docs: add models

docs/introduction/index.rst

@@ -6,7 +6,7 @@ Introduction
    :caption: Table of Contents
    :maxdepth: 1
 
-   task
    data
-   model
+   task
+   model/index
    benchmark

docs/introduction/model/coclustering.rst

@@ -0,0 +1,18 @@
+============
+CoClustering
+============
+
+Co-Clustering [#COC]_ predicts ratings by clustering users and items. 
+
+.. math::
+
+    \hat r_{ij}=A^{COC}_{gh}+(A^R_i-A^{RC}_g)+(A^C_j-A^{CC}_h)
+
+
+Training
+========
+
+References
+==========
+
+.. [#COC] George, Thomas, and Srujana Merugu. "A scalable collaborative filtering framework based on co-clustering." Data Mining, Fifth IEEE international conference on. IEEE, 2005.

docs/introduction/model/index.rst

@@ -0,0 +1,70 @@
+======
+Models
+======
+
+
++----------------------+------------------------------+------------------+
+| Model                | Data                         | Task             |
++----------------------+----------+----------+--------+--------+---------+
+|                      | explicit | implicit | weight | rating | ranking |
++======================+==========+==========+========+========+=========+
+| BaseLine             | √        |          |        | √      | √       |
++----------------------+----------+----------+--------+--------+---------+
+| NMF [#NMF]_          | √        |          |        | √      | √       |
++----------------------+----------+----------+--------+--------+---------+
+| SVD                  | √        |          |        | √      | √       |
++----------------------+----------+----------+--------+--------+---------+
+| SVD++ [#SVDPP]_      | √        |          |        | √      | √       |
++----------------------+----------+----------+--------+--------+---------+
+| KNN [#KNN]_          | √        |          |        | √      | √       |
++----------------------+----------+----------+--------+--------+---------+
+| CoClustering [#COC]_ | √        |          |        | √      | √       |
++----------------------+----------+----------+--------+--------+---------+
+| SlopeOne [#SO]_      | √        |          |        | √      | √       |
++----------------------+----------+----------+--------+--------+---------+
+| ItemPop              | √        | √        |        |        | √       |
++----------------------+----------+----------+--------+--------+---------+
+| WRMF [#WRMF]_        | √        | √        | √      |        | √       |
++----------------------+----------+----------+--------+--------+---------+
+| BPR [#BPR]_          | √        | √        |        |        | √       |
++----------------------+----------+----------+--------+--------+---------+
+
+Apparently, these models using implicit feedbacks are more general since explicit feedbacks could be converted to implicit feedbacks and item ranking could be done by rating prediction.
+
+
+Non-Personalized Models
+=======================
+
+Personalized Models
+===================
+
+.. toctree::
+   :caption: Personalized Models
+   :maxdepth: 1
+
+   matrix_factorization
+   knn
+   coclustering
+   slopeone
+
+
+References
+==========
+
+.. [#Surprise] Hug, Nicolas. Surprise, a Python library for recommender systems. http://surpriselib.com, 2017.
+
+.. [#LibRec] G. Guo, J. Zhang, Z. Sun and N. Yorke-Smith, LibRec: A Java Library for Recommender Systems, in Posters, Demos, Late-breaking Results and Workshop Proceedings of the 23rd Conference on User Modelling, Adaptation and Personalization (UMAP), 2015.
+
+.. [#NMF] Luo, Xin, et al. "An efficient non-negative matrix-factorization-based approach to collaborative filtering for recommender systems." IEEE Transactions on Industrial Informatics 10.2 (2014): 1273-1284.
+
+.. [#SO] "Slope one predictors for online rating-based collaborative filtering." Proceedings of the 2005 SIAM International Conference on Data Mining. Society for Industrial and Applied Mathematics, 2005.
+
+.. [#COC] George, Thomas, and Srujana Merugu. "A scalable collaborative filtering framework based on co-clustering." Data Mining, Fifth IEEE international conference on. IEEE, 2005.
+
+.. [#WRMF] Hu, Yifan, Yehuda Koren, and Chris Volinsky. "Collaborative filtering for implicit feedback datasets." Data Mining, 2008. ICDM'08. Eighth IEEE International Conference on. Ieee, 2008.
+
+.. [#KNN] Desrosiers, Christian, and George Karypis. "A comprehensive survey of neighborhood-based recommendation methods." Recommender systems handbook. Springer, Boston, MA, 2011. 107-144.
+
+.. [#SVDPP] Koren, Yehuda. "Factorization meets the neighborhood: a multifaceted collaborative filtering model." Proceedings of the 14th ACM SIGKDD international conference on Knowledge discovery and data mining. ACM, 2008.
+
+.. [#BPR] Rendle, Steffen, et al. "BPR: Bayesian personalized ranking from implicit feedback." Proceedings of the twenty-fifth conference on uncertainty in artificial intelligence. AUAI Press, 2009.

