Tanimoto coefficient distance measure

The Tanimoto similarity coefficient (which is not a true distance measure) is defined by

d(x,y) = x.y / ((|x|*|x|) + (|y|*|y|)- x.y)

for bit vectors x and y.

Now compare that with the cosine similarity coefficent,

 d(x,y) = x.y / (|x| * |y|)

The denominators differ by a x.y term. The Tanimoto and cosine similarity coefficients would be the same if x.y is zero.

Geometrically, x.y is zero if and only if x and y are perpendicular.

Since x and y are bit vectors (i.e. whose values in each dimension can only be 0 or 1), x.y equalling zero means

x1*y1 + x2*y2 + ... + xn*yn = 0

If xi*yi = 1*1 = 1, then the whole sum would be positive. For the whole sum to be zero, no term xi*yi can equal 1. They must all equal 0:

x1*y1 = 0
x2*y2 = 0
...
xn*yn = 0

In other words, if xi is 1, then yi must be 0, and vice versa.

So there are tons of examples where the Tanimoto similarity is equal to the cosine similarity:

x = (0,1,0,1)
y = (1,0,0,0)

for instance.

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