For two vectors \( \boldsymbol{a} \) and \( \boldsymbol{b} \) Cosine similarity is based on their inner product and defined as
$$ \mbox{similarity}(\boldsymbol{a}, \boldsymbol{b}) = cos(\theta) = \frac{ \mathbf{a} \cdot \mathbf{b} } {\left| \left| a \right| \right| \cdot \left| \left| b \right| \right| } = \frac{ \sum\limits_{i=1}^{n}{a_i b_i} }{ \sqrt{\sum\limits_{i=1}^{n}{a_i^2}} \sqrt{\sum\limits_{i=1}^{n}{b_i^2}} } $$The resulting similarity value ranges from −1 meaning exactly opposite, to 1 meaning exactly the same, with 0 indicating orthogonality. Since we use \( \mbox{tf} \ast \mbox{idf} \) weights, which cannot be negative, the cosine similarity between two word bags representing the input time series sets will range from 0 to 1.