There are two tasks supported by gorse: item ranking and similar recommendation. The rating prediction is also implemented as a previous stage for item ranking when the feedback is explicit.

## Rating Prediction¶

Given a user and an item, the recommender system is required to predict the rating. The famous Netflix Prize  is a competition to predict ratings. It works with explicit feedbacks but might be intractable in practice since there are few explicit feedbacks in the real world.

### Definition¶

Predict the rating $$\hat r_{ui}$$ that user $$u$$ gives to item $$i$$.

### Evaluation¶

A rating prediction model will be evaluated on a test dataset $$D = \{ <u,i,r_{ui}> \}$$.

#### Root Mean Sqaure Error¶

$\text{RMSE}=\sqrt{\frac{1}{|D|}\sum_{<u,i,r_{ui}>\in D}(\hat r_{ui}-r_{ui})^2}$

#### Mean Absolute Error¶

$\text{MAE}=\frac{1}{|D|}\sum_{<u,i,r_{ui}>\in D}|\hat r_{ui}-r_{ui}|$

Warning

Rating prediction has been implemented by gorse, but serves for item ranking task. You can’t get rating prediction directly from RESTful APIs.

## Item Ranking¶

In this case, a list of items are recommended to a user. For explicit feedbacks, items could be ranked by their predicted ratings. For implicit feedbacks, item ranking could be done by ranking models. The ranking model could be point-wise (WRMF ), pair-wise (BPR ) or list-wise (CLiMF ).

### Definition¶

Generate a list of top-N items $$\hat I_u$$ for user $$u$$, where $$N=|\hat I_u|$$. The i-th item in the list is $$\hat I_{u,i}$$.

### Evaluation¶

For each user $$u$$, there is a list of items $$I^+_u$$ (from test dataset) that user really interested in. Items that user $$u$$ has been interacted must be exclued from $$\hat I_u$$ and $$I^+_u$$.

#### Precision¶

$\text{Precision}(u)=\frac{\mid I^+_u\mid\cap\mid \hat I_u \mid}{\mid \hat I_u \mid}$

#### Recall¶

$\text{Recall}(u)=\frac{\mid I^+_u \mid \cap \mid \hat I_u \mid}{\mid I^+_u \mid}$

#### Mean Reciprocal Rank¶

$\text{MRR}(u)=\frac{1}{\mid U\mid}\sum_{u\in U}\frac{1}{\text{rank}(u)}$
$\begin{split}\text{rank}(u) = \begin{cases} \min_{\hat I_{u,i}, \in I^+_u} i & \exists i, \hat I_{u,i} \in I^+_u \\ +\infty, & \forall i, \hat I_{u,i} \notin I^+_u \\ \end{cases}\end{split}$

#### Mean Average Precision¶

$\text{AP}(u)=\frac{1}{|I^+_u|}\sum^N_{i=1}\mathcal 1(\hat I_{u,i} \in I^+_u)\text{Precision}(u)_i$

#### Normalized Discounted Cumulative Gain¶

$\text{NDCG}(u)=\frac{1}{N}\sum^N_{i=1}\frac{\mathcal 1(\hat I_{u,i} \in I^+_u)}{\log_2(1+i)}$

## Similar Recommendation¶

Recommend similar items ranked by _similarity to a item.