Check out the screencast if you need to recall anything here!
Let's work through an example of item-item collaborative filtering similar to what will be used in PA7:
It will help to make a copy and follow along in this spreadsheet.
Check out the solutions spreadsheet here.
Let's say we have this matrix of movie reviews from various users:
| U1 | U2 | U3 | U4 | |
|---|---|---|---|---|
| M1 | 1 | 5 | 3 | |
| M2 | 5 | 4 | ||
| M3 | 2 | 4 | 1 | |
| M4 | 2 | 4 | ||
| M5 | 4 | 3 | 4 | |
| M6 | 1 | 3 |
We have a new user with the following preferences:
U = [M1: ??, M2: ??, M3: 5, M4: 2, M5: 3, M6: ??]
Out of the movies they have not seen (M1, M2, and M6) we want to recommend the film that they are likely to rate the highest.
Start by binarizing the ratings! We will use the following cutoffs (same as PA7):
0
Unrated: 0
3
We partially fill out this table for you. Please fill out the missing binarization for M1 and M6, as well as the New User.
Binarized Matrix:
| U1 | U2 | U3 | U4 | |
|---|---|---|---|---|
| M1 | -1 | 1 | 1 | 0 |
| M2 | 0 | 0 | 1 | 1 |
| M3 | -1 | 1 | 0 | -1 |
| M4 | 0 | -1 | 1 | 0 |
| M5 | 0 | 1 | 1 | 1 |
| M6 | -1 | 0 | 1 | 0 |
New User:
| movie | binarized rating |
|---|---|
| M1 | 0 |
| M2 | 0 |
| M3 | 1 |
| M4 | -1 |
| M5 | 1 |
| M6 | 0 |
For this part (as in PA7) we will use raw cosine similarity, NOT mean-centered overlapping item cosine similarity. Note that in PA7 we do not use mean-centering at all.
Recall the formula for the cosine similarity of two vectors:
Use the binarized vectors when computing the cosine similarity. We provide a few of the calculations for you, fill in the similarities for
Note this is a symmetric matrix, that is
Compute the cosine similarity over all the MOVIE (item) vectors. These are the rows of the above matrix.
| M1 | M2 | M3 | M4 | M5 | M6 | |
|---|---|---|---|---|---|---|
| M1 | 1 | 0.41 | 0.67 | 0 | 0.67 | 0.82 |
| M2 | 1 | -0.41 | 0.5 | 0.82 | 0.5 | |
| M3 | 1 | -0.41 | 0 | 0.41 | ||
| M4 | 1 | 0 | 0.5 | |||
| M5 | 1 | 0.41 | ||||
| M6 | 1 |
$\texttt{sim}(M1, M3)$ =$\frac{(-1)(-1) + (1)(1) + (1)(0) + (0)(-1)}{\sqrt{(-1)^2 + 1^2 + 1^2 + 0^2}\cdot\sqrt{(-1)^2 + 1^2 + 0^2 + (-1)^2}} = \frac{2}{\sqrt{3} \cdot \sqrt{3}} = 0.66667$
$\texttt{sim}(M1, M5)$ =$\frac{(-1)(0) + (1)(1) + (1)(1) + (0)(1)}{\sqrt{(-1)^2 + 1^2 + 1^2 + 0^2}\cdot\sqrt{0^2 + 1^2 + 1^2 + 1^2}} = \frac{2}{\sqrt{3} \cdot \sqrt{3}} = 0.66667$
$\texttt{sim}(M5, M6)$ =$\frac{(0)(-1) + (1)(0) + (1)(1) + (1)(0)}{\sqrt{0^2 + 1^2 + 1^2 + 1^2}\cdot\sqrt{(-1)^2 + 0^2 + 1^2 + 0^2}} = \frac{1}{\sqrt{3} \cdot \sqrt{2}} = 0.40825$
Based on the New User's provided ratings for movies 3, 4, and 5, predict how they would rate movies 1, 2, and 6.
We will take the weighted average over the binarized ratings of all movies rated by the new user. We will weigh on similarity.
For example the predicted rating for movie 2 is:
Now you calculate for M1 and M6.
$\texttt{Rating}(M1) = (0.67)(1) + (0)(-1) + (0.67)(1) = 1.33$
$\texttt{Rating}(M6) = (0.41)(1) + (0.5)(-1) + (0.41)(1) = 0.32$
Now that we have the expected ratings of the user for the movies they have not seen we need to actually recommend a movie. Recommend the movie with the highest predicted rating!
M1 had the highest predicted rating, so we'll recommend that one!
We are uploading a Github tutorial to canvas after the lab if you're looking for a refresher
Enjoy the discussion!