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I have two lists: A and B where Size(A) = 4 and size(B) = 10. I would like to find the Jaccard similarity between these two lists.

Suppose A = {Tom, George, John, Jennifer} and B={Tom, Jessica, Angel, Hanna, Tom, John, Michele, Edward, Alex, Tom}

As far as I know, Jaccard is measured by = (Intersect A, B)/(Union A,B) and I read here, that jaccard similarity is the number of common attributes divided by the number of attributes that exists in at least one of the two objects that is: p/p+q+r where "p" is number of common attributes, "q" is # of attributes 1 for A and 0 for B while "r" is # of attributes 0 for A and 1 for B.

My question Considering the above formula (if it is correct), should I count the repeated items in list B (that is "Tom" in this example)? so, is the following correct:

Jaccard(A,B) = 2/2+2+6 = 20% 

or should I ignore repeated item "Tom", and write like this:

Jaccard(A,B) = 2/2+2+8 = 16.6% 

I really get confused, please help.

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  • $\begingroup$ It is your choice. Originally Jaccard similarity was formulated for binary data, where an attribute is either present or absent, and not counted. Tom therefore cannot appear in a set more than once. If it appears - discard the duplicates. However, you might prefer your second approach as well: what you did is like introducing three "Tom" attributes: Tom1, Tom2, Tom3; set1 ans set2 agree once, i.e. on Tom1, the other two "Toms" are found in set2 but were never "initialized" in set1. $\endgroup$
    – ttnphns
    Sep 30, 2015 at 16:53
  • $\begingroup$ Thanks for your comment.. do you think the first approach is better? $\endgroup$
    – mOna
    Sep 30, 2015 at 16:57
  • $\begingroup$ You might prefer other coefficients instead of Jaccard, for example Kulczynski-2. It uses two different bases (denominators), and in your case it will be (2/4 + 2/k)/2, where k=8 if you prefer to count Tom once or k=10 if you prefer to count it thrice. $\endgroup$
    – ttnphns
    Sep 30, 2015 at 17:01
  • $\begingroup$ I don't know what is "better". Better in what situation? $\endgroup$
    – ttnphns
    Sep 30, 2015 at 17:02
  • $\begingroup$ @ttnphns: actually, these two list are explicit & implicit user profiles. A: explicit opinion of users on cast (their favite cast) and B: Implicit casts based on movies she liked... my goal is to see how much users select a movie based on their explicit preferences on Features of movie (such as cast/director/genre).. $\endgroup$
    – mOna
    Sep 30, 2015 at 22:18

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