Cosine Measure

A popular measure of similarity for text (which normalizes the features by the covariance matrix) clustering is the cosine of the angle between two vectors. The cosine measure is given by

$$s^{(\mathrm{C})} (\mathbf{x}_a,\mathbf{x}_b) = \frac{\mathbf{x}_... ...ger} \mathbf{x}_b} {\Vert\mathbf{x}_a\Vert _2 \cdot \Vert\mathbf{x}_b\Vert _2}$$ (4.2)

and captures a scale invariant understanding of similarity. An even stronger property is that the cosine similarity does not depend on the length:
$s^{(\mathrm{C})} (\alpha \mathbf{x}_a,\mathbf{x}_b) = s^{(\mathrm{C})} (\mathbf{x}_a,\mathbf{x}_b)$ for $\alpha > 0$. This allows documents with the same composition, but different totals to be treated identically which makes this the most popular measure for text documents. Also, due to this property, samples can be normalized to the unit sphere for more efficient processing [DM01].