Item-Based Collaborative Filtering

Predict a missing rating with item-based collaborative filtering.

The idea: a user will rate a new item the way they rated items similar to it. Two items are similar when the same users rate them in the same way.

You get a ratings matrix R of shape (n_users, n_items), where 0 means "not rated". To predict user u's rating of item i:

1. Compute the cosine similarity between the column of item i and the column of each other item j, using the full columns (missing ratings count as 0).
2. Take the items j != i that user u has rated.
3. Return the similarity-weighted average of the user's ratings of those items:

prediction = sum_j sim(i, j) * R[u, j] / sum_j |sim(i, j)|

If the denominator is 0, return 0.0. If an item column is all zeros, its similarity to everything is 0.

Function signature

def predict_rating(R, user, item):

Example

Input

R = np.array([[5, 3, 0],
              [4, 0, 4],
              [1, 1, 5],
              [0, 4, 4]])
predict_rating(R, user=0, item=2)

Output

3.88067791

Reasoning

User 0 rated item 0 with 5 and item 1 with 3. Item 2's similarity to item 0 is 0.4292 and to item 1 is 0.5455. The prediction is (0.4292 * 5 + 0.5455 * 3) / (0.4292 + 0.5455) = 3.881, closer to 3 because item 1 is more like item 2.

Constraints and notes

• Ratings are non-negative, with 0 meaning missing.
• Do not use the user's own rating of item, even if it exists.
• Return a plain float.