Rating-Based Book Recommendation System

Question

Consider the user book-rating matrix given below. The ratings increase from 1 to 5. Absence of a rating is indicated by empty box.

$\begin{tabular}{|c|c|c|c|c|c|c|c|c|} \hline User/Book & b1 & b2 & b3 & b4 & b5 & b6 & b7\\ \hline U1 & 4 & 1 & _ & 5 & _ & _ & 2\\ \hline U2 & 5 & 1 & _ & 2 & 3 & _ & 4\\ \hline U3 & _ & 1 & _ & 4 & 4 & 1 & 1\\ \hline U4 & 3 & _ & 1 & 1 & _ & 5 & 3\\ \hline \end{tabular}$

A new book b8 is published and U4 liked it. Identify which user is most likely interested in the book b8 using collaborative filtering approach and cosine similarity as a similarity measure.

Solution

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Here, we need to find Cosine Similarity of all other users with U4 and then recommend b8 to the user most similar to U4.

$\begin{align*} \boxed{Cosine \ Similarity = \frac{x.y}{\lVert x \rVert \lVert y \rVert} } \end{align*}$

$\begin{align*} \left CS(U1,U4) = \frac{U1.U4}{\lVert U1 \rVert \lVert U4 \rVert} = \frac{(4,1,0,5,0,0,2).(3,0,1,1,0,5,3)}{6.78 * 6.71} = \frac{23}{45.49} = 0.51 \end{align*}$

$\begin{align*} \left CS(U2,U4) = \frac{U2.U4}{\lVert U2 \rVert \lVert U4 \rVert} = \frac{(5,1,0,2,3,0,4).(3,0,1,1,0,5,3)}{7.42 * 6.71} = \frac{29}{49.79} = 0.58 \end{align*}$

$\begin{align*} \left CS(U3,U4) = \frac{U3.U4}{\lVert U3 \rVert \lVert U4 \rVert} = \frac{(0,1,0,4,4,1,1).(3,0,1,1,0,5,3)}{5.92 * 6.71} = \frac{12}{39.72} = 0.3 \end{align*}$

Thus, we can see that U4 and U2 are most similar and hence, b8 should be recommended to U2.

Question

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