Recommender Systems — Similarity & Prediction

Part A — Collaborative Filtering Prediction

Active user ā has mean rating $\bar{r}_a = 3.6$. Three neighbours:

Prediction formula: $pred(a,j) = \bar{r}_a + g\sum_i w(a,i)\,(r_{i,j} - \bar{r}_i)$, where $g = 1/\sum_i |w(a,i)|$.

Part B — Evaluation Metrics

A model produced the following predictions on five held-out ratings:

$\text{MAE} = \frac{1}{n}\sum_i|p_i - r_i|, \quad \text{MSE} = \frac{1}{n}\sum_i(p_i - r_i)^2$

Part C — Cosine Similarity

Two users' rating vectors across 4 items: A = [2, 0, 3, 1] and B = [1, 2, 2, 0].

$sim(A,B) = \frac{\mathbf{a}\cdot\mathbf{b}}{\|\mathbf{a}\|\,\|\mathbf{b}\|}$

Task 1: Using all three neighbours, compute the weighted deviation sum Σ w(a,i)(rᵢⱼ − r̅ᵢ).

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Or enter your answer (±2% tolerance)

Task 2: Compute the normalising factor g = 1 / Σ|w(a,i)| (round to 3 decimal places).

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Task 3: Compute the predicted rating pred(a, j) (round to 3 decimal places).

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Task 4: Calculate the MAE (Mean Absolute Error) for the five predictions in Part B.

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Task 5: Calculate the MSE (Mean Squared Error) for the five predictions in Part B.

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Task 6: Compute the cosine similarity sim(A, B) between A = [2, 0, 3, 1] and B = [1, 2, 2, 0] (round to 3 decimal places).

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