谷本系数距离测度 | 谷本系数距离测度

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两个对象可以具有相同的余弦和Tanimoto系数距离测度吗，

Can two objects have identical cosine and Tanimoto coefficient distance measure, where

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

和

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

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

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

分母之间的区别是x.y项.如果x.y为零，则Tanimoto和余弦相似系数将相同.

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

在几何上，当且仅当x和y垂直时，x.y为零.

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

由于x和y是位向量(即每个维度中的值只能为0或1)，所以x.y等于零表示

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

如果xi * yi = 1 * 1 = 1，则总和为正.为了使总和为零，没有项 xi * yi可以等于1.它们都必须等于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:

所以

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

换句话说，如果xi为1，则yi必须为0，反之亦然.

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

因此，有很多例子中Tanimoto相似度等于余弦相似度:

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)

例如.

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