# Notation standards, References¶

In the documentation, you will find the following notation:

• $$R$$ : the set of all ratings.
• $$R_{train}$$, $$R_{test}$$ and $$\hat{R}$$ denote the training set, the test set, and the set of predicted ratings.
• $$U$$ : the set of all users. $$u$$ and $$v$$ denotes users.
• $$I$$ : the set of all items. $$i$$ and $$j$$ denotes items.
• $$U_i$$ : the set of all users that have rated item $$i$$.
• $$U_{ij}$$ : the set of all users that have rated both items $$i$$ and $$j$$.
• $$I_u$$ : the set of all items rated by user $$u$$.
• $$I_{uv}$$ : the set of all items rated by both users $$u$$ and $$v$$.
• $$r_{ui}$$ : the true rating of user $$u$$ for item $$i$$.
• $$\hat{r}_{ui}$$ : the estimated rating of user $$u$$ for item $$i$$.
• $$b_{ui}$$ : the baseline rating of user $$u$$ for item $$i$$.
• $$\mu$$ : the mean of all ratings.
• $$\mu_u$$ : the mean of all ratings given by user $$u$$.
• $$\mu_i$$ : the mean of all ratings given to item $$i$$.
• $$\sigma_u$$ : the standard deviation of all ratings given by user $$u$$.
• $$\sigma_i$$ : the standard deviation of all ratings given to item $$i$$.
• $$N_i^k(u)$$ : the $$k$$ nearest neighbors of user $$u$$ that have rated item $$i$$. This set is computed using a similarity metric.
• $$N_u^k(i)$$ : the $$k$$ nearest neighbors of item $$i$$ that are rated by user $$u$$. This set is computed using a similarity metric.

References

Here are the papers used as references in the documentation. Links to pdf files where added when possible. A simple Google search should lead you easily to the missing ones :)

 [GM05] Thomas George and Srujana Merugu. A scalable collaborative filtering framework based on co-clustering. 2005. URL: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.113.6458&rep=rep1&type=pdf.
 [Kor08] Yehuda Koren. Factorization meets the neighborhood: a multifaceted collaborative filtering model. 2008. URL: http://www.cs.rochester.edu/twiki/pub/Main/HarpSeminar/Factorization_Meets_the_Neighborhood-_a_Multifaceted_Collaborative_Filtering_Model.pdf.
 [Kor10] Yehuda Koren. Factor in the neighbors: scalable and accurate collaborative filtering. 2010. URL: http://courses.ischool.berkeley.edu/i290-dm/s11/SECURE/a1-koren.pdf.
 [KBV09] Yehuda Koren, Robert Bell, and Chris Volinsky. Matrix factorization techniques for recommender systems. 2009.
 [LS01] Daniel D. Lee and H. Sebastian Seung. Algorithms for non-negative matrix factorization. 2001. URL: http://papers.nips.cc/paper/1861-algorithms-for-non-negative-matrix-factorization.pdf.
 [LM07] Daniel Lemire and Anna Maclachlan. Slope one predictors for online rating-based collaborative filtering. 2007. URL: http://arxiv.org/abs/cs/0702144.
 [LZXZ14] Xin Luo, Mengchu Zhou, Yunni Xia, and Qinsheng Zhu. An efficient non-negative matrix factorization-based approach to collaborative filtering for recommender systems. 2014.
 [RRSK10] Francesco Ricci, Lior Rokach, Bracha Shapira, and Paul B. Kantor. Recommender Systems Handbook. 1st edition, 2010.
 [SM08] Ruslan Salakhutdinov and Andriy Mnih. Probabilistic matrix factorization. 2008. URL: http://papers.nips.cc/paper/3208-probabilistic-matrix-factorization.pdf.
 [ZWFM96] Sheng Zhang, Weihong Wang, James Ford, and Fillia Makedon. Learning from incomplete ratings using non-negative matrix factorization. 1996. URL: http://www.siam.org/meetings/sdm06/proceedings/059zhangs2.pdf.