Is Jaccard similarity/distance suitable for non-binary, quantitative data? I have a dataset with each row a country and 10 columns with numerical features like GDP,Electrcity consumption, GNI etc. I am trying to use distance metrics to find similarity between the countries and ultimately cluster them. I have tried quite a few distance metrics like Euclidean, Minkowski, canberra, jaccard etc. In case of jaccard (implementation in pdist in scipy) I don't think the resulting dissimilarity matrix makes sense as I have all 1's in the matrix other than 0 along diagonal. I read more on jaccard and it seems to use set union and intersection in the computation. So am I wrong to apply it in case of continous variables? I have a read a lot on jaccard and it seems to be useful only when data is represented in terms of 0/1 (present/absent). Please guide  :)
 A: Jaccard similarity is, in general, valid for any pair of sets https://en.wikipedia.org/wiki/Jaccard_index
Given two sets $A$ and $B$:
$$
J=\frac{|A \cap B|}{|A \cup B|}
$$
No requirement is given about the elements of $A$ and $B$. In general, it can be seen as the relative (Lebesgue) measure, between the intersection and the union of the two sets. Under this interpretation, it may be applied to all pairs of elements of a measurable space $X$. When $X$ is a Borel space, with $\sigma$-algebra $\Sigma$, the measure can be also probabilistic:
$$
J=\frac{\mu(A \cap B)}{\mu(A \cup B)}
$$
given that $A, B \in \Sigma$.
A: Originally, Jaccard similarity is defined on binary data only. However, its idea (as correctly displayed by @ping in their answer) could be attempted to extend over to quantitative (scale) data. In many sources, Ruzicka similarity is being seen as such equivalent of Jaccard. A screenshot from the document of my SPSS macro !PROXQNT (can be found on my web-page, "Various proximities" collection):

Besides this, one should also keep in mind that in case of binary data, Jaccard sim = Ruzicka sim (= 1 - Soergel dis) = Similarity ratio = Ellenberg sim.
Therefore per backward logic, Similarity ratio and Ellenberg similarity can be considered too, as other candidates for the equivalence towards Jaccard.

