Q$3.$ Collaborative Filtering

$3$a$.$ Complete the function cossim in the class Collaborative $[10$ pts$]$

To Do:

$1.$Impute the unrated entries in self.Mr to the user's average rating then subtract by the user mean, call this matrix X$.$

$2.$Calculate cosine similarity for all item$-$item pairs. Don't forget to rescale the cosine similarity to be $0$~$1.$

You might encounter divide by zero warning $($numpy will fill nan value for that entry$).$ In that case, you can fill those with appropriate values.

Hint: Let's say a movie item has not been rated by anyone. When you calculate similarity of this vector to anoter, you will get $0=[0,0,0,....,0].$ When you normalize this vector, you'll get divide by zero warning and it will make nan value in self.sim matrix. Theoretically what should the similarity value for $$when $=0?$ What about $$ when $=0$ and $$ is an any vector?

Hint: You may use scipy.spatial.distance.cosine, but it will be slow because its cosine function does vector$-$vector operation whereas you can implement matrix$-$matrix operation using numpy to calculate all cosines all at once $($it can be $100$ times faster than vector$-$vector operation in our data$).$ Also pay attention to the definition. The scipy.spatial.distance provides distance, not similarity.