Why don't we include 0 matches while calculati

问题:

I am working on a program based on Jaccard Distance, and I need to calculate the Jaccard Distance between two binary bit vectors. I came across the following on the net:

 If p1 = 10111 and p2 = 10011,

 The total number of each combination attributes for p1 and p2:

 M11 = total number of attributes where p1 & p2 have a value 1,
 M01 = total number of attributes where p1 has a value 0 & p2 has a value 1,
 M10 = total number of attributes where p1 has a value 1 & p2 has a value 0,
 M00 = total number of attributes where p1 & p2 have a value 0.
 Jaccard similarity coefficient = J = 
 intersection/union = M11/(M01 + M10 + M11) 
 = 3 / (0 + 1 + 3) = 3/4,

 Jaccard distance = J' = 1 - J = 1 - 3/4 = 1/4, 
 Or J' = 1 - (M11/(M01 + M10 + M11)) = (M01 + M10)/(M01 + M10 + M11)
 = (0 + 1)/(0 + 1 + 3) = 1/4

Now, while calculating the coefficient, why was "M00" not included in the denominator? Can anyone please explain?

回答1:

The Jacquard index of A and B is |A∩B|/|A∪B| = |A∩B|/(|A| + |B| - |A∩B|).

We have: |A∩B| = M11, |A| = M11 + M10, |B| = M11 + M01.

So |A∩B|/(|A| + |B| - |A∩B|) = M11 / (M11 + M10 + M11 + M01 - M11) = M11 / (M10 + M01 + M11).

This Venn diagram may help: