5
$\begingroup$

I am trying to understand the difference between Jaccard and Cosine. However, there seem to be a disagreement in the answers provided in Applications and differences for Jaccard similarity and Cosine Similarity.

I am seeking if anyone could step me through the calculations of the Jaccard Similarity in this Cosine Similarity example from https://bioinformatics.oxfordjournals.org/content/suppl/2009/10/24/btp613.DC1/bioinf-2008-1835-File004.pdf

Given:

enter image description here

Question: How do we compute the Jaccard Similarity index between t1 and t2?

Thank you.

$\endgroup$
5
$\begingroup$

Cosine similarity is for comparing two real-valued vectors, but Jaccard similarity is for comparing two binary vectors (sets). So you cannot compute the standard Jaccard similarity index between your two vectors, but there is a generalized version of the Jaccard index for real valued vectors which you can use in this case:

$J_g(\Bbb{a}, \Bbb{b}) =\frac{\sum_i min(\Bbb{a}_i, \Bbb{b}_i)}{\sum_i max(\Bbb{a}_i, \Bbb{b}_i)}$

So for your examples of $t_1 = (1, 1, 0, 1), t_2 = (2, 0, 1, 1)$, the generalized Jaccard similarity index can be computed as follows:

$J(t_1, t_2) = \frac{1+0+0+1}{2+1+1+1} = 0.4$

Alternatively you can treat your bag-of-words vector as a binary vector, where a value $1$ indicates a words presence and $0$ indicates a words absence i.e. $t_1 = (1, 1, 0, 1), t_2 = (1, 0, 1, 1)$. From there, you can compute the original Jaccard similarity index:

$J(t_1, t_2) = \frac{2}{2+1+1} = 0.5$

| improve this answer | |
$\endgroup$
  • 1
    $\begingroup$ @jkyh please see my edit to this answer -- I misinterpreted your situation initially and have added extra clarification $\endgroup$ – timleathart Dec 22 '16 at 2:02

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.