docs/introduction/model/knn.rst

@@ -0,0 +1,103 @@
+===
+KNN
+===
+
+KNN
+---
+
+Neighbors-based models [#KNN]_ predict ratings from similar items or similar users. There are two kinds of neighbors-based models: user-based and item-based, which means whether predictions are made by similar users or items. In general, item-based model is better than user-based model since items' characters are more consistent than users' preferences.
+
+.. _similarity:
+
+Similarity
+^^^^^^^^^^
+
+Similarity metrics define the definition of nearest neighbors. Items rated by two different users :math:`u` and :math:`v` are represented by :math:`I_u` and :math:`I_v`. Users rated two different items :math:`i` and :math:`j` are represented by :math:`U_i` and :math:`U_j`. There are several most used similarity functions:
+
+Cosine
+""""""
+
+.. math::
+
+    \cos(u,v)=\frac{\sum\limits_{k\in|I_u\cap I_v|}r_{uk}\cdot r_{vk}}{\sqrt{\sum\limits_{k\in|I_u\cap I_v|}r_{uk}^2}\cdot\sqrt{\sum\limits_{k\in|I_u\cap I_v|}r_{vk}^2}}
+
+Pearson
+"""""""
+
+Pearson similarity is similar to cosine similarity but ratings are subtracted by means first.
+
+.. math::
+
+    \text{pearson}(a,b)=\frac{\sum\limits_{k\in|I_a\cap I_b|}(r_{ak}-\tilde r_a)\cdot (r_{bk}-\tilde r_b)}{\sqrt{\sum\limits_{k\in|I_a\cap I_b|}(r_{ak}-\tilde r_a)^2}\cdot\sqrt{\sum\limits_{k\in|I_a\cap I_b|}(r_{bk}-\tilde r_b)^2}}
+
+where :math:`\tilde r_a` is the mean of ratings rated by the user :math:`a`:
+
+.. math::
+
+    \tilde r_a = \sum_{k\in I_a} r_{ak}
+
+Mean Square Distance
+""""""""""""""""""""
+
+
+The *Mean Square Distance* is
+
+.. math::
+
+    \text{msd}(a,b)=\frac{1}{|I_a\cap I_b|}\sum_{k\in|I_a\cap I_b|}(r_{ak}-r_{bk})^2
+
+Then, the *Mean Square Distance Similarity* is
+
+.. math::
+
+    \text{msd_sim}(u, v) = \frac{1}{\text{msd}(u, v) + 1}
+
+
+Predict
+^^^^^^^
+
+A rating could be predict by k nearest neighbors $\mathcal N_k(i)$ (k users or items with max k similarity).
+
+.. math::
+
+    \hat r_{ui}=\frac{\sum_{v\in \mathcal N_k(u)}\text{sim}(u,v)r_{vi}}{\sum_{v\in \mathcal N_k(u)}\text{sim}(u,v)}
+
+
+The basic KNN prediction has some problem, There are more advanced methods which achieve higher accuracy.
+
+KNN with Mean
+"""""""""""""
+
+$\tilde r_l$ is the mean of l-th user's (or item's) ratings.
+
+.. math::
+
+    \hat r_{ij}=\tilde r_i+\frac{\sum_{l\in \mathcal N_k(i)}\text{sim}(i,l)(r_{lj}-\tilde r_l)}{\sum_{l\in \mathcal N_k(i)}\text{sim}(i,l)}
+    
+KNN with Z-score
+""""""""""""""""
+
+$\sigma(r_i)$ is the standard deviation of l-th user's (or item's) ratings.
+
+.. math::
+
+    \hat r_{ij}=\tilde r_i+\sigma(r_i)\frac{\sum_{l\in \mathcal N_k(i)}\text{sim}(i,l)\frac{r_{lj}-\tilde r_l}{\sigma(r_l)}}{\sum_{l\in \mathcal N_k(i)}\text{sim}(i,l)}
+
+
+KNN with Baseline
+"""""""""""""""""
+
+$b_l$ is the baseline comes from the baseline model $\hat r_{ij}=b+b_i+b_j+p_i^Tq_j$.
+
+.. math::
+
+    \hat r_{ij}=b_i+\frac{\sum_{l\in \mathcal N_k(i)}\text{sim}(i,l)(r_{lj}- b_l)}{\sum_{l\in \mathcal N_k(i)}\text{sim}(i,l)}
+
+The KNN model with baseline is the best model since biases are used.
+
+
+
+References
+==========
+
+.. [#KNN] Desrosiers, Christian, and George Karypis. "A comprehensive survey of neighborhood-based recommendation methods." Recommender systems handbook. Springer, Boston, MA, 2011. 107-144.