Cosine similarity

Cosine similarity (distance) - distance between two vectors. $CosSim=\frac{{\sum }_{i=1}^n{x}_i{y}_i}{\sqrt{{\sum }_{i=1}^n{x}_i^2}\sqrt{{\sum }_{i=1}^n{y}_i^2}}$

If you’re trying to calculate similarity between user’s ratings you will encounter problem of users having subjective understanding of what is 1 star or 5 stars. To mitigate this problem Adjusted cosine similarity can be used. However, in case of sparse data, when most of users don’t rate most of the items (movies, products) adjusted cosine similarity can be misleading. In this case (item-based) Pearson similarity can be used.