User-Based Collaborative Filtering

Predict how a user would rate an item from similar users' ratings.

R is a users × items matrix with 0 meaning "not rated". Implement predict_rating(R, u, i, k):

• similarity between users = cosine over items both rated (0 if none shared)
• among users who rated item i (excluding u), take the k most similar with similarity > 0 (ties by lower index)
• predict $\bar r_u + \frac{\sum s_{uv}(r_{vi} - \bar r_v)}{\sum |s_{uv}|}$, where $\bar r$ is a user's mean over their rated items
• if no neighbours qualify, return $\bar r_u$