=(-2.0)+(-2.5)+4.0=-0.5<0\). Obviously, this predictive value is irrational because it does not meet the ranges. More examples were demonstrated in [14; 32].  
In real systems, users always give ratings according to their personalized rating criterion; some are strict, and some are lenient. Ideally, the baseline estimation will be accurate if we fully consider each user's personalization. However, it is time costly and unrealistic in real systems. Therefore, we focused on how to discover a unified personalized factor with commonality to improve the baseline estimation predictive accuracy and attempted to apply it to improve the performance of related CF recommendation algorithms, considering not only global deviation from system ratings but also local deviation from users' personalized ratings.  
We observed that users in many recommender systems can be classified into four kinds by their personalized rating ranges.  
1. _Normal user_. The actual rating range of this kind of user usually covers the recommender system's rating range. For example, assume that the system's rating range is between [1; 5], those normal users' rating range may be the same as [1; 5].
2. _Strict user_. In this situation, users have relatively strict criteria and make a relatively lower rating than others. They rate items strictly, and few cover the highest rating range.
3. _Lenient user_. These users habitually give relatively higher ratings than others, and their actual rating range usually does not cover the lowest range.
4. _Middle user_. Users of this kind neither give higher ratings nor give lower ratings, and their actual rating range is the middle, e.g., [2; 4] while the system's range is [1; 5].  
We performed rating statistics over five real datasets, including Flixster, MiniFilm, FilmTrust, Movielens (10 M), and Movielens (Latest Small), to verify the existence of these four kinds of personalized rating behaviors. Figure 1 shows the rating statistics. During the observation, we defined that the normal users' ratings cover all the system's rating scale, the strict users' ratings do not cover the highest boundary, and the lenient users' ratings do not cover the lowest, while the middle users' ratings do not include both the highest boundary and lowest boundary. From the statistics, we can see that the four situations exist in all datasets. The observation shows that most users were lenient or normal, some were middle, while a few users were strict, and proves that the real rating range of users may be less than the system's rating range. Therefore, it is necessary to consider local deviation during the baseline estimation.